2.1. The DD algorithm

The noisy DD algorithm is the best previously known efficient algorithm for exact recovery [Reference Gebhard, Johnson, Loick and Rolvien19]. SPEX uses DD as a subroutine for diagnosing a small fraction of individuals. Later, we compare the number of tests needed by SPEX with the one needed by DD. The DD algorithm from [Reference Gebhard, Johnson, Loick and Rolvien19] utilises the constant column design $\mathbf{G}_{\mathrm{cc}}$ .Thus, each individual independently joins an equal number $\Delta$ of random tests. Given the displayed test results, DD first declares certain individuals as uninfected by thresholding the number of negatively displayed tests. More precisely, DD declares as uninfected any individual that appears in at least $\alpha \Delta$ negatively displayed tests, with $\alpha$ a diligently chosen threshold. Having identified the respective individuals as uninfected, DD looks out for tests $a$ that display a positive result and that only contain a single individual $x$ that has not been identified as uninfected yet. Since such tests $a$ hint at $x$ being infected, in its second step, DD declares as infected any individual $x$ that appears in at least $\beta \Delta$ positively displayed tests $a$ where all other individuals $y\in \partial a\setminus x$ were declared uninfected by the first step. Once again, $\beta$ is a carefully chosen threshold. Finally, DD declares as uninfected all remaining individuals.

The DD algorithm exactly recovers the infected set w.h.p. provided the total number $m$ of tests is sufficiently large such that the aforementioned thresholds $\alpha, \beta$ exist. The required number of tests, which comes out in terms of a mildly delicate optimisation problem, was determined in [Reference Gebhard, Johnson, Loick and Rolvien19]. Let

(2.1) \begin{align} q_0^-&=\exp ({-}d)p_{00}+(1-\exp ({-}d))p_{10},&q_0^+&=\exp ({-}d)p_{01}+(1-\exp ({-}d))p_{11}. \end{align}

Theorem 2.1 ([Reference Gebhard, Johnson, Loick and Rolvien19, Theorem 2.2]). Let $\varepsilon \gt 0$ and with $\alpha \in (p_{10},q_0^-)$ and $\beta \in (0,\exp ({-}d)p_{11})$ , let

\begin{align*} c_{\mathrm{\texttt{DD}}}&=\min _{\alpha, \beta, d} \max \left \{{c_{\mathrm{\texttt{DD}},1}(\alpha, d),c_{\mathrm{\texttt{DD}},2}(\alpha, d),c_{\mathrm{\texttt{DD}},3}(\beta, d),c_{\mathrm{\texttt{DD}},4}(\alpha, \beta, d)}\right \},\qquad \text{ where}\\ c_{\mathrm{\texttt{DD}},1}(\alpha, d)&=\frac{\theta }{(1-\theta )D_{\mathrm{KL}}\left ({{{\alpha }\|{p_{10}}}}\right )},\qquad c_{\mathrm{\texttt{DD}},2}(\alpha, d)=\frac{1}{dD_{\mathrm{KL}}\left ({{{\alpha }\|{q_0^-}}}\right )},\\ & \qquad c_{\mathrm{\texttt{DD}},3}(\beta, d)=\frac{\theta }{d(1-\theta )D_{\mathrm{KL}}\left ({{{\beta }\|{p_{11}\exp ({-}d)}}}\right )},\\ c_{\mathrm{\texttt{DD}},4}(\alpha, \beta, d) & =\max _{(1-\alpha )\vee \beta \leq z\leq 1}\left (d(1-\theta )\left (D_{\mathrm{KL}}\left ({{{z}\|{q_0^+}}}\right )\right.\right.\\ &\qquad \left.\left. +\unicode{x1D7D9}\left \{{\beta \gt \frac{z\exp ({-}d)p_{01}}{q_0^+}}\right \}zD_{\mathrm{KL}}\left ({{{\beta /z}\|{\exp ({-}d)p_{01}/q_0^+}}}\right )\right )\right )^{-1}. \end{align*}

If $m\geq (1+\varepsilon )c_{\mathrm{\texttt{DD}}}k\ln (n/k)$ , then there exists $\Delta \gt 0$ and $0\leq \alpha, \beta \leq 1$ such that the DD algorithm outputs $\boldsymbol{\sigma }$ w.h.p.

The distinct feature of DD is its simplicity. However, the thresholding that DD applies does seem to leave something on the table. For a start, whether DD identifies a certain individual $x$ as infected depends only on the results of tests that have a distance of at most three from $x$ in the graph $\mathbf{G}_{\mathrm{cc}}$ . Moreover, it seems wasteful that DD takes only those positively displayed tests into consideration where all but one individual were already identified as uninfected.

2.2. Belief propagation

BP is a message passing algorithm that is expected to overcome these deficiencies. In fact, heuristic arguments suggest that BP might be the ultimate recovery algorithm for a wide class of inference algorithms on random graphs [Reference Zdeborová and Krzakala42]. That said, rigorous analyses of BP are few and far between.

Following the general framework from [Reference Mézard and Montanari30], in order to apply BP to a group testing design $G=(V\cup F,E)$ , we equip each test $a\in F$ with a weight function

(2.2) \begin{align} \psi _a=\psi _{G,\boldsymbol{\sigma }^{\prime\prime},a}&\,:\,\{0,1\}^{\partial a}\to \mathbb{R}_{\geq 0},& \sigma _{\partial a_i}&\mapsto \begin{cases} \unicode{x1D7D9}\{\|\sigma \|_{1}=0\}p_{00}+\unicode{x1D7D9}\{\|\sigma \|_{1}\gt 0\}p_{10}&\text{ if } a\in F^-\\ \unicode{x1D7D9}\{\|\sigma \|_{1}=0\}p_{01}+\unicode{x1D7D9}\{\|\sigma \|_{1}\gt 0\}p_{11}&\text{ if } a\in F^+. \end{cases} \end{align}

Thus, $\psi _a$ takes as argument a $\{0,1\}$ -vector $\sigma =(\sigma _x)_{x\in \partial a}$ indexed by the individuals that take part in test $a$ . The weight $\psi _a(\sigma )$ equals the probability of observing the result that $a$ displays if the infection status were $\sigma _x$ for every individual $x\in \partial a$ . In other words, $\psi _a$ encodes the posterior under the $\boldsymbol{p}$ -channel. Given $G,\boldsymbol{\sigma }^{\prime\prime}$ , the weight functions give rise to the total weight of $\sigma \in \{0,1\}^V$ by letting

(2.3) \begin{align} \psi (\sigma )=\psi _{G,\boldsymbol{\sigma }^{\prime\prime}}(\sigma )&=\unicode{x1D7D9}\left \{{\|\sigma \|_1=k}\right \}\prod _{a\in F}\psi _{a}(\sigma _{\partial a}). \end{align}

Thus, we just multiply up the contributions (2.2) of the various tests and add in the prior assumption that precisely $k$ individuals are infected. The total weight (2.3) induces a probability distribution

(2.4) \begin{align} \mu _{G,\boldsymbol{\sigma }^{\prime\prime}}(\sigma )&=\psi _{G,\boldsymbol{\sigma }^{\prime\prime}}(\sigma )/Z_{G,\boldsymbol{\sigma }^{\prime\prime}},&\text{where}&&Z_{G,\boldsymbol{\sigma }^{\prime\prime}}&=\sum _{\sigma \in \{0,1\}^V}\psi _{G,\boldsymbol{\sigma }^{\prime\prime}}(\sigma ). \end{align}

A simple application of Bayes’ rule shows that $\mu _G$ matches the posterior of the ground truth $\boldsymbol{\sigma }$ given the test results.

BP is a heuristic to calculate the marginals of $\mu _{G,\boldsymbol{\sigma }^{\prime\prime}}$ or, in light of Fact 2.2, the posterior probabilities ${\mathbb{P}}[\boldsymbol{\sigma }_{x_i}=1\mid \boldsymbol{\sigma }^{\prime\prime}]$ . To this end, BP associates messages with the edges of $G$ . Specifically, for any adjacent individual/test pair $x,a$ , there is a message $\mu _{x\to a,t}(\cdot)$ from $x$ to $a$ and another one $\mu _{a\to x,t}(\cdot)$ in the reverse direction. The messages are updated in rounds and therefore come with a time parameter $t\in \mathbb{Z}_{\geq 0}$ . Moreover, being probability distributions on $\{0,1\}$ , the messages always satisfy

(2.5) \begin{align} \mu _{x\to a,t}(0)+\mu _{x\to a,t}(1)=\mu _{a\to x,t}(0)+\mu _{a\to x,t}(1)=1. \end{align}

The intended semantics is that $\mu _{x\to a,t}(1)$ estimates the probability that $x$ is infected given all known information except the result of test $a$ . Analogously, $\mu _{a\to x,t}(1)$ estimates the probability that $x$ is infected if we disregard all other tests $b\in \partial x\setminus \{a\}$ .

This slightly convoluted interpretation of the messages facilitates simple heuristic formulas for computing the messages iteratively. To elaborate, in the absence of a better a priori estimate, at time $t=0$ , we simply initialise in accordance with the prior, that is,

(2.6) \begin{align} \mu _{x\to a,0}(0)&=1-k/n&\mu _{x\to a,0}(1)&=k/n. \end{align}

Subsequently, we use the weight function (2.2) to update the messages: inductively for $t\geq 1$ and $r\in \{0,1\}$ , let

(2.7) \begin{align} \mu _{a\to x,t}(r)&\propto \begin{cases} p_{11}&\text{if }r=1,\,a\in F^+,\\ p_{11}+(p_{01}-p_{11})\prod _{y\in \partial a\setminus \{x\}}\mu _{y\to a,t-1}(0)&\text{if }r=0,\,a\in F^+,\\ p_{10}& \text{ if }r=1,\,a\in F^-,\\ p_{10}+(p_{00}-p_{10})\prod _{y\in \partial a\setminus \{x\}}\mu _{y\to a,t-1}(0)&\text{ if }r=0,\,a\in F^-, \end{cases} \end{align}

(2.8) \begin{align} \mu _{x\to a,t}(r)&\propto \left ({\frac{k}{n}}\right )^r\left ({1-\frac{k}{n}}\right )^{1-r}\prod _{b\in \partial x\setminus \{a\}}\mu _{b\to x,t-1}(r). \end{align}

The $\propto$ -symbol hides the necessary normalisation to ensure that the messages satisfy (2.5). Furthermore, the $(k/n)^r$ and $(1-k/n)^{1-r}$ -prefactors in (2.8) encode the prior that precisely $k$ individuals are infected. The expressions (2.7)–(2.8) are motivated by the hunch that for most tests $a$ , the values $(\boldsymbol{\sigma }_y)_{y\in \partial a}$ should be stochastically dependent primarily through their joint membership in test $a$ . An excellent exposition of BP can be found in [Reference Mézard and Montanari30].

How do we utilise the BP messages to infer the actual infection status of each individual? The idea is to perform the update (2.7)–(2.8) for a ‘sufficiently large’ number of rounds, say, until an approximate fixed point is reached. The (heuristic) BP estimate of the posterior marginals after $t$ rounds then reads

(2.9) \begin{align} \mu _{x,t}(r)&\propto \left ({\frac{k}{n}}\right )^r\left ({1-\frac{k}{n}}\right )^{1-r}\prod _{b\in \partial x}\mu _{b\to x,t-1}(r)&&(r\in \{0,1\}). \end{align}

Thus, by comparison to (2.8), we just take the incoming messages from all tests $b\in \partial x$ into account. In summary, we ‘hope’ that after sufficiently many updates, we have $\mu _{x,t}(r)\approx{\mathbb{P}}[\boldsymbol{\sigma }_x=r\mid \boldsymbol{\sigma }^{\prime\prime}]$ . We could then, for instance, declare the $k$ individuals with the greatest $\mu _{x,t}(1)$ infected and everybody else uninfected.

For later reference, we point out that the BP updates (2.7)–(2.8) and (2.9) can be simplified slightly by passing to log-likelihood ratios. Thus, define

(2.10) \begin{align} \eta _{x\to a,t}&=\ln \frac{\mu _{G,\boldsymbol{\sigma }^{\prime\prime},x\to a,t}(1)}{\mu _{G,\boldsymbol{\sigma }^{\prime\prime},x\to a,t}(0)},& \eta _{G,\boldsymbol{\sigma }^{\prime\prime},a\to x,t}&=\ln \frac{\mu _{G,\boldsymbol{\sigma }^{\prime\prime},a\to x,t}(1)}{\mu _{G,\boldsymbol{\sigma }^{\prime\prime},a\to x,t}(0)}, \end{align}

with the initialisation $\eta _{G,\boldsymbol{\sigma }^{\prime\prime},x\to a,0}=\ln (k/(n-k))\sim (\theta -1)\ln n$ from (2.6). Then (2.7)–(2.9) transform into

(2.11) \begin{align} \eta _{a\to x,t}&=\begin{cases} \ln p_{11}-\ln \left \lbrack{p_{11}+(p_{01}-p_{11})\prod _{y\in \partial a\setminus \{x\}}\frac{1}{2}\left ({1-\tanh \left ({\frac{1}{2}\eta _{y\to a,t-1}}\right )}\right )}\right \rbrack &\text{ if }a\in F^+,\\ \ln p_{10}-\ln \left \lbrack{p_{10}+(p_{00}-p_{10})\prod _{y\in \partial a\setminus \{x\}}\frac{1}{2}\left ({1-\tanh \left ({\frac{1}{2}\eta _{y\to a,t-1}}\right )}\right )}\right \rbrack &\text{ if }a\in F^-,\\ \end{cases} \end{align}

(2.12) \begin{align} \eta _{x\to a,t}&=(\theta -1)\ln n+\sum _{b\in \partial x\setminus \{a\}}\eta _{b\to x,t},,\quad \eta _{x,t}=(\theta -1)\ln n+\sum _{b\in \partial x}\eta _{b\to x,t}. \end{align}

In this formulation, BP ultimately diagnoses the $k$ individuals with the largest $\eta _{x,t}$ as infected.

