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On resilient graph spanners
 IN ESA
, 2013
"... We introduce and investigate a new notion of resilience in graph spanners. Let S be a spanner of a graph G. Roughly speaking, we say that a spanner S is resilient if all its pointtopoint distances are resilient to edge failures. Namely, whenever any edge in G fails, then as a consequence of this ..."
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We introduce and investigate a new notion of resilience in graph spanners. Let S be a spanner of a graph G. Roughly speaking, we say that a spanner S is resilient if all its pointtopoint distances are resilient to edge failures. Namely, whenever any edge in G fails, then as a consequence of this failure all distances do not degrade in S substantially more than in G (i.e., the relative distance increases in S are very close to those in the underlying graph G). In this paper we show that sparse resilient spanners exist, and that they can be computed efficiently.
Dynamic approximate allpairs shortest paths: Breaking the O(mn) barrier and derandomization
 In Proc. FOCS
, 2013
"... We study dynamic (1 + )approximation algorithms for the allpairs shortest paths problem in unweighted undirected nnode medge graphs under edge deletions. The fastest algorithm for this problem is a randomized algorithm with a total update time of Õ(mn) and constant query time by Roditty and Zwi ..."
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We study dynamic (1 + )approximation algorithms for the allpairs shortest paths problem in unweighted undirected nnode medge graphs under edge deletions. The fastest algorithm for this problem is a randomized algorithm with a total update time of Õ(mn) and constant query time by Roditty and Zwick [33] (FOCS 2004). The fastest deterministic algorithm is from a 1981 paper by Even and Shiloach [23]; it has a total update time of O(mn2) and constant query time. We improve these results as follows: (1) We present an algorithm with a total update time of Õ(n5/2) and constant query time that has an additive error of two in addition to the 1 + multiplicative error. This beats the previous Õ(mn) time when m = Ω(n3/2). Note that the additive error is unavoidable since, even in the static case, an O(n3−δ)time (a socalled truly subcubic) combinatorial algorithm with 1 + multiplicative error cannot have an additive error less than 2 − , unless we make a major breakthrough for Boolean matrix multiplication [19] and many other longstanding problems [39]. The algorithm can also be turned into a (2 + )approximation algorithm (without an
Fully dynamic (1 + ǫ)approximate matchings
 In Proceedings of the 54th Anual IEEE Symposium on Foundations of Computer Science, FOCS
, 2013
"... ar ..."