Fast and Simple Connectivity in Graph Timelines

Abstract

In this paper we study the problem of answering connectivity queries about a graph timeline. A graph timeline is a sequence of undirected graphs $G_1,\ldots ,G_t$ on a common set of vertices of size n such that each graph is obtained from the previous one by an addition or a deletion of a single edge. We present data structures, which preprocess the timeline and can answer the following queries:

• $\mathtt {forall}(u,v,a,b)$ – does the path $u\rightarrow v$ exist in each of $G_a,\ldots ,G_b$?

• $\mathtt {exists}(u,v,a,b)$ – does the path $u\rightarrow v$ exist in any of $G_a,\ldots ,G_b$?

• $\mathtt {forall2}(u,v,a,b)$ – do there exist two edge-disjoint paths connecting u and v in each of $G_a,\ldots ,G_b$?

We show data structures that can answer $\mathtt {forall}$ and $\mathtt {forall2}$ queries in $O(\log n)$ time after preprocessing in $O(m+t\log n)$ time. Here by m we denote the number of edges that remain unchanged in each graph of the timeline. For the case of $\mathtt {exists}$ queries, we show how to extend an existing data structure to obtain a preprocessing/query trade-off of $\langle O(m+\min (nt, t^{2-\alpha })), O(t^\alpha )\rangle$ and show a matching conditional lower bound.

A. Karczmarz—Supported by the grant NCN2014/13/B/ST6/01811 of the Polish Science Center. Partially supported by FET IP project MULTIPLEX 317532.

J. Łącki —Jakub Łącki is a recipient of the Google Europe Fellowship in Graph Algorithms, and this research is supported in part by this Google Fellowship.