Under the assumptions of Theorem 2.1, the DD algorithm can be viewed as the special case of BP with $t=2$ applied to $\mathbf{G}_{\mathrm{cc}}$ . Indeed, the analysis of DD evinces that on $\mathbf{G}_{\mathrm{cc}}$ , the largest $k$ BP estimates (2.9) with $t\geq 2$ correctly identify the infected individuals w.h.p.Footnote ³

It is therefore an obvious question whether BP on the constant column design fits the bill of Theorem 1.1. Clearly, BP remedies the obvious deficiencies of DD by taking into account information from a larger radius around an individual (if we iterate beyond $t=2$ ). Also in contrast to DD’s hard thresholding, the update rules (2.7)–(2.8) take information into account in a more subtle, soft manner. Nonetheless, we do not expect that BP applied to the constant column design meets the information-theoretically optimal bound from Theorem 1.1. In fact, there is strong evidence that BP does not suffice to meet the information-threshold for all $\theta$ even in the noiseless case [Reference Coja-Oghlan, Gebhard, Hahn-Klimroth, Wein and Zadik10]. The fundamental obstacle appears to be the ‘cold’ initialisation (2.6), which (depending on the parameters) can cause the BP messages to approach a meaningless fixed point. Yet for symmetry reasons on the constant column design, no better starting point than the prior (2.6) springs to mind; after all, $\mathbf{G}_{\mathrm{cc}}$ is a nearly biregular random graph, and thus, all individuals look alike. To overcome this issue, we will employ a different type of test design that enables a warm start for BP. This technique goes by the name of spatial coupling.

2.3. Spatial coupling

The thrust of spatial coupling is to blend a randomised construction, in our case, the constant column design, with a spatial arrangement so as to provide a propitious starting point for BP. Originally hailing from coding theory, spatial coupling has also been used in previous work on noiseless group testing [Reference Coja-Oghlan, Gebhard, Hahn-Klimroth and Loick9]. In fact, the construction that we use is essentially identical to that from [Reference Coja-Oghlan, Gebhard, Hahn-Klimroth and Loick9] (with suitably adapted parameters).

To set up the spatially coupled test design $\mathbf{G}_{\mathrm{sc}}$ , we divide the set $V=V_n=\{x_1,\ldots, x_n\}$ of individuals into

(2.13) \begin{align} \ell =\lceil \sqrt{\ln n}\rceil \end{align}

pairwise disjoint compartments $V[1],\ldots, V[\ell ]\subseteq V$ such that $\lfloor n/\ell \rfloor \leq |V[i]|\leq \lceil n/\ell \rceil$ . We think of these compartments as being spatially arranged so that $V[i+1]$ comes ‘to the right’ of $V[i]$ for $1\leq i\lt \ell$ . More precisely, we arrange the compartments in a ring topology such that $V[\ell ]$ is followed again by $V[1]$ . Hence, for notational convenience, let $V[\ell +j]=V[j]$ and $V[1-j]=V[\ell -j+1]$ for $1\leq j\leq \ell$ . Additionally, we introduce $\ell$ compartments $F[1],\ldots, F[\ell ]$ of tests arranged in the same way: think of $F[i]$ as sitting ‘above’ $V[i]$ . We assume that the total number $m$ of tests in $F[1]\cup \ldots \cup F[\ell ]$ is divisible by $\ell$ and satisfies $m=\Theta (k\ln (n/k))$ . Hence, let each compartment $F[i]$ contain precisely $m/\ell$ tests. As in the case of the individuals, we let $F[\ell +j]=F[j]$ for $0\lt j\leq \ell$ . Additionally, let

(2.14) \begin{align} s=\lceil \ln \ln n\rceil \quad \text{ and} \quad \Delta =\Theta (\ln n) \end{align}

be integers such that $\Delta$ is divisible by $s$ . Construct $\mathbf{G}_{\mathrm{sc}}$ by letting each $x\in V[i]$ join precisely $\Delta /s$ tests from $F[i+j-1]$ for $j=1,\ldots, s$ . These tests are chosen uniformly without replacement and independently for different $x$ and $j$ . Additionally, $\mathbf{G}_{\mathrm{sc}}$ contains a compartment $F[0]$ of

(2.15) \begin{align} m_0&=2c_{\mathrm{\texttt{DD}}}\frac{ks}{\ell }\ln (n/k)=o(k\ln (n/k)) \end{align}

tests. Every individual $x$ from the first $s$ compartments $V[1],\ldots, V[s]$ joins an equal number $\Delta _0$ of tests from $F[0]$ . These tests are drawn uniformly without replacement and independently. For future reference, we let

(2.16) \begin{align} c=c_n=\frac{m}{k\ln (n/k)},\hspace{0.5cm}d=d_n=\frac{k\Delta }{m}; \end{align}

the aforementioned assumptions on $m,\Delta$ ensure that $c,d=\Theta (1)$ and the total number of tests of $\mathbf{G}_{\mathrm{sc}}$ comes to

(2.17) \begin{align} \sum _{i=0}^{\ell }|F[i]|&=(c+o(1))k\ln (n/k). \end{align}

In summary, $\mathbf{G}_{\mathrm{sc}}$ consists of $\ell$ equally sized compartments $V[i]$ , $F[i]$ of tests plus one extra serving $F[0]$ of tests. Each individual $V[i]$ joins random tests in the $s$ consecutive compartments $F[i+j-1]$ with $1\leq j\leq s$ . Additionally, the individuals in the first $s$ compartments $V[1]\cup \cdots \cup V[s]$ , which we refer to as the seed, also join the tests in $F[0]$ .

We will discover momentarily how $\mathbf{G}_{\mathrm{sc}}$ facilitates inference via BP. But first let us make a note of some basic properties of $\mathbf{G}_{\mathrm{sc}}$ . Recall that $\boldsymbol{\sigma }\in \{0,1\}^V$ , which encodes the true infection status of each individual, is chosen uniformly and independently of $\mathbf{G}_{\mathrm{sc}}$ from all vectors of Hamming weight $k$ . Let $V_1=\{x\in V\,:\,\boldsymbol{\sigma }_x=1\}$ , $V_0=V\setminus V_1$ , and let $V_r[i]=V_r\cap V[i]$ be the set of individuals with infection status $r\in \{0,1\}$ in compartment $i$ . Furthermore, recall that $\boldsymbol{\sigma }^{\prime}_a\in \{0,1\}$ denotes the actual result of test $a\in F=F[0]\cup \cdots \cup F[\ell ]$ and that $\boldsymbol{\sigma }^{\prime\prime}_a$ signifies the displayed result of $a$ as per (1.2). For $r\in \{0,1\}$ and $0\leq i\leq \ell$ , let

\begin{align*} F_r[i]&=F_r\cap F[i],&F_r^+[i]&=F_r[i]\cap F^+,&F_r^-[i]&=F_r[i]\cap F^-. \end{align*}

Thus, the subscript indicates the actual test result, while the superscript indicates the displayed result. In Section 3.1, we will prove the following.

2.4. Approximate recovery

We are going to exploit the spatial structure of $\mathbf{G}_{\mathrm{sc}}$ in a manner reminiscent of domino toppling. To get started, we will run DD on the seed $V[1]\cup \cdots \cup V[s]$ and the tests $F[0]$ only; this is our first domino. The choice (2.15) of $m_0$ ensures that DD diagnoses all individuals in $V[1]\cup \cdots \cup V[s]$ correctly w.h.p. The seed could then be used as an informed starting point from which we could iterate BP to infer the status of the individuals in $V[s+1]\cup \ldots \cup V[\ell ]$ . However, this algorithm appears to be difficult to analyse. Instead, we will show under that the assumptions of Theorem 1.1 a modified, ‘paced’ version of BP that diagnoses one compartment (or ‘domino’) at a time and then re-initialises the messages ultimately classifies all but $o(k)$ individuals correctly. Let us flesh this strategy out in detail.

2.4.2. A combinatorial condition

To simplify the analysis of the message passing step in Section 2.4.3, we observe that certain individuals can be identified as likely uninfected purely on combinatorial grounds. More precisely, consider $x\in V[i]$ for $s\lt i\leq \ell$ . If $x$ is infected, then any test $a\in \partial x$ is actually positive. Hence, we expect that $x$ appears in about $p_{11}\Delta$ tests that display a positive result. In fact, the choice (2.14) of $s$ ensures that w.h.p. even within each separate compartment $F[i+j-1]$ , $1\leq j\leq s$ the individual $x$ appears in about $p_{11}\Delta /s$ positively displayed tests. Thus, let

(2.23) \begin{align} V^+[i]&=\left \{{x\in V[i]\,:\,\sum _{j=1}^s\left |{|\partial x\cap F^+[i+j-1]|-\Delta p_{11}/s}\right |\leq \ln ^{4/7}n}\right \}. \end{align}

The following proposition confirms that all but $o(k/s)$ infected individuals $x\in V[i]$ belong to $V^+[i]$ . Additionally, the proposition determines the approximate size of $V^+[i]$ . For notational convenience, we define

\begin{align*} V_0^+[i]&=V^+[i]\cap V_0,&V_1^+[i]&=V^+[i]\cap V_1,&V^+&=\bigcup _{s\lt i\leq \ell }V^+[i]. \end{align*}

Recall $q_0^+$ from (2.1). The proof of the following proposition can be found in Section 3.2.

Algorithm 1 SPARC, steps 1–2

Proposition 2.5. W.h.p., we have

(2.24) \begin{align} \sum _{i=s+1}^\ell |V_1[i]\setminus V^+[i]|&\leq k\exp ({-}\Omega (\ln ^{1/7}n))&&\text{ and} \end{align}

(2.25) \begin{align} \left |{V^+[i]\setminus V_1[i]}\right |&\leq \frac{n}{\ell }\exp \left ({-\Delta D_{\mathrm{KL}}\left ({{{p_{11}}\|{q_0^+}}}\right )+O(\ln ^{4/7}n)}\right ) &&\text{ for all } s+1\leq i\leq \ell . \end{align}

2.4.3. Belief propagation redux

The main phase of the SPARC algorithm employs a simplified version of the BP update rules (2.7)–(2.8) to diagnose one compartment $V[i]$ , $s\lt i\leq \ell$ , after the other. The textbook way to employ BP would be to diagnose the seed, initialise the BP messages emanating from the seed accordingly, and then run BP updates until the messages converge. However, towards the proof of Theorem 1.1, this way of applying BP seems too complicated to analyse. Instead, SPARC relies on a ‘paced’ version of BP. Rather than updating the messages to convergence from the seed, we perform one round of message updates, then diagnose the next compartment, re-initialise the messages to coincide with the newly diagnosed compartment, and proceed to the next undiagnosed compartment.

We work with the log-likelihood versions of the BP messages from (2.10) to (2.12). Suppose we aim to process compartment $V[i]$ , $s\lt i\leq \ell$ , having completed $V[1],\ldots, V[i-1]$ already. Then for a test $a\in F[i+j-1]$ , $j\in [s]$ , and an adjacent variable $x\in V[i+j-s]\cup V[i+j-1]$ , we initialise

(2.26) \begin{align} \eta _{x\to a,0}&=\begin{cases} -\infty &\text{ if }\tau _x=0,\\ +\infty &\text{ if }\tau _x=1,\\ \ln (k/(n-k))&\text{ if }\tau _x=*. \end{cases} \end{align}

The third case above occurs if and only if $x\in V[i]\cup \cdots \cup V[i+j-1]$ , that is, if $x$ belongs to an as yet undiagnosed compartment. For the compartments that have been diagnosed already, we set $\eta _{x\to a,0}$ to $\pm \infty$ , depending on whether $x$ has been classified as infected or uninfected.

Let us now investigate the ensuing messages $\eta _{a\to x,1}$ for $x\in V[i]$ and tests $a\in \partial x\cap F[i+j-1]$ . A glimpse at (2.11) reveals that for any test $a$ that contains an individual $y\in V[i+j-s]\cup \cdots \cup V[i-1]$ with $\tau _y=1$ , we have $\eta _{a\to x,1}=0$ . This is because (2.26) ensures that $\eta _{y\to a,0}=\infty$ and $\tanh (\infty )=1$ . Hence, the test $a$ contains no further information as to the status of $x$ . Therefore, we call a test $a\in F[i+j-1]$ informative towards $V[i]$ if $\tau _y=0$ for all $y\in \partial a\cap (V[i+j-s]\cup \cdots \cup V[i-1])$ .

