
Fully Dynamic Maximal Independent Set in Expected PolyLog Update Time
In the fully dynamic maximal independent set (MIS) problem our goal is t...
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Deterministic Rounding of Dynamic Fractional Matchings
We present a framework for deterministically rounding a dynamic fraction...
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The Expander Hierarchy and its Applications to Dynamic Graph Algorithms
We introduce a notion for hierarchical graph clustering which we call th...
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Decremental StronglyConnected Components and SingleSource Reachability in NearLinear Time
Computing the StronglyConnected Components (SCCs) in a graph G=(V,E) is...
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Dynamic LowStretch Trees via Dynamic LowDiameter Decompositions
Spanning trees of low average stretch on the nontree edges, as introduc...
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Fast Dynamic Cuts, Distances and Effective Resistances via Vertex Sparsifiers
We present a general framework of designing efficient dynamic approximat...
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Incremental Edge Orientation in Forests
For any forest G = (V, E) it is possible to orient the edges E so that n...
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FullyDynamic Graph Sparsifiers Against an Adaptive Adversary
Designing dynamic graph algorithms against an adaptive adversary is a major goal in the field of dynamic graph algorithms. While a few such algorithms are known for spanning trees, matchings, and singlesource shortest paths, very little was known for an important primitive like graph sparsifiers. The challenge is how to approximately preserve so much information about the graph (e.g., allpairs distances and all cuts) without revealing the algorithms' underlying randomness to the adaptive adversary. In this paper we present the first nontrivial efficient adaptive algorithms for maintaining spanners and cut sparisifers. These algorithms in turn imply improvements over existing algorithms for other problems. Our first algorithm maintains a polylog(n)spanner of size Õ(n) in polylog(n) amortized update time. The second algorithm maintains an O(k)approximate cut sparsifier of size Õ(n) in Õ(n^1/k) amortized update time, for any k>1, which is polylog(n) time when k=log(n). The amortized update time of both algorithms can be made worstcase by paying some subpolynomial factors. Prior to our result, there were nearoptimal algorithms against oblivious adversaries (e.g. Baswana et al. [TALG'12] and Abraham et al. [FOCS'16]), but the only nontrivial adaptive dynamic algorithm requires O(n) amortized update time to maintain 3 and 5spanner of size O(n^1+1/2) and O(n^1+1/3), respectively [Ausiello et al. ESA'05]. Our results are based on two novel techniques. First of all, we show a generic blackbox reduction that allows us to assume that the graph undergoes only edge deletions and, more importantly, remains an expander with almostuniform degree. The second is a new technique called proactive resampling. [Abstract was shortened]
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