While a natural fit for modeling and understanding mobile networks, time-varying graphs remain poorly understood. Indeed, many of the usual concepts of static graphs have no obvious counterpart in time-varying ones. In this paper, we introduce the notion of temporal reachability graphs. A (tau,delta)-reachability graph is a time-varying directed graph derived from an existing connectivity graph. An edge exists from one node to another in the reachability graph at time t if there exists a journey (i.e., a spatiotemporal path) in the connectivity graph from the first node to the second, leaving after t, with a positive edge traversal time tau, and arriving within a maximum delay delta. We make three contributions. First, we develop the theoretical framework around temporal reachability graphs. Second, we harness our theoretical findings to propose an algorithm for their efficient computation. Finally, we demonstrate the analytic power of the temporal reachability graph concept by applying it to synthetic and real-life datasets. On top of defining clear upper bounds on communication capabilities, reachability graphs highlight asymmetric communication opportunities and offloading potential.
Adaptations of traditional static graph distance metrics and algorithms to time-varying graphs have yielded many different concepts. For example, Orda propose a shortest path algorithm for TVGs based on different waiting policies (unrestricted, forbidden, and source waiting) @cite_10 . Our work corresponds to the policy, in which a message may wait for an unlimited amount of time anywhere along its path through the TVG. Bui Xuan have proposed efficient algorithms for calculating shortest (in number of hops), fastest (in path traversal time), and foremost (i.e., earliest arrival) paths in TVGs @cite_1 . All these algorithms are designed to compute the shortest paths to all destination from a source and a fixed starting time. Our algorithms, in contrast, computes the reachibility graph by estimating the shortest paths for all possible starting times.
@cite_1 @cite_10
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The decremental single-source shortest paths (SSSP) problem concerns maintaining the distances between a given source node s to every node in an n-node m-edge graph G undergoing edge deletions. While its static counterpart can be easily solved in near-linear time, this decremental problem is much more challenging even in the undirected unweighted case. In this case, the classic O(mn) total update time of Even and Shiloach (JACM 1981) has been the fastest known algorithm for three decades. With the loss of a (1 + aepsi;)-approximation factor, the running time was recently improved to O(n2+o(1)) by Bernstein and Roditty (SODA 2011), and more recently to O(n1.8+o(1) + m1+o(1)) by Henzinger, Krinninger, and Nanongkai (SODA 2014). In this paper, we finally bring the running time of this case down to near-linear: We give a (1 + aepsi;)-approximation algorithm with O(m1+o(1)) total update time, thus obtaining near-linear time. Moreover, we obtain O(m1+o(1)log W) time for the weighted case, where the edge weights are integers from 1 to W. The only prior work on weighted graphs in o(mn log W) time is the O(mn0.986 log W)-time algorithm by Henzinger, Krinninger, and Nanongkai (STOC 2014) which works for the general weighted directed case. In contrast to the previous results which rely on maintaining a sparse emulator, our algorithm relies on maintaining a so-called sparse (d, aepsi;)-hop set introduced by Cohen (JACM 2000) in the PRAM literature. A (d, aepsi;)-hop set of a graph G = (V, E) is a set E' of weighted edges such that the distance between any pair of nodes in G can be (1 + aepsi;)-approximated by their d-hop distance (given by a path containing at most d edges) on G'=(V, E a#x222A; E'). Our algorithm can maintain an (no(1), aepsi;)-hop set of near-linear size in near-linear time under edge deletions. It is the first of its kind to the best of our knowledge. To maintain the distances on this hop set, we develop a monotone bounded-hop Even-Shiloach tree. It results from extending and combining the monotone Even-Shiloach tree of Henzinger, Krinninger, and Nanongkai (FOCS 2013) with the bounded-hop SSSP technique of Bernstein (STOC 2013). These two new tools might be of independent interest. We present a deterministic (1+o(1))-approximation O(n1 2+o(1)+D1+o(1))-time algorithm for solving the single-source shortest paths problem on distributed weighted networks (the CONGEST model); here n is the number of nodes in the network and D is its (hop) diameter. This is the first non-trivial deterministic algorithm for this problem. It also improves (i) the running time of the randomized (1+o(1))-approximation O(n1 2D1 4+D)-time algorithm of Nanongkai [STOC 2014] by a factor of as large as n1 8, and (ii) the O(є−1logє−1)-approximation factor of Lenzen and Patt-Shamir’s O(n1 2+є+D)-time algorithm [STOC 2013] within the same running time. Our running time matches the known time lower bound of Ω(n1 2 logn + D) [Das STOC 2011] modulo some lower-order terms, thus essentially settling the status of this problem which was raised at least a decade ago [Elkin SIGACT News 2004]. It also implies a (2+o(1))-approximation O(n1 2+o(1)+D1+o(1))-time algorithm for approximating a network’s weighted diameter which almost matches the lower bound by [PODC 2012]. In achieving this result, we develop two techniques which might be of independent interest and useful in other settings: (i) a deterministic process that replaces the “hitting set argument” commonly used for shortest paths computation in various settings, and (ii) a simple, deterministic, construction of an (no(1), o(1))-hop set of size O(n1+o(1)). We combine these techniques with many distributed algorithmic techniques, some of which from problems that are not directly related to shortest paths, e.g. ruling sets [ STOC 1987], source detection [Lenzen, Peleg PODC 2013], and partial distance estimation [Lenzen, Patt-Shamir PODC 2015]. Our hop set construction also leads to single-source shortest paths algorithms in two other settings: (i) a (1+o(1))-approximation O(no(1))-time algorithm on congested cliques, and (ii) a (1+o(1))-approximation O(no(1)logW)-pass O(n1+o(1)logW)-space streaming algorithm, when edge weights are in 1, 2, …, W . The first result answers an open problem in [Nanongkai, STOC 2014]. The second result partially answers an open problem raised by McGregor in 2006 [ sublinear.info , Problem 14]. While a natural fit for modeling and understanding mobile networks, time-varying graphs remain poorly understood. Indeed, many of the usual concepts of static graphs have no obvious counterpart in time-varying ones. In this paper, we introduce the notion of temporal reachability graphs. A (tau,delta)-reachability graph is a time-varying directed graph derived from an existing connectivity graph. An edge exists from one node to another in the reachability graph at time t if there exists a journey (i.e., a spatiotemporal path) in the connectivity graph from the first node to the second, leaving after t, with a positive edge traversal time tau, and arriving within a maximum delay delta. We make three contributions. First, we develop the theoretical framework around temporal reachability graphs. Second, we harness our theoretical findings to propose an algorithm for their efficient computation. Finally, we demonstrate the analytic power of the temporal reachability graph concept by applying it to synthetic and real-life datasets. On top of defining clear upper bounds on communication capabilities, reachability graphs highlight asymmetric communication opportunities and offloading potential. We obtain the following results related to dynamic versions of the shortest-paths problem: Reductions that show that the incremental and decremental single-source shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest-paths problem. We also obtain slightly weaker results for the corresponding unweighted problems. A randomized fully-dynamic algorithm for the all-pairs shortest-paths problem in directed unweighted graphs with an amortized update time of @math (we use @math to hide small poly-logarithmic factors) and a worst case query time is O(n3 4). A deterministic O(n2log n) time algorithm for constructing an O(log n)-spanner with O(n) edges for any weighted undirected graph on n vertices. The algorithm uses a simple algorithm for incrementally maintaining single-source shortest-paths tree up to a given distance.