Let $\boldsymbol{W}_{i,j}(\tau )$ be the set of all informative $a\in F[i+j-1]$ . Then any $a\in \boldsymbol{W}_{i,j}(\tau )$ receives $\eta _{y\to a,0}=-\infty$ from all individuals $y$ that have been diagnosed already, that is, all $y\in V[h]$ with $i+j-s\leq h\lt i$ . Another glance at the update rule shows that the corresponding terms simply disappear from the product on the r.h.s. of (2.11) because $\tanh ({-}\infty )=-1$ . Consequently, only the factors corresponding to undiagnosed individuals $y\in V[i]\cup \cdots \cup V[i+j-1]$ remain. Hence, with $r=\unicode{x1D7D9}\{a\in F^+\}$ , the update rule (2.11) simplifies to

(2.27) \begin{align} \eta _{a\to x,1}&= \ln p_{1r}-\ln \left \lbrack{p_{1r}+(p_{0r}-p_{1r})\left ({1-k/n}\right )^{\left |{\partial a\cap (V[i]\cup \cdots \cup V[i+j-1])}\right |-1}}\right \rbrack . \end{align}

The only random element in the expression (2.27) is the number $\left |{\partial a\cap (V[i]\cup \cdots \cup V[i+j-1])}\right |$ of members of test $a$ from compartments $V[i]\cup \cdots \cup V[i+j-1]$ . But by the construction of $\mathbf{G}_{\mathrm{sc}}$ , this number has a binomial distribution with mean

\begin{align*} \mathbb{E}\left |{\partial a\cap (V[i]\cup \cdots \cup V[i+j-1])}\right |&=\frac{j\Delta n}{ms}+o(1)=\frac{djn}{ks}+o(1)&&\text{[using (2.16)]}. \end{align*}

Since the fluctuations of $\left |{\partial a\cap (V[i]\cup \cdots \cup V[i+j-1])}\right |$ are of smaller order than the mean, we conclude that w.h.p., (2.27) can be well approximated by a deterministic quantity:

(2.28) \begin{align} \eta _{a\to x,1}&=\begin{cases} w_j^++o(1),&\text{ if }a\in F^+,\\ -w_j^-+o(1),&\text{ if }a\in F^- \end{cases},\qquad \text{ where } \end{align}

(2.29) \begin{align} w_j^+&=\ln \frac{p_{11}}{p_{11}+(p_{01}-p_{11})\exp ({-}dj/s)}\geq 0,& \!\!\! w_j^-&=-\ln \frac{p_{10}}{p_{10}+(p_{00}-p_{10})\exp ({-}dj/s)}\geq 0. \end{align}

Note that in the case $p_{10} = 0$ , the negative test weight $W_j^-$ evaluates to $w_j^- = \infty$ , indicating that individual contained in negative test definitely are uninfected. Finally, the messages (2.28) lead to the BP estimate of the posterior marginal of $x$ via (2.12), that is, by summing on all informative tests $a\in \partial x$ . To be precise, letting

\begin{align*} \boldsymbol{W}_{x,j}^{\pm }(\tau )&=\partial x\cap \boldsymbol{W}_{i,j}(\tau )\cap F^{\pm } \end{align*}

be the positively/negatively displayed informative tests adjacent to $x$ and setting

(2.30) \begin{align} \boldsymbol{W}_x^+(\tau )& = \sum _{j=1}^{s} w_j^+\left |{\boldsymbol{W}_{x,j}^+(\tau )}\right |,& \boldsymbol{W}_x^-(\tau )&=\sum _{j=1}^{s}w_j^- \left |{\boldsymbol{W}_{x,j}^-(\tau )}\right |, \end{align}

we obtain

(2.31) \begin{align} \eta _{x,1}=&\boldsymbol{W}_x^+(\tau )-\boldsymbol{W}_x^-(\tau )+\text{ `lower order fluctuations}^{\prime}. \end{align}

One issue with the formula (2.28) is the analysis of the ‘lower order fluctuations’, which come from the random variables $|\partial a\cap (V^+[i]\cup \cdots \cup V^+[i+s-1])|$ . Of course, one could try to analyse these deviations carefully by resorting to some kind of a normal approximation. But for our purposes, this is unnecessary. It turns out that we may simply use the sum on the r.h.s. of (2.31) to identify which individuals are infected. Specifically, instead of computing the actual BP approximation $\eta _{x,1}$ after one round of updating, we just compare $\boldsymbol{W}_x^+(\tau )$ and $\boldsymbol{W}_x^-(\tau )$ with the values that we would expect these random variables to take if $x$ were infected. These conditional expectations work out to be

(2.32) \begin{align} W^+&=p_{11}\Delta \sum _{j=1}^{s}\exp (d(j-s)/s)w_j^+,& W^-&= \begin{cases} p_{10}\Delta \sum _{j=1}^{s}\exp (d(j-s)/s)w_j^- &\text{ if } p_{10}\gt 0\\ 0 &\text{ otherwise. } \end{cases} \end{align}

Thus, SPARC will diagnose $V[i]$ by comparing $\boldsymbol{W}_x^{\pm }(\tau )$ with $W^{\pm }$ . Additionally, SPARC takes into account that infected individuals likely belong to $V^+[i]$ , as we learned from Proposition 2.5.

Algorithm 2 SPARC, steps 3–9.

Let

(2.33) \begin{align} \zeta =(\ln \ln \ln n)^{-1} \end{align}

be a term that tends to zero slowly enough to absorb error terms. The following proposition, which we prove in Section 3.3, summarises the analysis of phase 3. Recall from (2.16) that $c=m/(k\ln (n/k))$ .

Proposition 2.6. Assume that for a fixed $\varepsilon \gt 0$ , we have

(2.34) \begin{align} c\gt c_{\mathrm{ex},2}(d)+\varepsilon . \end{align}

Then w.h.p., the output $\tau$ of SPARC satisfies

\begin{equation*}\sum _{x\in V^+}\unicode {x1D7D9}\left \{{\tau _x\neq \boldsymbol {\sigma }_x}\right \}\leq k\exp \left ({-\Omega \left ({\frac {\ln n}{(\ln \ln n)^5}}\right )}\right ).\end{equation*}

The proof of Proposition 2.6, which can be found in Section 3.3, is the centrepiece of the analysis of SPARC. The proof is based on a large deviations analysis that bounds the number of individuals $x\in V^+[i]$ whose corresponding sums $\boldsymbol{W}_x^{\pm }(\tau )$ deviate from their conditional expectations given $\boldsymbol{\sigma }_x$ . We have all the pieces in place to complete the proof of the first theorem.

Proof of Theorem 1.1 (upper bound on approximate recovery by SPARC). Setting $d=d_{\mathrm{Sh}}$ from (1.17) and invoking (1.18), we see that the theorem is an immediate consequence of Propositions 2.4, 2.5, and 2.6.

2.5. Exact recovery

As we saw in Section 1.3, the threshold $c_{\mathrm{ex},1}(d,\theta )$ encodes a local stability condition. This condition is intended to ensure that w.h.p., no other ‘solution’ $\sigma \neq \boldsymbol{\sigma }$ with $\|\sigma -\boldsymbol{\sigma }\|_1=o(k)$ of a similarly large posterior likelihood exists. In fact, because $\mathbf{G}_{\mathrm{sc}}$ enjoys fairly good expansion properties, the test results provide sufficient clues for us to home in on $\boldsymbol{\sigma }$ once we get close, provided the number of tests is as large as prescribed by Theorem 1.3. Thus, the idea behind exact recovery is to run SPARC first and then apply local corrections to fully recover $\boldsymbol{\sigma }$ ; a similar strategy was employed in the noiseless case in [Reference Coja-Oghlan, Gebhard, Hahn-Klimroth and Loick9].

Though this may sound easy and a simple greedy strategy does indeed do the trick in the noiseless case [Reference Coja-Oghlan, Gebhard, Hahn-Klimroth and Loick9], in the presence of noise, it takes a good bit of care to get the local search step right. Hence, as per (1.11), suppose that $c,d$ from (2.16) satisfy $c\gt \max \{c_{\mathrm{ex},2}(d),c_{\mathrm{ex},1}(d,\theta )\}+\varepsilon$ . Also suppose that we already ran SPARC to obtain $\tau \in \{0,1\}^V$ with $\|\tau -\boldsymbol{\sigma }\|_1=o(k)$ (as provided by Proposition 2.6). How can we set about learning the status of an individual $x$ with perfect confidence?

Assume for the sake of argument that $\tau _y=\boldsymbol{\sigma }_y$ for all $y$ that share a test $a\in \partial x$ with $x$ . If $a$ contains another infected individual $y\neq x$ , then unfortunately, nothing can be learned from $a$ about the status of $x$ . In this case, we call the test $a$ tainted. By contrast, if $\tau _y=0$ for all $y\in \partial a\setminus \{x\}$ , that is, if $a$ is untainted, then the displayed result $\boldsymbol{\sigma }^{\prime\prime}_a$ hinges on the infection status of $x$ itself. Hence, the larger the number of untainted positively displayed $a\in \partial x$ , the likelier $x$ is infected. Consequently, to accomplish exact recovery, we are going to threshold the number of untainted positively displayed $a\in \partial x$ . But crucially, to obtain an optimal algorithm, we cannot just use a scalar, one-size-fits-all threshold. Instead, we need to carefully choose a threshold function that takes into account the total number of untainted tests $a\in \partial x$ .

To elaborate, let

(2.35) \begin{align} \boldsymbol{Y}_x&=\left |{\left \{{a\in \partial x\setminus F[0]\,:\,\forall y\in \partial a\setminus \{x\}\,:\,\boldsymbol{\sigma }_y=0}\right \}}\right | \end{align}

be the total number of untainted tests $a\in \partial x$ ; to avoid case distinctions, we omit seed tests $a\in F[0]$ . Routine calculations reveal that $\boldsymbol{Y}_x$ is well approximated by a binomial variable with mean $\exp ({-}d)\Delta$ . Therefore, the fluctuations of $\boldsymbol{Y}_x$ can be estimated via the Chernoff bound. Specifically, the numbers of infected/uninfected individuals with $\boldsymbol{Y}_x=\alpha \Delta$ can be approximated as

(2.36) \begin{align} \mathbb{E}\left |{\left \{{x\in V_1\,:\,\boldsymbol{Y}_x=\alpha \Delta }\right \}}\right |&=k\exp \left ({-\Delta D_{\mathrm{KL}}\left ({{{\alpha }\|{\exp ({-}d)}}}\right )+o(\Delta )}\right ), \end{align}

(2.37) \begin{align} \mathbb{E}\left |{\left \{{x\in V_0\,:\,\boldsymbol{Y}_x=\alpha \Delta }\right \}}\right |&=n\exp \left ({-\Delta D_{\mathrm{KL}}\left ({{{\alpha }\|{\exp ({-}d)}}}\right )+o(\Delta )}\right ). \end{align}

Consequently, since $k=\lceil n^\theta \rceil =o(n)$ ‘atypical’ values of $\boldsymbol{Y}_x$ occur more frequently in healthy than in infected individuals. In fact, recalling (1.6), we learn from a brief calculation that for $\alpha \not \in \mathscr{Y}(c,d,\theta )$ not a single $x\in V_1$ with $\boldsymbol{Y}_x=\alpha \Delta$ exists w.h.p. Hence, if $\boldsymbol{Y}_x/\Delta \not \in \mathscr{Y}(c,d,\theta )$ , we deduce that $x$ is uninfected.

For $x$ such that $\boldsymbol{Y}_x/\Delta \in \mathscr{Y}(c,d,\theta )$ , more care is required. In this case, we need to compare the number

(2.38) \begin{align} \boldsymbol{Z}_x&=\left |{\left \{{a\in \partial ^+ x\setminus F[0]\,:\,\forall y\in \partial a\setminus \{x\}\,:\,\boldsymbol{\sigma }_y=0}\right \}}\right | \end{align}

of positively displayed untainted tests to the total number $\boldsymbol{Y}_x$ of untainted tests. Since the test results are put through the $\boldsymbol{p}$ -channel independently, $\boldsymbol{Z}_x$ is a binomial variable given $\boldsymbol{Y}_x$ . The conditional mean of $\boldsymbol{Z}_x$ equals $p_{11}\boldsymbol{Y}_x$ if $x$ is infected and $p_{01}\boldsymbol{Y}_x$ otherwise. Therefore, the Chernoff bound shows that

(2.39) \begin{align} {\mathbb{P}}\left \lbrack{\boldsymbol{Z}_x=\alpha \beta \Delta \mid \boldsymbol{Y}_x=\alpha \Delta }\right \rbrack &=\exp \left ({-\alpha \Delta D_{\mathrm{KL}}\left ({{{\beta }\|{p_{\boldsymbol{\sigma }_x1}}}}\right )+o(\Delta )}\right ). \end{align}

In light of (2.39), we set up the definition (1.9) of $c_{\mathrm{ex},1}(d,\theta )$ so that $\mathfrak z(\cdot)$ can be used as a threshold function to tell infected from uninfected individuals. Indeed, given $\boldsymbol{Y}_x,\boldsymbol{Z}_x$ , we should declare $x$ uninfected if either $\boldsymbol{Y}_x/\Delta \not \in \mathscr{Y}(c,d,\theta )$ or $\boldsymbol{Z}_x/\boldsymbol{Y}_x\lt \mathfrak z(\boldsymbol{Z}_x/\Delta )$ and infected otherwise; then the choice of $\mathfrak z(\cdot)$ and $c_{\mathrm{ex}}(\theta )$ would ensure that all individuals get diagnosed correctly w.h.p. Figure 2 displays a characteristic specimen of the function $\mathfrak z(\cdot)$ and the corresponding rate function from (1.9).

Figure 2. The threshold function $\mathfrak z(\cdot)$ (red) on the interval $\mathscr{Y}(c_{\mathrm{ex},1}(d,\theta ),d,\theta )$ and the resulting large deviations rate $c_{\mathrm{ex},1}(d,\theta )d(1-\theta )({D_{\mathrm{KL}}({{{\alpha }\|{\exp ({-}d)}}})+\alpha D_{\mathrm{KL}}({{{\mathfrak z(\alpha )}\|{p_{01}}}})})$ (black) with $\theta =1/2$ , $p_{00}=0.972$ , $p_{11}=0.9$ at the optimal choice of $d$ .

Yet trying to distil an algorithm from these considerations, we run into two obvious obstacles. First, the threshold $\mathfrak z(\cdot)$ may be hard to compute precisely. Similarly, the limits of the interval $\mathscr{Y}(c,d,\theta )$ may be irrational (or worse). The following proposition, which we prove in Section 4.1, remedies these issues.

The second obstacle is that in the above discussion, we assumed that $\tau _y=\boldsymbol{\sigma }_y$ for all $y\in \partial a\setminus \{x\}$ . But all that the analysis of SPARC provides is that w.h.p. $\tau _y\neq \boldsymbol{\sigma }_y$ for at most $o(k)$ individuals $y$ . To cope with this issue, we resort to the expansion properties of $\mathbf{G}_{\mathrm{sc}}$ . Roughly speaking, we show that w.h.p. for any small set $S$ of individuals (such as $\{y\,:\,\tau _y\neq \boldsymbol{\sigma }_y\}$ ), the set of individuals $x$ that occur in ‘many’ tests that also contain a second individual from $S$ is significantly smaller than $S$ . As a consequence, as we apply the thresholding procedure repeatedly, the number of misclassified individuals decays geometrically. We thus arrive at the following algorithm.

Algorithm 3 The SPEX algorithm.

Proposition 2.8. Suppose that $c\gt c_{\mathrm{ex}}(d,\theta )+\varepsilon$ for a fixed $\varepsilon \gt 0$ . There exists $\varepsilon ^{\prime}\gt 0$ , a rational interval $\mathscr{I}$ and a rational step function $\mathscr{Z}$ such that w.h.p. for all $1\leq i\lt \ln n$ , we have

\begin{equation*}\|\boldsymbol {\sigma }-\tau ^{(i+1)}\|_1\leq \frac {1}{3}\|\boldsymbol {\sigma }-\tau ^{(i)}\|_1.\end{equation*}

We prove Proposition 2.8 in Section 4.2.

