Decremental All-Pairs ALL Shortest Paths and Betweenness Centrality

Abstract

Abstract. We consider the all pairs all shortest paths (APASP) problem, which maintains the shortest path dag rooted at every vertex in a directed graph G = (V, E) with positive edge weights. For this problem we present a decremental algorithm (that supports the deletion of a vertex, or weight increases on edges incident to a vertex). Our algorithm runs in amortized O(ν * 2 · log n) time per update, where n = |V |, and ν * bounds the number of edges that lie on shortest paths through any given vertex. Our APASP algorithm can be used for the decremental computation of betweenness centrality (BC), a graph parameter that is widely used in the analysis of large complex networks. No nontrivial decremental algorithm for either problem was known prior to our work. Our method is a generalization of the decremental algorithm of Demetrescu and Italiano [3] for unique shortest paths, and for graphs with ν * = O(n), we match the bound in [3]. Thus for graphs with a constant number of shortest paths between any pair of vertices, our algorithm maintains APASP and BC scores in amortized time O(n 2 · log n) under decremental updates, regardless of the number of edges in the graph.

Parallel Algorithm for Incremental Betweenness Centrality on Large Graphs

“…Kas' algorithm was further optimized by Wei et al [25]. Many other approaches also store the intermediate all-pairs shortest path information in various ways [26], [27], [28], [29]. Kourtellis et al [11] extend Green's algorithm [8] and deploy a Hadoop implementation on a large cluster to support large graphs.…”

Fully Dynamic Betweenness Centrality

“…NPRdec extends this result to APASP by introducing the tuple-system to replace the need to maintain every SP and LSP in DI. We now briefly review this system, referring the reader to [19] for more details. Let d(x, y) denote the shortest path length from x to y.…”

“…We will use components from PR such as the abstract representation of the level system using PDGs (see Section 4) and the flag bit β for a triple, the Marked-Tuples scheme introduced in NPRdec (see [19,21] for more details), and the maintenance of level graphs from Thorup. In this section, we describe all data structures used by our algorithm. In Table 2 we summarize the structures we use, including those inherited from [19,21,28]. The new components we introduce in this paper to achieve efficiency for fully dynamic APASP, are described in section 4.2 and listed in Table 2, Part D.…”

“…ff-update. As in [8,19,21], the update of a node occurs in a sequence of two steps: a cleanup phase and a fixup phase. Here, we call this update ff-update and it is a sequence of 2 calls: ff-cleanup and ff-fixup.…”

“…Heuristics for dynamic betweenness centrality with good experimental performance are given in [10,16,26], but none provably improve on Brandes. The only earlier exact dynamic BC algorithms that provably improve on Brandes on some classes of graphs are the recent separate incremental and decremental 1 algorithms in [18,19]. Recently, we give in [21] (see also [22]) the first fully dynamic algorithm for BC (the PR method) that is provably faster than Brandes for the class of dense graphs (where m is close to n 2 ) with succinct single-source SP dags.…”