43,505 research outputs found
(1 + )-Approximate shortest paths in dynamic streams
Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of 2â1 versus n1+1/, for an integer parameter . (In fact, existing solutions also incur an extra factor of 1 + in the stretch for weighted graphs, and an additional factor of logO(1) n in the space.) The only existing solution of the second type uses n1/2âO(1/) passes over the stream (for space O(n1+1/)), and applies only to unweighted graphs. In this paper we show that (1+)-approximate single-source shortest paths can be computed with ËO (n1+1/) space using just constantly many passes in unweighted graphs, and polylogarithmically many passes in weighted graphs. Moreover, the same result applies for multi-source shortest paths, as long as the number of sources is O(n1/). We achieve these results by devising efficient dynamic streaming constructions of (1 + , )-spanners and hopsets. On our way to these results, we also devise a new dynamic streaming algorithm for the 1-sparse recovery problem. Even though our algorithm for this task is slightly inferior to the existing algorithms of [26, 11], we believe that it is of independent interest. 2012 ACM Subject Classification Theory of computation ! Streaming models; Theory of computation ! Streaming, sublinear and near linear time algorithms; Theory of computation ! Shortest paths; Theory of computation ! Sparsification and spanner
Fully-dynamic Approximation of Betweenness Centrality
Betweenness is a well-known centrality measure that ranks the nodes of a
network according to their participation in shortest paths. Since an exact
computation is prohibitive in large networks, several approximation algorithms
have been proposed. Besides that, recent years have seen the publication of
dynamic algorithms for efficient recomputation of betweenness in evolving
networks. In previous work we proposed the first semi-dynamic algorithms that
recompute an approximation of betweenness in connected graphs after batches of
edge insertions.
In this paper we propose the first fully-dynamic approximation algorithms
(for weighted and unweighted undirected graphs that need not to be connected)
with a provable guarantee on the maximum approximation error. The transfer to
fully-dynamic and disconnected graphs implies additional algorithmic problems
that could be of independent interest. In particular, we propose a new upper
bound on the vertex diameter for weighted undirected graphs. For both weighted
and unweighted graphs, we also propose the first fully-dynamic algorithms that
keep track of such upper bound. In addition, we extend our former algorithm for
semi-dynamic BFS to batches of both edge insertions and deletions.
Using approximation, our algorithms are the first to make in-memory
computation of betweenness in fully-dynamic networks with millions of edges
feasible. Our experiments show that they can achieve substantial speedups
compared to recomputation, up to several orders of magnitude
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