2.6. The constant column lower bound

Having completed the discussion of the algorithms SPARC and SPEX for Theorems 1.1 and 1.3, we turn to the proof of Theorem 1.4 on the information-theoretic lower bound for the constant column design. The goal is to show that for $m\lt (c_{\mathrm{ex}}(\theta )-\varepsilon )k\ln (n/k)$ , no algorithm, efficient or otherwise, can exactly recover $\boldsymbol{\sigma }$ on $\mathbf{G}_{\mathrm{cc}}$ .

To this end, we are going to estimate the probability of the actual ground truth $\boldsymbol{\sigma }$ under the posterior given the test results. We recall from Fact 2.2 that the posterior coincides with the Boltzmann distribution from (2.4). The following proposition, whose proof can be found in Section 5.1, estimates the Boltzmann weight $\psi _{\mathbf{G}_{\mathrm{cc}},\boldsymbol{\sigma }^{\prime\prime}}(\boldsymbol{\sigma })$ of the ground truth. Recall from (2.16) that $d=k\Delta /m$ . Also recall the weight function from (2.3).

Proposition 2.10. W.h.p., the weight of $\boldsymbol{\sigma }$ satisfies

(2.40) \begin{align} \frac{1}{m}\ln \psi _{\mathbf{G}_{\mathrm{cc}},\boldsymbol{\sigma }^{\prime\prime}}(\boldsymbol{\sigma })&=-\exp ({-}d)h(p_{00})-(1-\exp ({-}d))h(p_{11})+O(m^{-1/2}\ln ^3). \end{align}

We are now going to argue that for $c\lt c_{\mathrm{ex}}(\theta )$ the partition function $Z_{\mathbf{G}_{\mathrm{cc}},\boldsymbol{\sigma }^{\prime\prime}}$ dwarfs the Boltzmann weight (2.40) w.h.p. The definition (1.11) of $c_{\mathrm{ex}}(\theta )$ suggests that there are two possible ways how this may come about. The first occurs when $c=m/(k\ln (n/k))$ is smaller than the local stability bound $c_{\mathrm{ex},1}(d,\theta )$ from (1.9). In this case, we will essentially put the analysis of SPEX into reverse gear. That is, we will show that there are plenty of individuals $x$ whose status could be flipped from infected to healthy or vice versa without reducing the posterior likelihood. In effect, the partition function will be far greater than the Boltzmann weight of $\boldsymbol{\sigma }$ .

Proposition 2.11. Assume that there exists $y\in \mathscr{Y}(c,d,\theta )$ such that there exists $z\in (p_{01},p_{11})$ such that

(2.41) \begin{align} cd(1-\theta )\left ({D_{\mathrm{KL}}\left ({{{y}\|{\exp ({-}d)}}}\right )+yD_{\mathrm{KL}}\left ({{{z}\|{p_{11}}}}\right )}\right )&\lt \theta &&\text{ and} \end{align}

(2.42) \begin{align} cd(1-\theta )\left ({D_{\mathrm{KL}}\left ({{{y}\|{\exp ({-}d)}}}\right )+yD_{\mathrm{KL}}\left ({{{z}\|{p_{01}}}}\right )}\right )&\lt 1. \end{align}

Then w.h.p., there exist $n^{\Omega (1)}$ pairs $(v,w)\in V_1\times V_0$ such that for the configuration $\boldsymbol{\sigma }^{[v,w]}$ obtained from $\boldsymbol{\sigma }$ by inverting the $v,w$ -entries, we have

(2.43) \begin{align} \mu _{\mathbf{G}_{\mathrm{cc}},\boldsymbol{\sigma }^{\prime\prime}}(\boldsymbol{\sigma }^{[v,w]})=\mu _{\mathbf{G}_{\mathrm{cc}},\boldsymbol{\sigma }^{\prime\prime}}(\boldsymbol{\sigma }). \end{align}

We prove Proposition 2.11 in Section 5.2.

The second case is that $c$ is smaller than the entropy bound $c_{\mathrm{ex},2}(d)$ from (1.10). In this case, we will show by way of a moment computation that $Z_{\mathbf{G}_{\mathrm{cc}},\boldsymbol{\sigma }^{\prime\prime}}$ exceeds the Boltzmann weight of $\boldsymbol{\sigma }$ w.h.p. More precisely, in Section 5.3, we prove the following.

Proposition 2.12. Let $\varepsilon \gt 0$ . If $c\lt c_{\mathrm{ex},2}(d)-\varepsilon$ , then

\begin{align*}{\mathbb{P}}\left \lbrack{ \ln Z_{\mathbf{G}_{\mathrm{cc}},\boldsymbol{\sigma }^{\prime\prime}}\geq k\ln (n/k)\left \lbrack{1-c/c_{\mathrm{ex},2}(d)+o(1)}\right \rbrack }\right \rbrack \gt 1-\varepsilon +o(1). \end{align*}

3.1. Proof of Proposition 2.3 (properties of $\mathbf{G}_{\mathrm{sc}}$ )

Property G1 is a routine consequence of the Chernoff bound and was previously established as part of (Reference Coja-Oghlan, Gebhard, Hahn-Klimroth and Loick9, Proposition 4.1). With respect to G2, we may condition on the event $\mathfrak{E}$ that the bound (2.18) from G1 is satisfied. Consider a test $a\in F[i]$ . Recall that $a$ comprises individuals from the compartments $V[i-s+1],\ldots, V[i]$ . Since the probability that a specific individual joins a specific test is given by (3.1) and since individuals choose the tests that they join independently, on $\mathfrak{E}$ for each $i-s+1\leq h\leq i$ , we have

(3.2) \begin{align} |V_1[h]\cap \partial a|\sim \textrm{Bin}\left ({\frac{k}{\ell }+O\left ({\sqrt{\frac{k}{\ell }}\ln n}\right ),\frac{\Delta \ell }{ms}+O\left ({\left ({\frac{\Delta \ell }{m s}}\right )^2}\right )}\right ). \end{align}

Combining (2.16) and (3.2), we obtain

(3.3) \begin{align} \mathbb{E}[|V_1[h]\cap \partial a|\mid \mathfrak{E}]&=\frac{\Delta k}{ms}+O\left ({\sqrt{\frac{\ell }{k}}\ln n}\right )=\frac{d}{s}+O\left ({k^{-1/2}\ln ^{3/2}n}\right ). \end{align}

Further, combining (3.2) and (3.3), we get

\begin{align*}{\mathbb{P}}\left \lbrack{V_1[h]\cap \partial a =\emptyset \mid \mathfrak{E}}\right \rbrack &=\exp \left \lbrack{\left ({\frac{k}{\ell }+O\left ({\sqrt{\frac{k}{\ell }}\ln n}\right )}\right )\ln \left ({1-\frac{\Delta \ell }{ms}+O\left ({\left ({\frac{\Delta \ell }{m s}}\right )^2}\right )}\right )}\right \rbrack\\& =\exp \left ({-\frac{d}{s}+O\left ({k^{-1/2}\ln ^{3/2}n}\right )}\right ). \end{align*}

Multiplying these probabilities up on $i-s+1\leq h\leq i$ , we arrive at the estimate

\begin{align*}{\mathbb{P}}\left \lbrack{V_1[h]\cap \partial a=\emptyset \mid \mathfrak{E}}\right \rbrack &=\exp \left ({-d+O\left ({k^{-1/2}\ln ^{8/5}n}\right )}\right ). \end{align*}

Hence,

(3.4) \begin{align} \mathbb{E}[\left |{F_0[i]}\right |\mid \mathfrak{E}]=\frac{m}{\ell }\exp ({-}d)+O\left(\sqrt m \ln ^{2}n\right). \end{align}

To establish concentration, observe that the set $\partial x$ of tests that a specific infected individual $x\in V_1[h]$ joins can change $|F_0[i]|$ by $\Delta$ at the most. Moreover, changing the neighbourhood $\partial x$ of a healthy individual cannot change the actual test results at all. Therefore, by Azuma–Hoeffding,

(3.5) \begin{align} {\mathbb{P}}\left \lbrack{\left |{|F_0[i]|-\mathbb{E}\left \lbrack{|F_0[i]| \mid{\mathscr{E}}}\right \rbrack }\right |\geq \sqrt m\ln ^2n \mid{\mathscr{E}}}\right \rbrack &\leq 2\exp \left ({-\frac{m\ln ^4n}{2k\Delta ^2}}\right )=o(n^{-3}). \end{align}

Thus, (3.4), (3.5), and G1 show that with probability $1-o(n^{-2})$ for all $1\leq i\leq \ell$ , we have

(3.6) \begin{align} \left |{F_0[i]}\right |&=\frac{m}{\ell }\exp ({-}d)+O\left(\sqrt m \ln ^{2}n\right),&\text{ and thus }&& \left |{F_1[i]}\right |&=\frac{m}{\ell }(1-\exp ({-}d))+O\left(\sqrt m \ln ^{2}n\right). \end{align}

Since the actual test results are subjected to the $\boldsymbol{p}$ -channel independently to obtain the displayed test results, the distributions of $|F_{0/1}^{\pm }[i]$ given $F_{0/1}[i]$ read

(3.7) \begin{align} &\left |{F_0^-[i]}\right |=\textrm{Bin}(|F_0[i]|,p_{00}), &\left |{F_0^+[i]}\right |=\textrm{Bin}(|F_0[i]|,p_{01}), \end{align}

(3.8) \begin{align} &\left |{F_1^-[i]}\right |=\textrm{Bin}(|F_1[i]|,p_{10}), & \left |{F_1^+[i]}\right |=\textrm{Bin}(|F_1[i]|,p_{11}). \end{align}

Thus, G2 follows from (3.6), (3.7), and Lemma 1.5 (the Chernoff bound).

3.2. Proof of Proposition 2.5 (plausible number of positive tests)

Any $x\in V_1[i]$ has a total of $\Delta /s$ neighbours in each of $F[i],\ldots, F[i+s-1]$ . Moreover, all tests $a\in \partial x$ are actually positive. Since the displayed result is obtained via the $\boldsymbol{p}$ -channel independently for every $a$ , the number of displayed positive neighbours $|\partial x\cap F^+[i+j-1]|$ is a binomial variable with distribution $\textrm{Bin}(\Delta /s,p_{11})$ . Since $\Delta =\Theta (\ln n)$ and $s=\Theta (\ln \ln n)$ , the first assertion (2.24) is immediate from Lemma 1.5.

Moving on to the second assertion, we condition on the event $\mathfrak{E}$ that the bounds (2.19)–(2.22) hold for all $i\in [\ell ]$ . Then Proposition 2.3 shows that ${\mathbb{P}}\left \lbrack{\mathfrak{E}}\right \rbrack =1-o(n^{-2})$ . Given $\mathfrak{E}$ , we know that

(3.9) \begin{align} |F^+[i]|&=\frac{q_0^+m}\ell +O\left(\sqrt m\ln ^3n\right),& |F^-[i]|&=\frac{(1-q_0^+)m}\ell +O\left(\sqrt m\ln ^3n\right). \end{align}

Now consider an individual $x\in V_0[i]$ . Also consider any test $a\in F[i+j-1]$ for $j\in [s]$ . Then the actual result $\boldsymbol{\sigma }^{\prime}_a$ of $a$ is independent of the event $\{x\in a\}$ because $x$ is uninfected. Since the displayed result $\boldsymbol{\sigma }^{\prime\prime}_a$ depends solely on $\boldsymbol{\sigma }^{\prime}_a$ , we conclude that $\boldsymbol{\sigma }^{\prime\prime}_a$ is independent of $\{x\in a\}$ as well. Therefore, (3.9) shows that on the event $\mathfrak{E}$ , the number of displayed positive tests that $x$ is a member has conditional distribution

(3.10) \begin{align} |\partial x\cap F^+[i+j-1]|\sim \textrm{Hyp}\left ({\frac{m}{\ell },\frac{q_0^+m}\ell +O\left(\sqrt m\ln ^3n\right),\frac{\Delta }{s}}\right ). \end{align}

Since the random variables $(|\partial x\cap F^+[i+j-1]|)_{1\leq j\leq s}$ are mutually independent, (2.25) follows from Lemma 1.6.

3.3. Proof of Proposition 2.6 (correctness of SPARC for $V^+$ )

We reduce the proof of Proposition 2.6 to three lemmas. The first two estimate the sums from (2.30) when evaluated at the actual ground truth $\boldsymbol{\sigma }$ .

Lemma 3.1. Assume that (2.34) is satisfied. w.h.p., we have

(3.11) \begin{align} \sum _{s\lt i\leq \ell }\sum _{x\in V_1^+[i]}\unicode{x1D7D9}\left \{{\boldsymbol{W}_x^+(\boldsymbol{\sigma }) \lt (1-\zeta /2) W^+\text{ or }\boldsymbol{W}_x^-(\boldsymbol{\sigma }) \gt (1+\zeta /2) W^-}\right \}\leq k\exp \left ({-\Omega \left ({\frac{\ln n}{(\ln \ln n)^4}}\right )}\right ). \end{align}

Lemma 3.2. Assume that (2.34) is satisfied. w.h.p., we have

(3.12) \begin{align} \sum _{s\leq i\lt \ell }\sum _{x\in V_{0}^+[i]}\unicode{x1D7D9}\left \{{\boldsymbol{W}_x^+(\boldsymbol{\sigma }) \geq (1-2\zeta ) W^+\text{ and }\boldsymbol{W}_x^-(\boldsymbol{\sigma }) \leq (1+2\zeta ) W^-}\right \}&\leq k^{1-\Omega (1)}. \end{align}

We defer the proofs of Lemmas 3.1 and 3.2 to Sections 3.4 and 3.5. While the proof of Lemma 3.1 is fairly routine, the proof of Lemma 3.2 is the linchpin of the entire analysis of SPARC, as it effectively vindicates the BP heuristics that we have been invoking so liberally in designing the algorithm.

