Graph-constrained group testing

Non-adaptive group testing involves grouping arbitrary subsets of n items into different pools and identifying defective items based on tests obtained for each pool. Motivated by applications in network tomography, sensor networks and infection propagation we formulate non-adaptive group testing problems on graphs. Unlike conventional group testing problems each group here must conform to the constraints imposed by a graph. For instance, items can be associated with vertices and each pool is any set of nodes that must be path connected. In this paper we associate a test with a random walk. In this context conventional group testing corresponds to the special case of a complete graph on n vertices. For interesting classes of graphs we arrive at a rather surprising result, namely, that the number of tests required to identify d defective items is substantially similar to that required in conventional group testing problems, where no such constraints on pooling is imposed. Specifically, if T(n) corresponds to the mixing time of the graph G, we show that with m = O(d2T2(n) log(n/d)) non-adaptive tests, one can identify the defective items. Consequently, for the Erdős-Renyi random graph G(n, p), as well as expander graphs with constant spectral gap, it follows that m = O(d2 log3 n) non-adaptive tests are sufficient to identify d defective items. We next consider a specific scenario that arises in network tomography and show that m = O(d3 log3 n) non-adaptive tests are sufficient to identify d defective items. We also consider noisy counterparts of the graph constrained group testing problem and develop parallel results for these cases.

[1]  David Bruce Wilson,et al.  Generating random spanning trees more quickly than the cover time , 1996, STOC '96.

[2]  Pavel A. Pevzner,et al.  Towards DNA Sequencing Chips , 1994, MFCS.

[3]  Eli Upfal,et al.  Probability and Computing: Randomized Algorithms and Probabilistic Analysis , 2005 .

[4]  Jack K. Wolf,et al.  Born again group testing: Multiaccess communications , 1985, IEEE Trans. Inf. Theory.

[5]  M. Sobel,et al.  Group testing to eliminate efficiently all defectives in a binomial sample , 1959 .

[6]  Nick G. Duffield,et al.  Network Tomography of Binary Network Performance Characteristics , 2006, IEEE Transactions on Information Theory.

[7]  Enrique Mallada,et al.  Compressive sensing over graphs , 2010, 2011 Proceedings IEEE INFOCOM.

[8]  R. Dorfman The Detection of Defective Members of Large Populations , 1943 .

[9]  Michael Mitzenmacher,et al.  Probability And Computing , 2005 .

[10]  Rajeev Motwani,et al.  Randomized algorithms , 1996, CSUR.

[11]  Piotr Indyk,et al.  Combining geometry and combinatorics: A unified approach to sparse signal recovery , 2008, 2008 46th Annual Allerton Conference on Communication, Control, and Computing.

[12]  Patrick Thiran,et al.  Using End-to-End Data to Infer Lossy Links in Sensor Networks , 2006, Proceedings IEEE INFOCOM 2006. 25TH IEEE International Conference on Computer Communications.

[13]  Mark Jerrum,et al.  Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains , 1987, International Workshop on Graph-Theoretic Concepts in Computer Science.

[14]  Amin Karbasi,et al.  Compressed sensing with probabilistic measurements: A group testing solution , 2009, 2009 47th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[15]  Patrick Thiran,et al.  The Boolean Solution to the Congested IP Link Location Problem: Theory and Practice , 2007, IEEE INFOCOM 2007 - 26th IEEE International Conference on Computer Communications.

[16]  Béla Bollobás,et al.  Random Graphs , 1985 .

[17]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[18]  D. Du,et al.  Combinatorial Group Testing and Its Applications , 1993 .

[19]  George Atia,et al.  Boolean Compressed Sensing and Noisy Group Testing , 2009, IEEE Transactions on Information Theory.

[20]  George Atia,et al.  Noisy group testing: An information theoretic perspective , 2009, 2009 47th Annual Allerton Conference on Communication, Control, and Computing (Allerton).