$(1+ε)$-Approximate Shortest Paths in Dynamic Streams

Abstract

Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied during the last decade. 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 2κ − 1 versus n 1+1/κ , for an integer parameter κ. (In fact, existing solutions also incur an extra factor of 1 + in the stretch for weighted graphs, and an additional factor of log O(1) n in the space.) The only existing solution of the second type uses n 1/2−O(1/κ) passes over the stream (for space O(n 1+1/κ )), and applies only to unweighted graphs.In this paper we show that (1 + )-approximate single-source shortest paths can be computed in this setting with Õ(n 1+1/κ ) space using just constantly many passes in unweighted graphs, and polylogarithmically many passes in weighted graphs (assuming and κ are constant). Moreover, in fact, the same result applies for multi-source shortest paths, as long as the number of sources is O(n 1/κ ).We achieve these results by devising efficient dynamic streaming constructions of (1 + , β)spanners and hopsets. We believe that these constructions are of independent interest.

Brief Announcement: (1+ε)-Approximate Shortest Paths in Dynamic Streams.

“…We then use the spanner thus constructed to compute (1 + ϵ )approximate S × V shortest paths from up to n 1/κ designated sources. (See Section 6 of [15] for more details.) Theorem 2.2.…”

“…We then use the hopset thus constructed to compute (1 + ϵ )approximate S × V shortest paths from up to n 1/κ designated sources. (See Section 8 of [15] for more details.) These algorithms extend the results of [10,11] from the static streaming setting to dynamic streaming one.…”

“…In this report we give a brief overview of our work on approximate shortest paths in dynamic streams. We refer the reader to the full version [15] for a detailed exposition. One of the most common theoretical models for addressing the challenge of processing massive graphs is the semi-streaming model of computation [1,17,27].…”