Additionally, to compare $\boldsymbol{W}_x^{\pm }(\boldsymbol{\sigma })$ with the algorithm’s estimate $\boldsymbol{W}_x^{\pm }(\tau )$ , we resort to the following expansion property of $\mathbf{G}_{\mathrm{sc}}$ .

Lemma 3.3. Let $0\lt \alpha, \beta \lt 1$ be such that $\alpha +\beta \gt 1$ . Then w.h.p. for any $T \subseteq V$ of size $|T|\leq \exp ({-}\ln ^\alpha n)k$ ,

\begin{align*} \left |{\left \{{x\in V\,:\,\sum _{a\in \partial x\setminus F[0]}\unicode{x1D7D9}\left \{{T\cap \partial a\setminus \left \{{x}\right \}\neq \emptyset }\right \}\geq \ln ^{\beta }n}\right \}}\right |\leq \frac{|T|}{8\ln \ln n}. \end{align*}

In a nutshell, Lemma 3.3 posits that for any ‘small’ set $T$ of individuals, there are even fewer individuals that share many tests with individuals from $T$ . Lemma 3.3 is an generalisation of (Reference Coja-Oghlan, Gebhard, Hahn-Klimroth and Loick9, Lemma 4.16). The proof, based on a routine union bound argument, is included in Appendix C for completeness.

Proof of Proposition 2.6 (correctness of SPARC for V⁺). Proposition 2.4 shows that for all individuals $x$ in the seed $V[1],\ldots, V[s]$ , we have $\tau _x=\boldsymbol{\sigma }_x$ w.h.p. Let $\mathscr{M}[i] = \left \{{x \in V^+[i]\,:\, \tau _x \neq \boldsymbol{\sigma }_x}\right \}$ be the set of misclassified individuals in $V^+[i]$ . We are going to show that w.h.p. for all $1\leq i\leq \ell$ and for large enough $n$ ,

(3.13) \begin{align} \left |{\mathscr{M}[i]}\right |\leq k\exp \left ({-\frac{\ln n}{(\ln \ln n)^{5}}}\right ). \end{align}

We proceed by induction on $i$ . As mentioned above, Proposition 2.4 ensures that $\mathscr{M}[1]\cup \cdots \cup \mathscr{M}[s]=\emptyset$ w.h.p. Now assume that (3.13) is correct for all $i\lt h\leq \ell$ ; we are going to show that (3.13) holds for $i=h$ as well. To this end, recalling $\zeta$ from (2.33) and $W^{\pm }$ from (2.32), we define for $p_{10} \gt 0$

\begin{align*} \mathscr{M}_1[h]&=\left \{{x\in V_1^+[h]\,:\,\boldsymbol{W}_x^+(\boldsymbol{\sigma }) \lt (1- \zeta /2) W^+\text{ or }\boldsymbol{W}_x^-(\boldsymbol{\sigma }) \gt (1+\zeta /2) W^-}\right \},\\ \mathscr{M}_2[h]&=\left \{{x\in V_0^+[h]\,:\,\boldsymbol{W}_x^+(\boldsymbol{\sigma }) \gt (1- 2\zeta ) W^+\text{ and }\boldsymbol{W}_x^-(\boldsymbol{\sigma }) \lt (1+2\zeta ) W^-}\right \},\\ \mathscr{M}_3[h]&=\left \{{x\in V^+[h]\,:\,|\boldsymbol{W}_x^+(\boldsymbol{\sigma })-\boldsymbol{W}_x^+(\tau )|+|\boldsymbol{W}_x^-(\boldsymbol{\sigma })-\boldsymbol{W}_x^-(\tau )|\gt \zeta (W^+\wedge W^-)/8}\right \} \end{align*}

and further for $p_{10} = 0$

\begin{align*} \mathscr{M}_1[h]&=\left \{{x\in V_1^+[h]\,:\,\boldsymbol{W}_x^+(\boldsymbol{\sigma }) \lt (1- \zeta /2) W^+}\right \},\\ \mathscr{M}_2[h]&=\left \{{x\in V_0^+[h]\,:\,\boldsymbol{W}_x^+(\boldsymbol{\sigma }) \gt (1- 2\zeta ) W^+\text{ and }\boldsymbol{W}_x^-(\boldsymbol{\sigma }) = 0 }\right \},\\ \mathscr{M}_3[h]&=\left \{{x\in V^+[h]\,:\,|\boldsymbol{W}_x^+(\boldsymbol{\sigma })-\boldsymbol{W}_x^+(\tau )|\gt \zeta (W^+)/8}\right \}. \end{align*}

We claim that $\mathscr{M}[h]\subseteq \mathscr{M}_1[h]\cup \mathscr{M}_2[h]\cup \mathscr{M}_3[h]$ . To see this, assume that $x\in \mathscr{M}[h]\setminus (\mathscr{M}_1[h]\cup \mathscr{M}_2[h])$ . Then for SPARC to misclassify $x$ , it must be the case that

\begin{align*} \left|\boldsymbol{W}_x^+(\tau )-\boldsymbol{W}_x^+(\boldsymbol{\sigma })\right|+|\boldsymbol{W}_x^-(\tau )-\boldsymbol{W}_x^-(\boldsymbol{\sigma })|\gt \frac{\zeta }{8}(W^+\wedge W^-), \end{align*}

and thus $x\in \mathscr{M}_3[h]$ .

Thus, to complete the proof, we need to estimate $|\mathscr{M}_1[h]|,|\mathscr{M}_2[h]|,|\mathscr{M}_3[h]|$ . Lemmas 3.1 and 3.2 readily show that

(3.14) \begin{align} |\mathscr{M}_1[h]|+|\mathscr{M}_2[h]|\leq k\exp \left ({-\Omega \left ({\frac{\ln n}{(\ln \ln n)^4}}\right )}\right ). \end{align}

Furthermore, in order to bound $|\mathscr{M}_3[h]|$ , we will employ Lemma 3.3. Specifically, consider $x\in \mathscr{M}_3[h]$ . Since $W^+\wedge W^-=\Omega (\Delta )=\Omega (\ln n)$ for $p_{10}\gt 0$ by (2.29) and (2.32), there exists $j\in [s]$ such that

\begin{align*} |\,|\boldsymbol{W}_{x,j}^+(\tau )|-|\boldsymbol{W}_{x,j}^+(\boldsymbol{\sigma })|\,|&\gt \ln ^{1/2}n&\text{ or}&&|\,|\boldsymbol{W}_{x,j}^-(\tau )|-|\boldsymbol{W}_{x,j}^-(\boldsymbol{\sigma })|\,|\gt \ln ^{1/2}n. \end{align*}

Since $W^+=\Omega (\ln n)$ for $p_{10}=0$ , there exists $j\in [s]$ such that

\begin{align*} |\,|\boldsymbol{W}_{x,j}^+(\tau )|-|\boldsymbol{W}_{x,j}^+(\boldsymbol{\sigma })|\,|&\gt \ln ^{1/2}n \end{align*}

Assume without loss of generality that the left inequality holds. Then there are at least $\ln ^{1/2}n$ tests $a\in \partial x\cap F^+[h+j-1]$ such that $\partial a\cap (\mathscr{M}[1]\cup \cdots \cup \mathscr{M}[h-1])\neq \emptyset$ . Therefore, Lemma 3.3 shows that

\begin{align*} |\mathscr{M}_3[h]|\leq \frac{|\mathscr{M}[h-s]\cup \cdots \cup \mathscr{M}[h-1]|}{8\ln \ln n}. \end{align*}

Hence, using induction to bound $|\mathscr{M}[h-s]|,\ldots, |\mathscr{M}[h-1]|$ and recalling from (2.14) that $s\leq 1+\ln \ln n$ , we obtain (3.13) for $i=h$ .

3.4. Proof of Lemma 3.1 (large deviations for infected individuals)

The thrust of Lemma 3.1 is to verify that the definition (2.23) of the set $V^+[i]$ faithfully reflects the typical statistics of positively/negatively displayed tests in the neighbourhood of an infected individual $x\in V_1[i]$ with $s\lt i\leq \ell$ . Recall from the definition of $\mathbf{G}_{\mathrm{sc}}$ that such an individual $x$ joins tests in $F[i+j-1]$ for $j\in [s]$ . Moreover, apart from $x$ itself, a test $a\in F[i+j-1]\cap \partial x$ recruits from $V[i+j-s],\ldots, V[i+j-1]$ . In particular, $a$ recruits participants from the $s-j$ compartments $V[i+j-s],\ldots, V[i-1]$ preceding $V[i]$ . Let

(3.15) \begin{align} \mathscr{U}_{i,j} & ={\left \{{a\in F[i+j-1]\,:\,(V_1[1]\cup \cdots \cup V_1[i-1])\cap \partial a=\emptyset }\right \}}\nonumber\\[3pt]& ={\left \{{a\in F[i+j-1]\,:\,\bigcup _{h=i+j-s}^{i-1}V_1[h]\cap \partial a=\emptyset }\right \}} \end{align}

be the set of tests in $F[i+j-1]$ that do not contain an infected individual from $V[i+j-s],\ldots, V[i-1]$ .

Claim 3.4. With probability $1-o(n^{-2})$ for all $s\leq i\lt \ell$ , $j\in [s]$ , we have

\begin{align*} |\mathscr{U}_{i,j}|&=\left ({1+O(n^{-\Omega (1)})}\right )\frac{m}{\ell }\cdot \exp (d(j-s)/s). \end{align*}

Proof. We condition on the event $\mathfrak{E}$ that G1 from Proposition 2.3 holds. Then for any $a\in F[i+j-1]$ and any $i+j-s\leq h\lt i$ , the number $V_1[h]\cap \partial a$ of infected individuals in $a$ from $V[h]$ is a binomial variable as in (3.2). Since the random variables $(V_1[h]\cap \partial a)_h$ are mutually independent, we therefore obtain

(3.16) \begin{align} {\mathbb{P}}\left \lbrack{(V_1[i+j-s]\cup \cdots V_1[i-1])\cap \partial a=\emptyset \mid \mathfrak{E}}\right \rbrack =\exp (d(j-s)/s)+O(n^{-\Omega (1)}). \end{align}

Hence,

(3.17) \begin{align} \mathbb{E}\left \lbrack{|\mathscr{U}_{i,j}|\mid \mathfrak{E}}\right \rbrack &=\frac{m}{\ell }\exp \left \lbrack{d(j-s)/s+O(n^{-\Omega (1)})}\right \rbrack . \end{align}

Further, changing the set $\partial x$ of tests that a single $x\in V_1$ joins can alter $|\mathscr{U}_{i,j}|$ by $\Delta$ at the most, while changing the set of neighbours of any $x\in V_0$ does not change $|\mathscr{U}_{i,j}|$ at all. Therefore, Azuma–Hoeffding shows that

(3.18) \begin{align} {\mathbb{P}}\left \lbrack{\left |{|\mathscr{U}_{i,j}|-\mathbb{E}[|\mathscr{U}_{i,j}|\mid \mathfrak{E}]}\right |\gt \sqrt m\ln ^2n\mid \mathfrak{E}}\right \rbrack &\leq 2\exp \left ({-\frac{m\ln ^4n}{2k\Delta ^2}}\right )=o(n^{-2}). \end{align}

The assertion follows from (3.17) to (3.18).

Let

(3.19) \begin{align} q_{1,j}^-&=p_{01}\exp (d(j-s)/s),&q_{1,j}^+&=p_{11}\exp (d(j-s)/s). \end{align}

Claim 3.5. For all $s\lt i\leq \ell$ , $x \in V_1[i]$ , and $j \in [s]$ , we have

\begin{align*}{\mathbb{P}}\left \lbrack{|\boldsymbol{W}_{{x},{j}}^+|\lt (1-\zeta /2)q_{1,j}^+\frac{\Delta }{s}}\right \rbrack +{\mathbb{P}}\left \lbrack{|\boldsymbol{W}_{{x},{j}}^-|\gt (1+\zeta /2)q_{1,j}^-\frac{\Delta }{s}}\right \rbrack &\leq \exp \left ({-\Omega \left ({\frac{\ln n}{(\ln \ln n)^4}}\right )}\right ). \end{align*}

Proof. Fix $s\leq i\lt \ell, 1\leq j\leq s$ and $x\in V_1[i]$ . In light of Proposition 2.3 and Claim 3.4, the event $\mathfrak{E}$ that G1 from Proposition 2.3 holds and that $|\mathscr{U}_{i,j}|=\exp (d(j-s)/s)\frac{m}{\ell }(1+O(n^{-\Omega (1)}))$ has probability

(3.20) \begin{align} {\mathbb{P}}\left \lbrack{\mathfrak{E}}\right \rbrack =1-o(n^{-2}). \end{align}

Let $\boldsymbol{U}_{i,j}=|\partial x\cap \mathscr{U}_{i,j}|$ . Given $\mathscr{U}_{i,j}$ , we have

\begin{align*} \boldsymbol{U}_{i,j}\sim \textrm{Hyp}\left ({\frac{m}{\ell }+O(1),\exp \left ({\frac{d(j-s)}{s}}\right )\frac{m}{\ell }(1+O(n^{-\Omega (1)})),\frac{\Delta }{s}}\right ). \end{align*}

Therefore, Lemma 1.6 shows that the event

\begin{equation*}\mathfrak {E}^{\prime}=\left \{{\left |{\boldsymbol {U}_{i,j}-\frac {\Delta }{s}\exp \left ({\frac {d(j-s)}{s}}\right )}\right |\geq \frac {\ln n}{(\ln \ln n)^2}}\right \}\end{equation*}

has conditional probability

(3.21) \begin{align} {\mathbb{P}}\left \lbrack{\mathfrak{E}^{\prime}\mid \mathfrak{E}}\right \rbrack \leq \exp \left ({-\Omega \left ({\frac{\ln n}{(\ln \ln n)^4}}\right )}\right ). \end{align}

Finally, given $\boldsymbol{U}_{i,j}$ the number $|\boldsymbol{W}_{x,j}^+|$ of positively displayed $a\in \partial x\cap \mathscr{U}_{i,j}$ has distribution $\textrm{Bin}(\boldsymbol{U}_{i,j},p_{11})$ ; similarly, $|\boldsymbol{W}_{x,j}^-|$ has distribution $\textrm{Bin}(\boldsymbol{U}_{i,j},p_{01})$ . Thus, since $\Delta =\Theta (\ln n)$ while $s=O(\ln \ln n)$ by (2.14), the assertion follows from (3.20), (3.21), and Lemma 1.5.

