131,585 research outputs found

    Distances and Isomorphism between Networks and the Stability of Network Invariants

    Full text link
    We develop the theoretical foundations of a network distance that has recently been applied to various subfields of topological data analysis, namely persistent homology and hierarchical clustering. While this network distance has previously appeared in the context of finite networks, we extend the setting to that of compact networks. The main challenge in this new setting is the lack of an easy notion of sampling from compact networks; we solve this problem in the process of obtaining our results. The generality of our setting means that we automatically establish results for exotic objects such as directed metric spaces and Finsler manifolds. We identify readily computable network invariants and establish their quantitative stability under this network distance. We also discuss the computational complexity involved in precisely computing this distance, and develop easily-computable lower bounds by using the identified invariants. By constructing a wide range of explicit examples, we show that these lower bounds are effective in distinguishing between networks. Finally, we provide a simple algorithm that computes a lower bound on the distance between two networks in polynomial time and illustrate our metric and invariant constructions on a database of random networks and a database of simulated hippocampal networks

    Route Planning in Transportation Networks

    Full text link
    We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4, previously published by Microsoft Research. This work was mostly done while the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at Microsoft Research Silicon Valle

    Implications of Selfish Neighbor Selection in Overlay Networks

    Full text link
    In a typical overlay network for routing or content sharing, each node must select a fixed number of immediate overlay neighbors for routing traffic or content queries. A selfish node entering such a network would select neighbors so as to minimize the weighted sum of expected access costs to all its destinations. Previous work on selfish neighbor selection has built intuition with simple models where edges are undirected, access costs are modeled by hop-counts, and nodes have potentially unbounded degrees. However, in practice, important constraints not captured by these models lead to richer games with substantively and fundamentally different outcomes. Our work models neighbor selection as a game involving directed links, constraints on the number of allowed neighbors, and costs reflecting both network latency and node preference. We express a node's "best response" wiring strategy as a k-median problem on asymmetric distance, and use this formulation to obtain pure Nash equilibria. We experimentally examine the properties of such stable wirings on synthetic topologies, as well as on real topologies and maps constructed from PlanetLab and AS-level Internet measurements. Our results indicate that selfish nodes can reap substantial performance benefits when connecting to overlay networks composed of non-selfish nodes. On the other hand, in overlays that are dominated by selfish nodes, the resulting stable wirings are optimized to such great extent that even non-selfish newcomers can extract near-optimal performance through naive wiring strategies.Marie Curie Outgoing International Fellowship of the EU (MOIF-CT-2005-007230); National Science Foundation (CNS Cybertrust 0524477, CNS NeTS 0520166, CNS ITR 0205294, EIA RI 020206

    Exact Distance Oracles for Planar Graphs

    Full text link
    We present new and improved data structures that answer exact node-to-node distance queries in planar graphs. Such data structures are also known as distance oracles. For any directed planar graph on n nodes with non-negative lengths we obtain the following: * Given a desired space allocation S[nlglgn,n2]S\in[n\lg\lg n,n^2], we show how to construct in O~(S)\tilde O(S) time a data structure of size O(S)O(S) that answers distance queries in O~(n/S)\tilde O(n/\sqrt S) time per query. As a consequence, we obtain an improvement over the fastest algorithm for k-many distances in planar graphs whenever k[n,n)k\in[\sqrt n,n). * We provide a linear-space exact distance oracle for planar graphs with query time O(n1/2+eps)O(n^{1/2+eps}) for any constant eps>0. This is the first such data structure with provable sublinear query time. * For edge lengths at least one, we provide an exact distance oracle of space O~(n)\tilde O(n) such that for any pair of nodes at distance D the query time is O~(minD,n)\tilde O(min {D,\sqrt n}). Comparable query performance had been observed experimentally but has never been explained theoretically. Our data structures are based on the following new tool: given a non-self-crossing cycle C with c=O(n)c = O(\sqrt n) nodes, we can preprocess G in O~(n)\tilde O(n) time to produce a data structure of size O(nlglgc)O(n \lg\lg c) that can answer the following queries in O~(c)\tilde O(c) time: for a query node u, output the distance from u to all the nodes of C. This data structure builds on and extends a related data structure of Klein (SODA'05), which reports distances to the boundary of a face, rather than a cycle. The best distance oracles for planar graphs until the current work are due to Cabello (SODA'06), Djidjev (WG'96), and Fakcharoenphol and Rao (FOCS'01). For σ(1,4/3)\sigma\in(1,4/3) and space S=nσS=n^\sigma, we essentially improve the query time from n2/Sn^2/S to n2/S\sqrt{n^2/S}.Comment: To appear in the proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms, SODA 201
    corecore