Fig. 10

Illustration for Lemma 3.3 using the same coloring as Fig. 2a. a An example for an epidemic where among the nodes of the transmission path \((v_1, v_2, v_3, v_5)\), the middle household contains no symptomatic node (only the asymptomatic node \(v_3\)), but the LS+ algorithm still finds the source. Indeed, at iteration 0 we set \(s_{c,0}=v_5\), after which we find that \(v_3\) is asymptomatic, and next that \(v_2\) is asymptomatic and \(v_4\) is symptomatic, with a lower symptom onset time then \(v_5\). Hence, in iteration 1 we set \(s_{c,1}=v_4\), and we find that \(v_3, v_2\) are asymptomatic and \(v_1\) is is symptomatic, with a lower symptom onset time then \(v_4\). Finally, in iteration 2 we set \(s_{c,2}=v_1\), and we find \(s_c'=v_1=s_{c,2}\), which implies that the algorithm stops, and returns the correct source \(v_1\). b An example for an epidemic where the LS+ algorithm would fail if we would update the candidate before the test queue becomes empty. Similarly to subfigure a, in iteration 0 of the algorithm first learns about asymptomatic node \(v_3\) and next about asymptomatic node \(v_2\) and symptomatic node \(v_4\). If the algorithm updates the candidate to \(v_4\) and continues further, instead of scheduling the tests of the household members of \(v_2\), then it is not hard to check that \(v_4\) will be the final estimate and the algorithm fails. However, if the algorithm waits until the test queue becomes empty and tests the household members of \(v_2\), then \(v_1\) becomes the next candidate and the algorithm finds the source