3.5. Proof of Lemma 3.2 (large deviations for healthy individuals)

To prove Lemma 3.2, we need to estimate the probability that an uninfected individual $x\in V[i]$ , $s\lt i\leq \ell$ , ‘disguises’ itself to look like an infected individual. In addition to the sets $\mathscr{U}_{i,j}$ from (3.15) towards the proof of Lemma 3.2, we also need to consider the sets

\begin{align*} \mathscr{P}_{i,j}&=F_1[i+j-1]\cap \mathscr{U}_{i,j},&\mathscr{N}_{i,j}&=F_0[i+j-1]\cap \mathscr{U}_{i,j},\\ \mathscr{P}^{\pm }_{i,j}&=F_1^{\pm }[i+j-1]\cap \mathscr{U}_{i,j},&\mathscr{N}^{\pm }_{i,j}&=F_0^{\pm }[i+j-1]\cap \mathscr{U}_{i,j}. \end{align*}

In words, $\mathscr{P}_{i,j}$ and $\mathscr{N}_{i,j}$ are the sets of actually positive/negative tests in $\mathscr{U}_{i,j}$ , that is, actually positive/negative tests $a\in F[i+j-1]$ that do not contain an infected individual from $V[i+j-s]\cup \cdots \cup V[i-1]$ . Additionally, $\mathscr{P}^{\pm }_{i,j},\mathscr{N}^{\pm }_{i,j}$ discriminate based on the displayed test results. We begin by estimating the likely sizes of these sets.

Claim 3.6. Let $s\lt i\leq \ell$ and $j\in [s]$ . Then with probability $1-o(n^{-2})$ , we have

(3.22) \begin{align} |\mathscr{P}_{i,j}|&=\left ({1+O(n^{-\Omega (1)})}\right )\frac{m}{\ell }\cdot \frac{\exp (dj/s)-1}{\exp (d)},&|\mathscr{N}_{i,j}|&=\left ({1+O(n^{-\Omega (1)})}\right )\frac{m}{\ell }\cdot \exp ({-}d), \end{align}

(3.23) \begin{align} |\mathscr{P}^+_{i,j}|&=\left ({1+O(n^{-\Omega (1)})}\right )\frac{m}{\ell }\cdot \frac{p_{11}({\exp} (dj/s)-1)}{\exp (d)},&|\mathscr{P}^-_{i,j}|&=\left ({1+O(n^{-\Omega (1)})}\right )\frac{m}{\ell }\cdot \frac{p_{10}({\exp} (dj/s)-1)}{\exp (d)}, \end{align}

(3.24) \begin{align} |\mathscr{N}^+_{i,j}|&=\left ({1+O(n^{-\Omega (1)})}\right )\frac{m}{\ell }\cdot p_{01}\exp ({-}d),&|\mathscr{N}^-_{i,j}|&=\left ({1+O(n^{-\Omega (1)})}\right )\frac{m}{\ell }\cdot p_{00}\exp ({-}d). \end{align}

Proof. Since $\mathscr{N}_{i,j}=F_0[i+j-1]$ , the second equation in (3.22) just follows from Proposition 2.3, G2. Furthermore, since $\mathscr{P}_{i,j}=\mathscr{U}_{i,j}\setminus \mathscr{N}_{i,j}$ , the first equation (3.22) is an immediate consequence of Claim 3.4. Finally, to obtain (3.23)–(3.24), we simply notice that given $|\mathscr{P}_{i,j}|,|\mathscr{N}_{i,j}|$ , we have

\begin{align*} |\mathscr{P}^+_{i,j}|&=\textrm{Bin}(|\mathscr{P}_{i,j}|,p_{11}),& \!\!|\mathscr{P}^-_{i,j}|&=\textrm{Bin}(|\mathscr{P}_{i,j}|,p_{10}),& \!\!|\mathscr{N}^+_{i,j}|&=\textrm{Bin}(|\mathscr{N}_{i,j}|,p_{01}),& \!\!|\mathscr{N}^-_{i,j}|&=\textrm{Bin}(|\mathscr{N}_{i,j}|,p_{00}). \end{align*}

Hence, (3.23)–(3.24) just follow from (3.22) and Lemma 1.5.

Let $\mathfrak U$ be the event that G1–G2 from Proposition 2.3 hold and that the estimates (3.22)–(3.24) hold. Then by Proposition 2.3 and Claim 3.6, we have

(3.25) \begin{align} {\mathbb{P}}\left \lbrack{\mathfrak U}\right \rbrack =1-o(n^{-2}). \end{align}

To facilitate the following computations, we let

(3.26) \begin{align} q_{0,j}^-&=\exp ({-}d)p_{00}+({\exp} (d(j-s)/s)-\exp ({-}d))p_{10},&q_{0,j}^+&=\exp ({-}d)p_{01}\nonumber\\& \quad +({\exp} (d(j-s)/s)-\exp ({-}d))p_{11}. \end{align}

Additionally, we introduce the shorthand $\lambda =(\ln \ln n)/\ln ^{3/7}n$ for the error term from the definition (2.23) of $V^+[i]$ . Our next step is to determine the distribution of the random variables $|\boldsymbol{W}_{{x},{j}}^{\pm }|$ that contribute to $\boldsymbol{W}_x^{\pm }(\boldsymbol{\sigma })$ from (2.30).

Claim 3.7. Let $s\lt i\leq \ell$ and $j\in [s]$ . Given $\mathfrak U$ for every $x \in V_{0}^+[i]$ , we have

(3.27) \begin{align} |\boldsymbol{W}_{{x},{j}}^-| \sim \textrm{Hyp} \left ({\left ({1 + O(n^{-\Omega (1)})}\right )\frac{mq_0^-}{\ell },\left ({1 + O(n^{-\Omega (1)})}\right ) \frac{mq_{0,j}^-}{2\ell },(1+O(\lambda ))\frac{\Delta }{s} p_{10}}\right ), \end{align}

(3.28) \begin{align} |\boldsymbol{W}_{{x},{j}}^+| \sim \textrm{Hyp} \left (\left ({1 + O(n^{-\Omega (1)})}\right )\frac{mq_0^+}{\ell },\left ({1 + O(n^{-\Omega (1)})}\right ) \frac{mq_{0,j}^+}{2\ell },\left ({1+O(\lambda )}\right )\frac{\Delta s}{p_{11}}\right ). \end{align}

Proof. The definition (2.23) of $V^+[i]$ prescribes that for any $x\in V_0^+[i]$ , we have $\partial x\cap F^+[i+j-1]=(p_{11}+O(\lambda ))\Delta /s$ . The absence or presence of $x\in V_0[i]$ in any test $a$ does not affect the displayed results of $a$ because $x$ is uninfected. Therefore, the conditional distributions of $|\boldsymbol{W}^{\pm }_{x,j}|$ read

\begin{align*} |\boldsymbol{W}_{{x},{j}}^{\pm }| \sim \textrm{Hyp} \left (|F^{\pm }[i+j-1]|,|\mathscr{N}_{i,j}^{\pm }|+|\mathscr{P}_{i,j}^{\pm }|,(1+O(\lambda ))\frac{\Delta }{s} p_{10}\right ). \end{align*}

Since on $\mathfrak U$ the bounds (2.19)–(2.22) and (3.22)–(3.24) hold, the assertion follows.

We are now ready to derive an exact expression for the probability that for $x\in V_0[i]$ , the value $\boldsymbol{W}_x^{+}$ gets (almost) as large as the value $W^+$ that we would expected to see for an infected individual. Recall the values $q_{1,j}^{\pm }$ from (3.19).

Claim 3.8. Let

(3.29) \begin{align} \mathscr{M}^+=&\min \frac{1}{s}\sum _{j=1}^{s} D_{\mathrm{KL}}\left ({{{z_j}\|{\frac{q_{0,j}^+}{q_0^+}}}}\right )\qquad \text{ s.t.}\quad \sum _{j=1}^{s}\left ({z_j-\frac{q_{1,j}^+}{p_{11}}}\right ) w_j^+ = 0,\qquad z_1,\ldots, z_{s}\in (0,1). \end{align}

Then for all $s \lt i \leq \ell$ and all $x\in V[i]$ , we have

\begin{align*}{\mathbb{P}}\left \lbrack{\boldsymbol{W}_{{x}}^+ \geq (1-2\zeta )W^+ \mid \mathfrak U,\,x\in V_0^+[i]}\right \rbrack &\leq \exp ({-}(1+o(1))p_{11}\Delta \mathscr{M}^+). \end{align*}

Proof. Since $\Delta =O(\ln n)$ and $s=O(\ln \ln n)$ by (2.14), we can write

(3.30) \begin{align}&{\mathbb{P}}\left \lbrack{\boldsymbol{W}_{{x}}^+(\boldsymbol{\sigma })\geq (1-2\zeta )W^+ \mid \mathfrak U,\,x\in V_{0}^+[{i}]}\right \rbrack \nonumber \\ &\leq \sum _{0\leq v_1, \ldots, v_s\leq \Delta /s}\unicode{x1D7D9}\left \{{\sum _{j=1}^nv_jw_j^+\geq (1-2\zeta )W^+}\right \}{\mathbb{P}}\left \lbrack{\forall j\in [s]\,:\,\boldsymbol{W}^+_{x,j}(\boldsymbol{\sigma })=v_j\mid \mathfrak U,\,x\in V_{0}^+[{i}]}\right \rbrack \nonumber \\ & \leq \exp (o(\Delta ))\max _{0\leq v_1, \ldots, v_s\leq \Delta /s}\unicode{x1D7D9}\!\!\left \{{\sum _{j=1}^nv_jw_j^+\geq (1-2\zeta )W^+}\right \}{\mathbb{P}}\left \lbrack{\forall j\in [s]\,:\,\boldsymbol{W}^+_{x,j}(\boldsymbol{\sigma })=v_j\mid \mathfrak U,\,x\in V_{0}^+[{i}]}\right \rbrack . \end{align}

Further, given $\mathfrak U$ and $x\in V_0^+[i]$ , the random variables $(\boldsymbol{W}^+_{x,j})_{j\in [s]}$ are mutually independent because $x$ joins tests in the compartments $F[i+j-1]$ , $j\in [s]$ , independently. Hence, Claim 3.7 shows that for any $v_1,\ldots, v_s$ ,

(3.31) \begin{align}{\mathbb{P}}\left \lbrack{\forall j\in [s]\,:\,\boldsymbol{W}^+_{x,j}(\boldsymbol{\sigma })=v_j\mid \mathfrak U,\,x\in V_{0}^+[{i}]}\right \rbrack =\prod _{j=1}^s{\mathbb{P}}\left \lbrack{\boldsymbol{W}^+_{x,j}(\boldsymbol{\sigma })=v_j\mid \mathfrak U,\,x\in V_{0}^+[{i}]}\right \rbrack . \end{align}

Thus, consider the optimisation problem

(3.32) \begin{align} \mathscr{M}^+_t=&\min \frac{1}{s}\sum _{j=1}^{s} D_{\mathrm{KL}}\left ({{{z_j}\|{\frac{q_{0,j}^+}{q_0^+}}}}\right )\qquad \text{ s.t.}\quad \sum _{j=1}^{s}z_j w_j^+ \geq (1-t)W^+,\quad z_1,\ldots, z_{s}\in [q_{0,j}^+/q_0^+,1]. \end{align}

Then combining (3.30) and (3.31) with Claim 3.7 and Lemma 1.6 and using the substitution $z_j=v_j/(p_{11}\Delta )$ , we obtain

(3.33) \begin{align}{\mathbb{P}}&\left \lbrack{\boldsymbol{W}_{{x}}^+\geq (1-2\zeta )W^+ \mid \mathfrak U,\,x\in V_{0}^+[{i}]}\right \rbrack \leq \exp ({-}\Delta p_{11}\mathscr{M}^+_{2\zeta }+o(\Delta )). \end{align}

We claim that (3.33) can be sharpened to

(3.34) \begin{align}{\mathbb{P}}&\left \lbrack{\boldsymbol{W}_{{x}}^+\geq (1-2\zeta )W^+ \mid \mathfrak U,\,x\in V_{0}^+[{i}]}\right \rbrack \leq \exp ({-}\Delta p_{11}\mathscr{M}^+_0+o(\Delta )). \end{align}

Indeed, consider any feasible solution $z_1,\ldots, z_s$ to $\mathscr{M}_\zeta ^+$ such that $\sum _{j\geq s}z_jw_j^+\lt W^+$ . Obtain $z^{\prime}_1,\ldots, z^{\prime}_s$ by increasing some of the values $z_1,\ldots, z_s$ such that $\sum _{j\leq s}z^{\prime}_jw_j^+=W^+$ . Then because the functions $z\mapsto D_{\mathrm{KL}}({z}\|{q_{0,j}^+/q_0^+})$ with $j\in [s]$ are equicontinuous on the compact interval $[0,1]$ , we see that

\begin{align*} \frac{1}{s}\sum _{j=1}^sD_{\mathrm{KL}}\left ({{{z_j}\|{q_{0,j}^+/q_0^+}}}\right )\geq \frac{1}{s}\sum _{j=1}^sD_{\mathrm{KL}}\left ({{{z^{\prime}_j}\|{q_{0,j}^+/q_0^+}}}\right )+o(1) \end{align*}

uniformly for all $z_1,\ldots, z_s$ and $z^{\prime}_1,\ldots, z^{\prime}_s$ . Thus, (3.34) follows from (3.33).

Finally, we notice that the condition $z_j\geq q_{0,j}^+/q_0^+$ in (3.32) is superfluous. Indeed, since $D_{\mathrm{KL}}({q_{0,j}^+/q_0^+}\|{q_{0,j}^+/q_0^+})=0$ , there is nothing to be gained from choosing $z_j\lt q_{0,j}^+/q_0^+$ . Furthermore, since the Kullback–Leibler divergence is strictly convex and (1.1) ensures that $q_{0,j}^+/q_0^+\lt q_{1,j}^+/p_{11}$ for all $j$ , the optimisation problem $\mathscr{M}^+_0$ attains a unique minimum, which is not situated at the boundary of the intervals $[q_{0,j}^+/q_0^+,1]$ . That said, the unique minimiser satisfies the weight constraint $\sum _{j\geq s}z_jw_j^+$ with equality; otherwise, we could reduce some $z_j$ , thereby decreasing the objective function value. In summary, we conclude that $\mathscr{M}_0^+=\mathscr{M}^+$ . Thus, the assertion follows from (3.34).

