Connectivity Labeling for Multiple Vertex Failures
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We present an efficient labeling scheme for answering connectivity queries in graphs subject to a specified number of vertex failures. Our first result is a randomized construction of a labeling function that assigns vertices O(f^3log^5 n)-bit labels, such that given the labels of F∪{s,t} where |F|≤ f, we can correctly report, with probability 1-1/poly(n), whether s and t are connected in G-F. However, it is possible that over all n^O(f) distinct queries, some are answered incorrectly. Our second result is a deterministic labeling function that produces O(f^7 log^13 n)-bit labels such that all connectivity queries are answered correctly. Both upper bounds are polynomially off from an Ω(f)-bit lower bound. Our labeling schemes are based on a new low degree decomposition that improves the Duan-Pettie decomposition, and facilitates its distributed representation. We make heavy use of randomization to construct hitting sets, fault-tolerant graph sparsifiers, and in constructing linear sketches. Our derandomized labeling scheme combines a variety of techniques: the method of conditional expectations, hit-miss hash families, and ϵ-nets for axis-aligned rectangles. The prior labeling scheme of Parter and Petruschka shows that f=1 and f=2 vertex faults can be handled with O(log n)- and O(log^3 n)-bit labels, respectively, and for f>2 vertex faults, Õ(n^1-1/2^f-2)-bit labels suffice.
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