Claim 3.9. Let

(3.35) \begin{align} \mathscr{M}^-=& \min \frac{1}{s}\sum _{j=1}^{s} D_{\mathrm{KL}}\left ({{{z_j}\|{\frac{q_{0,j}^-}{q_0^-}}}}\right )\qquad \text{ s.t.}\quad \sum _{j=1}^{s}\left ({z_j-\frac{q_{1,j}^-}{p_{01}}}\right ) w_j^- = 0,\qquad z_1,\ldots, z_{s}\in (0,1). \end{align}

Then for all $s \lt i \leq \ell$ and all $x\in V[i]$ , we have

\begin{align*}{\mathbb{P}}\left \lbrack{\boldsymbol{W}_{{x}}^- \leq (1+2\zeta )W^- \mid \mathfrak U,\,x\in V_0^+[i]}\right \rbrack &\leq \exp ({-}(1+o(1))p_{10}\Delta \mathscr{M}^-). \end{align*}

Recall that by convention $0 \cdot \infty = 0$ . Thus for $p_{10}=0$ , the condition of (3.35) boils down to $z_j ={q_{1,j}^-}/{p_{01}}$ , and the optimisation becomes trivial.

Clearly, as a next step, we need to solve the optimisation problems (3.29) and (3.35). We defer this task to Section 3.6, where we prove the following.

Lemma 3.10. Let $\boldsymbol{X}$ have distribution ${\textrm{Be}}({\exp} ({-}d))$ , and let $\boldsymbol{Y}$ be the (random) channel output given input $\boldsymbol{X}$ . Then

\begin{align*} p_{11}\mathscr{M}^++p_{10}\mathscr{M}^-=&\frac{I(\boldsymbol{X},\boldsymbol{Y})}d-D_{\mathrm{KL}}\left ({{{p_{11}}\|{q_0^+}}}\right ). \end{align*}

Proof of Lemma 3.2 (large deviations for healthy individuals). In light of Claims 3.8 and 3.9 and Lemma 3.10 to work out that all but $o(k)$ positive individuals are identified correctly, using Markov’s inequality, we need to verify that

(3.36) \begin{align} \left |{V_0^+}\right | \exp \left ({-}\Delta \left (p_{11} \mathscr{M}^+ + p_{10}\mathscr{M}^-\right ) \right ) \lt & k \end{align}

Taking the logarithm of (3.36) and simplifying, we arrive at

(3.37) \begin{align} & \ln (n)\left (1-cd(1-\theta ) \left (D_{\mathrm{KL}}\left ({{{p_{11}}\|{(1-\exp ({-}d))p_{11}+\exp ({-}d)p_{01}}}}\right )+p_{11}\mathscr{M}^++p_{10}\mathscr{M}^-\right ) \right )\nonumber\\& \qquad\qquad\qquad\lt \theta \ln (n). \end{align}

Thus, we need to show that

(3.38) \begin{align} cd\left \lbrack{D_{\mathrm{KL}}\left ({{{p_{11}}\|{(1-\exp ({-}d))p_{11}+\exp ({-}d)p_{01}}}}\right )+p_{11}\mathscr{M}^++p_{10}\mathscr{M}^-}\right \rbrack \gt &1. \end{align}

This boils down to $cI(\boldsymbol{X},\boldsymbol{Y})\gt 1$ , which in turn is identical to (2.34).

3.6. Proof of Lemma 3.10

We tackle the optimisation problems $\mathscr{M}^{\pm }$ via the method of Lagrange multipliers. Since the objective functions are strictly convex, these optimisation problems possess unique stationary points. As the parameters from (3.19) satisfy $q_{1,j}^+/p_{11}=\exp (d(j-s)/s)$ , the optimisation problem (3.29) gives rise to the following Lagrangian.

Claim 3.11. The Lagrangian

\begin{align*} \mathscr{L}^+ =& \sum _{j=1}^s D_{\mathrm{KL}}\left ({{{z_j}\|{\frac{q_{0,j}^+}{q_0^+}}}}\right ) + \lambda w_j^+\left ({z_j - \exp ({-}d(s-j)/s)}\right ) \end{align*}

has the unique stationary point $z_j=\exp ({-}d(s-j)/s)$ , $\lambda =-1$ .

Proof. Since the objective function $\sum _{j=1}^s D_{\mathrm{KL}}\left ({{{z_j}\|{q_{1,j}^+/p_{11}}}}\right )$ is strictly convex, we just need to verify that $\lambda =-1$ and $z_j=\exp ({-}d(s-j)/s)$ is a stationary point. To this end, we calculate

(3.39) \begin{align} \frac{\partial \mathscr{L}^+}{\partial z_j} = & \ln \frac{z_j^+}{1-z_j^+}- \ln \frac{q_{1,j}^+}{p_{11}-q_{1,j}^+} + \lambda w_j^+,& \frac{\partial \mathscr{L}^+}{\partial \lambda } =& \sum _{j=1}^s \left ({z_j - \exp \left ({-d(s-j)/s}\right )}\right ) w_j^+. \end{align}

Substituting in the definition (2.29) of the weights $w_j^+$ and the definition (3.19) of $p_{11},q_{1,j}^+$ and simplifying, we obtain

\begin{align*} \frac{\partial L^+}{\partial z_j}\bigg |_{\substack{z_j=\exp ({-}d(s-j)/s)\\\lambda =-1}} & =\ln \frac{\exp ({-}d(s-j)/s)}{1-\exp ({-}d(s-j)/s)} - \ln \frac{p_{11}({\exp} \left ({dj/s}\right ) -1) + p_{01}}{p_{11}({\exp} (d)-1)+p_{01}}\\& +\ln \left (\!{1-\frac{p_{11}({\exp} (d j/s) -1) + p_{01}}{p_{11}({\exp} (d)-1)+p_{01}} }\right ) {-}\ln \frac{p_{11}}{p_{11} + (p_{01}-p_{11}) \exp ({-}d j/s )}=0. \end{align*}

Finally, (3.39) shows that setting $z_j=\exp ({-}d(s-j)/s)$ ensures that $\partial \mathscr{L}^+/\partial \lambda =0$ as well.

Analogous steps prove the corresponding statement for $\mathscr{M}^-$ .

Claim 3.12. Assume $p_{10} \gt 0$ . The Lagrangian

\begin{align*} \mathscr{L}^- =& \sum _{j=1}^s D_{\mathrm{KL}}\left ({{{z_j}\|{\frac{q_{0,j}^-}{p_{01}}}}}\right ) + \lambda w_j^-\left ({z_j - \exp ({-}d(s-j)/s)}\right ) \end{align*}

has the unique stationary point $z_j=\exp ({-}d(s-j)/s)$ , $\lambda =-1$ .

Having identified the minimisers of $\mathscr{M}^{\pm }$ , we proceed to calculate the optimal objective function values. Note that for $\mathscr{M}^-$ , the minimisers $z_j$ for the cases $p_{10}\gt 0$ and $p_{10} = 0$ coincide.

Claim 3.13. Let

\begin{align*} \lambda ^+ & =\ln (q_0^+)=\ln (p_{01}\exp ({-}d)+p_{11}(1-\exp ({-}d))), \lambda ^- =\ln (q_0^-)\\[3pt]& =\ln (p_{00}\exp ({-}d)+p_{10}(1-\exp ({-}d))). \end{align*}

Then

\begin{align*} p_{11}d\exp (d)\mathscr{M}^+=&(\lambda ^++d)\left ({p_{11}\left ({(d-1)\exp (d)+1}\right )-p_{01}}\right )+p_{01}\ln (p_{01})\\& -p_{11}\left ({(d-1)\exp (d)+1}\right )\ln (p_{11})-d(d-1)\exp (d)p_{11}+O(1/s),\\ p_{10}d\exp (d)\mathscr{M}^-=&(\lambda ^-+d)\left ({p_{10}\left ({(d-1)\exp (d)+1}\right )-p_{00}}\right )+p_{00}\ln (p_{00})\\ & -p_{10}\left ({(d-1)\exp (d)+1}\right )\ln (p_{10})-d(d-1)\exp (d)p_{10}+O(1/s)\quad \text{if } p_{10}\gt 0. \end{align*}

Proof. We perform the calculation for $\mathscr{M}^+$ in detail. Syntactically identical steps yield the expression for $\mathscr{M}^-$ , the only required change being the obvious modification of the indices of the channel probabilities. Substituting the optimal solution $z_j=\exp ({-}d(s-j)/s)$ and the definitions (3.19) and (2.1), (3.19), and (3.26) of $q^+_{1,j},q_0^+,q_{0,j}^+$ into (3.29), we obtain

(3.40) \begin{align} \mathscr{M}^+=&\frac{1}{s}\sum _{j=1}^{s}D_{\mathrm{KL}}\left ({{{\exp ({-}d(s-j)/s)}\|{\frac{p_{01}+({\exp} (dj/s)-1)p_{11}}{p_{01}+({\exp} (d)-1)p_{11}}}}}\right )=\mathscr{I}^++O(1/s),\qquad \text{ where}\\ \mathscr{I}^+=&\int _0^1D_{\mathrm{KL}}\left ({{{\exp (d(x-1))}\|{\frac{p_{11}({\exp} (dx)-1)+p_{01}}{p_{11}({\exp} (d)-1)+p_{01}}}}}\right ){\mathrm d} x;\nonumber \end{align}

the $O(1/s)$ -bound in (3.40) holds because the derivative of the integrand $x\mapsto D_{\mathrm{KL}}\left ({{{\exp (d(x-1))}\|{\frac{p_{11}({\exp} (dx)-1)+p_{01}}{p_{11}({\exp} (d)-1)+p_{01}}}}}\right )$ is bounded on $[0,1]$ . Replacing the Kullback–Leibler divergence by its definition, we obtain $\mathscr{I}^+=\mathscr{I}^+_1+\mathscr{I}^+_2$ , where

\begin{align*} \mathscr{I}^+_1&=\int _{0}^{1} \exp \left ({d(x-1)}\right ) \ln \frac{\exp \left ({d(x-1)}\right ) ( p_{11}({\exp} \left ({d}\right )-1)+p_{01} ) }{p_{11}({\exp} \left ({dx}\right )-1)+p_{01} }{\mathrm d} x,\\ \mathscr{I}^+_2&=\int _{0}^{1} (1-\exp \left ({d(x-1)}\right )) \ln \left \lbrack{ \frac{ 1- \exp \left ({d(x-1)}\right ) }{ 1 - \frac{p_{11}({\exp} \left ({dx}\right )-1)+p_{01}}{p_{11}({\exp} \left ({d}\right )-1)+p_{01}} } }\right \rbrack{\mathrm d} x. \end{align*}

Splitting the logarithm in the first integrand, we further obtain $\mathscr{I}^+_1=\mathscr{I}_{11}^++\mathscr{I}_{12}^+$ , where

\begin{align*} \mathscr{I}^+_{11}&=\int _{0}^{1} \exp \left ({d(x-1)}\right ) \ln \left \lbrack{ \exp \left ({d(x-1)}\right ) ( p_{11}({\exp} \left ({d}\right )-1)+p_{01} )}\right \rbrack{\mathrm d} x,\\ \mathscr{I}^+_{12}&=-\int _{0}^{1} \exp \left ({d(x-1)}\right ) \ln \left \lbrack{ p_{11}({\exp} \left ({dx}\right )-1)+p_{01} }\right \rbrack{\mathrm d} x. \end{align*}

Setting $\Lambda ^+=\ln ( p_{11}({\exp} \left ({d}\right )-1)+p_{01})=\lambda ^++d$ and introducing $ u = \exp \left ({d(x-1)}\right )$ , we calculate

(3.41) \begin{align} \mathscr{I}_{11}^+&=\frac{1}{d} \left \lbrack{\int _{\exp \left ({-d}\right )}^{1} \ln ( u){\mathrm d} u + \int _{\exp \left ({-d}\right )}^{1} \Lambda ^+{\mathrm d} u }\right \rbrack =\frac{1}{d} \left \lbrack{(d+1)\exp ({-}d) -1 + (1 - \exp \left ({-d}\right ) ) \Lambda ^+}\right \rbrack . \end{align}

Concerning $\mathscr{I}_{12}^+$ , we once again substitute $ u = \exp \left ({d(x-1)}\right )$ to obtain

(3.42) \begin{align} \nonumber \mathscr{I}_{12}^+&=-\frac{1}{d}\int _{\exp ({-}d)}^1\ln \left ({p_{11}\exp (d)u+p_{01}-p_{11}}\right ){\mathrm d} u\\[4pt] &= - \frac{1}{d} \left \lbrack{ \left ({\frac{ p_{01}}{p_{11}}\exp \left ({-d}\right )-\exp \left ({-d}\right )+1}\right )\Lambda ^+ - \exp \left ({-d}\right )\frac{p_{01}}{p_{11}} \ln (p_{01}) + \exp \left ({-d}\right )-1 }\right \rbrack \end{align}

Proceeding to $\mathscr{I}_2$ , we obtain

(3.43) \begin{align} \nonumber \mathscr{I}_2^+ &= \int _{0}^{1} (1- \exp \left ({d(x-1)}\right )) \ln \frac{ p_{11} ({\exp} \left ({d}\right ) - 1) + p_{01} }{ p_{11}\exp (d) }{\mathrm d} x\nonumber\\ & = \int _{0}^{1} (1- \exp \left ({d(x-1)}\right )) \ln \left ({ p_{11} ({\exp} (d) - 1) + p_{01} }\right ){\mathrm d} x\nonumber\\ & \quad - \int _{0}^{1} (1- \exp \left ({d(x-1)}\right )) \ln ( p_{11}\exp (d) ){\mathrm d} x\nonumber \\ &=\frac{ (\Lambda ^+ - \ln ( p_{11}) - d ) ( d-1 + \exp \left ({-d}\right ) )}{d}. \end{align}

Finally, recalling that $\mathscr{I}^+=\mathscr{I}_1+\mathscr{I}_2=\mathscr{I}^+_{11}+\mathscr{I}^+_{12}+\mathscr{I}_2^+$ and combining (3.40)–(3.43) and simplifying, we arrive at the desired expression for $\mathscr{M}^+$ .

Proof of Lemma 3.10 . We have

\begin{align*} I(\boldsymbol{X},\boldsymbol{Y})&=H(\boldsymbol{Y})-H(\boldsymbol{Y}|\boldsymbol{X})=h(p_{00}\exp ({-}d)+p_{10}(1-\exp ({-}d)))-\exp ({-}d)h(p_{00})\\& \quad -(1-\exp ({-}d))h(p_{11}). \end{align*}

Hence, Claim 3.13 yields

\begin{align*} p_{11}\mathscr{M}^++p_{10}\mathscr{M}^-=&-\frac{h(p_{00})}{d\exp (d)}-\frac{(1-\exp ({-}d))h(p_{11})}{d}+h(p_{11})\\[3pt] &+\frac{(p_{11}-p_{01})\lambda ^++(p_{10}-p_{00})\lambda ^-}{d\exp (d)}+\frac{d-1}d\left \lbrack{p_{11}\lambda ^++p_{10}\lambda ^-}\right \rbrack \\[3pt] &=-\frac{h(p_{00})}{d\exp (d)}-\frac{(1-\exp ({-}d))h(p_{11})}{d}-D_{\mathrm{KL}}\left ({{{p_{11}}\|{q_0^+}}}\right )-\frac{\lambda ^+}d\exp (\lambda ^+)\\ & \quad -\frac{\lambda ^-}d\exp (\lambda ^-)\\[3pt] &=\frac{I(\boldsymbol{X},\boldsymbol{Y})}d-D_{\mathrm{KL}}\left ({{{p_{11}}\|{q_0^+}}}\right ), \end{align*}

as desired.

4.1. Proof of Proposition 2.7

Assume that $c\gt c_{\mathrm{ex},1}(d,\theta )+\varepsilon$ , and let $c^{\prime}=c_{\mathrm{ex},1}(d,\theta )+\varepsilon /2$ . Since $c\gt c^{\prime}+\varepsilon /2$ , the definitions (1.11) of $c_{\mathrm{ex},1}(d,\theta )$ and (1.6) of $\mathscr{Y}(c^{\prime},d,\theta )$ ensure that for small enough $\delta \gt 0$ , we can find an open interval $\mathscr{I}\subseteq \mathscr{Y}(c^{\prime},d,\theta )$ with rational boundary points such that Z1 is satisfied.

Let $\,\,\,\bar{\!\!\!\mathscr{I}}\,$ be the closure of $\mathscr{I}$ . Then by the definition of $c_{\mathrm{ex},1}(d,\theta )$ , there exists a function $z\,:\, \,\,\,\bar{\!\!\!\mathscr{I}}\,\to [p_{01},p_{11}]$ such that for all $y\in \,\,\,\bar{\!\!\!\mathscr{I}}\,$ we have

(4.1) \begin{align} cd(1-\theta )\left \lbrack{D_{\mathrm{KL}}\left ({{{y}\|{\exp ({-}d)}}}\right )+yD_{\mathrm{KL}}\left ({{{z(y)}\|{p_{11}}}}\right )}\right \rbrack &=\theta . \end{align}

In fact, because the Kullback–Leibler divergence is strictly convex, the equation (4.1) defines $z(y)$ uniquely. The inverse function theorem implies that the function $z(y)$ is continuous and therefore uniformly continuous on $\,\,\,\bar{\!\!\!\mathscr{I}}\,$ . Additionally, once again, because the Kullback–Leibler divergence is convex and $c\gt c_{\mathrm{ex},1}(d,\theta )$ , for all $y\in \,\,\,\bar{\!\!\!\mathscr{I}}\,$ , we have

\begin{align*} cd(1-\theta )\left \lbrack{D_{\mathrm{KL}}\left ({{{y}\|{\exp ({-}d)}}}\right )+yD_{\mathrm{KL}}\left ({{{z(y)}\|{p_{01}}}}\right )}\right \rbrack &\gt 1. \end{align*}

Therefore, there exists $\hat \delta =\hat \delta (c,d,\theta )\gt 0$ such that for all $y\in \,\,\,\bar{\!\!\!\mathscr{I}}\,$ , we have

(4.2) \begin{align} cd(1-\theta )\left \lbrack{D_{\mathrm{KL}}\left ({{{y}\|{\exp ({-}d)}}}\right )+yD_{\mathrm{KL}}\left ({{{z(y)}\|{p_{01}}}}\right )}\right \rbrack &\gt 1+\hat \delta . \end{align}

Combining (4.1) and (4.2), we find a continuous $\hat z \,:\, \,\,\,\bar{\!\!\!\mathscr{I}}\,\to [p_{01},p_{11}]$ such that for small enough $\delta \gt 0$ for all $y\in [0,1]$ , we have

(4.3) \begin{align} cd(1-\theta )\left \lbrack{D_{\mathrm{KL}}\left ({{{y}\|{\exp ({-}d)}}}\right )+yD_{\mathrm{KL}}\left ({{{\hat z(y)}\|{p_{11}}}}\right )}\right \rbrack &\gt \theta +2\delta, \qquad \text{ and} \end{align}

(4.4) \begin{align} cd(1-\theta )\left \lbrack{D_{\mathrm{KL}}\left ({{{y}\|{\exp ({-}d)}}}\right )+yD_{\mathrm{KL}}\left ({{{\hat z(y)}\|{p_{01}}}}\right )}\right \rbrack &\gt 1+2\delta . \qquad\qquad\end{align}

Additionally, by uniform continuity for any given $0\lt \varepsilon ^{\prime}\lt \delta /2$ (which may depend arbitrarily on $\delta$ and $\mathscr{I}$ ), we can choose $\delta ^{\prime}\gt 0$ small enough so that

(4.5) \begin{align} |\hat z(y)-\hat z(y^{\prime})|&\lt \varepsilon ^{\prime}/2&\text{for all } y,y^{\prime}\in \,\,\,\bar{\!\!\!\mathscr{I}}\, \text{ with} |y-y^{\prime}|\lt \delta ^{\prime}. \end{align}

Finally, let $y_0,\ldots, y_\nu$ with $\nu =\nu (\delta ^{\prime},\varepsilon ^{\prime})\gt 0$ be a large enough number of equally spaced points in $\,\,\,\bar{\!\!\!\mathscr{I}}\,=[y_0,y_\nu ]$ . Then for each $i$ , we pick $\mathscr{Z}(y_i)\in [p_{01},p_{11}]\cap \mathbb{Q}$ such that $|\hat z(y_i)-\mathscr{Z}(y_i)|$ is small enough. Extend $\mathscr{Z}$ to a step function $\,\,\,\bar{\!\!\!\mathscr{I}}\,\to \mathbb{Q}\cap [0,1]$ by letting $\mathscr{Z}(y)=\mathscr{Z}(y_{i-1})$ for all $y\in (y_{i-1},y_i)$ for $1\leq i\leq \nu$ . Since $y\mapsto \hat z(y)$ is uniformly continuous, we can choose $\nu$ large enough so that (4.3)–(4.5) imply

\begin{align*} cd(1-\theta )\left \lbrack{D_{\mathrm{KL}}\left ({{{y}\|{\exp ({-}d)}}}\right )+yD_{\mathrm{KL}}\left ({{{\mathscr{Z}(y)}\|{p_{11}}}}\right )}\right \rbrack &\gt \theta +\delta &&\text{for all }y\in \,\,\bar{\!\!\!\mathscr{I}},\\ cd(1-\theta )\left \lbrack{D_{\mathrm{KL}}\left ({{{y}\|{\exp ({-}d)}}}\right )+yD_{\mathrm{KL}}\left ({{{\mathscr{Z}(y)}\|{p_{01}}}}\right )}\right \rbrack &\gt 1+\delta &&\text{ for all }y\in \,\,\bar{\!\!\!\mathscr{I}},\\ |\mathscr{Z}(y)-\mathscr{Z}(y^{\prime})|&\lt \varepsilon ^{\prime}&&\text{ for all } y,y^{\prime}\in \,\,\bar{\!\!\!\mathscr{I}}\, \text{ such that } |y-y^{\prime}|\lt \delta ^{\prime}, \end{align*}

as claimed.

4.2. Proof of Proposition 2.8

As in the proof of Proposition 2.6 in Section 3.3, we will first investigate an idealised scenario where we assume that the ground truth $\boldsymbol{\sigma }$ is known. Then we will use the expansion property provided by Lemma 3.3 to connect this idealised scenario with the actual steps of the algorithm.

In order to study the idealised scenario, for $x\in V[i]$ and $j\in [s]$ , we introduce

\begin{align*} \boldsymbol{Y}_{x,j}&= \left |{ \left \{{ a \in F[i+j-1]\cap \partial x\,:\, V_1\cap \partial a\subseteq \left \{{x}\right \}}\right \}}\right |,& \boldsymbol{Z}_{x,j}&= \left |{ \left \{{ a \in F^+[i+j-1]\cap \partial x\,:\, V_1\cap \partial a\subseteq \left \{{x}\right \}}\right \}}\right |,\\ \boldsymbol{Y}_x&= \sum _{j=1}^s\boldsymbol{Y}_{x,j},&\boldsymbol{Z}_x&= \sum _{j=1}^s\boldsymbol{Z}_{x,j}. \end{align*}

Thus, $\boldsymbol{Y}_{x,j}$ is the number of untainted tests in compartment $F[i+j-1]$ that contain $x$ , that is, test that does not contain another infected individual. Moreover, $\boldsymbol{Z}_{x,j}$ is the number of positively displayed untainted tests. Finally, $\boldsymbol{Y}_x,\boldsymbol{Z}_x$ are the sums of these quantities on $j\in [s]$ . The following lemma provides an estimate of the number of individuals $x$ with a certain value of $\boldsymbol{Y}_x$ .

Lemma 4.1. w.h.p., for all $0\leq Y\leq \Delta$ and all $i\in [\ell ]$ , we have

(4.6) \begin{align} \sum _{x\in V_0[i]}\unicode{x1D7D9}\{\boldsymbol{Y}_x=Y\}\leq n\exp \left ({-\Delta D_{\mathrm{KL}}\left ({{{Y/\Delta }\|{\exp ({-}d)}}}\right )+o(\Delta )}\right ), \end{align}

(4.7) \begin{align} \sum _{x\in V_1[i]}\unicode{x1D7D9}\{\boldsymbol{Y}_x=Y\}\leq k\exp \left ({-\Delta D_{\mathrm{KL}}\left ({{{Y/\Delta }\|{\exp ({-}d)}}}\right )+o(\Delta )}\right ). \end{align}

Proof. Let $1\leq i\leq \ell$ and consider any $x\in V[i]$ . Further, obtain $\mathbf{G}_{\mathrm{sc}}-x$ from $\mathbf{G}_{\mathrm{sc}}$ by deleting individual $x$ (and, of course, removing $x$ from all tests). Additionally, obtain $\mathbf{G}_{\mathrm{sc}}^{\prime}$ from $\mathbf{G}_{\mathrm{sc}}-x$ by re-inserting $x$ and assigning $x$ to $\Delta /s$ random tests in the compartments $F[i+j-1]$ for $j\in [s]$ as per the construction of the spatially coupled test design. Then the random test designs $\mathbf{G}_{\mathrm{sc}}$ and $\mathbf{G}_{\mathrm{sc}}^{\prime}$ are identically distributed.

Let $\mathfrak{E}$ be the event that $\mathbf{G}_{\mathrm{sc}}$ enjoys properties G1 and G2 from Proposition 2.3. Then Proposition 2.3 shows that

(4.8) \begin{align} {\mathbb{P}}\left \lbrack{\mathfrak{E}}\right \rbrack &=1-o(n^{-2}). \end{align}

Moreover, given $\mathfrak{E}$ for every $j\in [s]$ , the number of tests in $F[i+j-1]$ that contain no infected individual aside from $x$ satisfies

(4.9) \begin{align} \sum _{a\in F[i+j-1]}\unicode{x1D7D9}\{\partial a\cap V_1\setminus \{x\}=\emptyset \}=(1+O(n^{-\Omega (1)}))\frac{m}{\ell }\exp ({-}d); \end{align}

this follows from the bounds on $F_0[i+j-1]$ provided by G2 and the fact that discarding $x$ can change the numbers of actually positive/negative tests by no more than $\Delta$ .

Now consider the process of re-inserting $x$ to obtain $\mathbf{G}_{\mathrm{sc}}^{\prime}$ . Then (4.9) shows that given $\mathfrak{E}$ , we have

\begin{align*} \boldsymbol{Y}_{x,j}&\sim \textrm{Hyp}\left (\frac{m}{\ell }+O(1),(1+O(n^{-\Omega (1)}))\frac{m}{\ell }\exp ({-}d),\frac{\Delta }{s}\right )&&(j\in [s]). \end{align*}

These hypergeometric variables are mutually independent given $\mathbf{G}_{\mathrm{sc}}-x$ . Therefore, Lemma 1.6 implies that on $\mathfrak{E}$ ,

(4.10) \begin{align} {\mathbb{P}}\left \lbrack \boldsymbol{Y}_x=Y\mid \mathbf{G}_{\mathrm{sc}}-x\right \rbrack &\leq \exp \left ({-}\Delta D_{\mathrm{KL}}\left ({Y/\Delta }\|{\exp ({-}d)}\right )+o(\Delta )\right ). \end{align}

This estimate holds independently of the infection status $\boldsymbol{\sigma }_x$ . Thus, the assertion follows from (4.8), (4.10), and Markov’s inequality.

As a next step, we argue that for $c$ beyond the threshold $c_{\mathrm{ex},1}(d,\theta )$ , the function $\mathscr{Z}$ from Proposition 2.7 separates the infected from the uninfected individuals w.h.p.