1,244 research outputs found

    Faster Path Queries in Colored Trees via Sparse Matrix Multiplication and Min-Plus Product

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    Route Planning in Transportation Networks

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    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

    Triangles and Girth in Disk Graphs and Transmission Graphs

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    Let S subset R^2 be a set of n sites, where each s in S has an associated radius r_s > 0. The disk graph D(S) is the undirected graph with vertex set S and an undirected edge between two sites s, t in S if and only if |st| <= r_s + r_t, i.e., if the disks with centers s and t and respective radii r_s and r_t intersect. Disk graphs are used to model sensor networks. Similarly, the transmission graph T(S) is the directed graph with vertex set S and a directed edge from a site s to a site t if and only if |st| <= r_s, i.e., if t lies in the disk with center s and radius r_s. We provide algorithms for detecting (directed) triangles and, more generally, computing the length of a shortest cycle (the girth) in D(S) and in T(S). These problems are notoriously hard in general, but better solutions exist for special graph classes such as planar graphs. We obtain similarly efficient results for disk graphs and for transmission graphs. More precisely, we show that a shortest (Euclidean) triangle in D(S) and in T(S) can be found in O(n log n) expected time, and that the (weighted) girth of D(S) can be found in O(n log n) expected time. For this, we develop new tools for batched range searching that may be of independent interest

    An Improved Algorithm for Incremental DFS Tree in Undirected Graphs

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    Depth first search (DFS) tree is one of the most well-known data structures for designing efficient graph algorithms. Given an undirected graph G=(V,E)G=(V,E) with nn vertices and mm edges, the textbook algorithm takes O(n+m)O(n+m) time to construct a DFS tree. In this paper, we study the problem of maintaining a DFS tree when the graph is undergoing incremental updates. Formally, we show: Given an arbitrary online sequence of edge or vertex insertions, there is an algorithm that reports a DFS tree in O(n)O(n) worst case time per operation, and requires O(min{mlogn,n2})O\left(\min\{m \log n, n^2\}\right) preprocessing time. Our result improves the previous O(nlog3n)O(n \log^3 n) worst case update time algorithm by Baswana et al. and the O(nlogn)O(n \log n) time by Nakamura and Sadakane, and matches the trivial Ω(n)\Omega(n) lower bound when it is required to explicitly output a DFS tree. Our result builds on the framework introduced in the breakthrough work by Baswana et al., together with a novel use of a tree-partition lemma by Duan and Zhan, and the celebrated fractional cascading technique by Chazelle and Guibas

    Social Influences in Recommendation Systems

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    Social networking sites such as Flickr and Facebook allow users to share content with family, friends, and interest groups. Also, tags can often assign to resources. In the previous research using few association rules FAR, we have seen that high-quality and efficient association-based tag recommendation is possible, but the set-up that we considered was very generic and did not take social information into account. The proposed method in the previous paper, FAR, in particular, exhibited a favorable trade-off between recommendation quality and runtime. Unfortunately, recommendation quality is unlikely to be optimal because the algorithms are not aware of any social information that may be available. Two proposed approaches take a more social view on tag recommendation regarding the issue: social contact variants and social groups of interest. The user data is varied and used as a source of associations. The adoption of social contact variants has two approaches. The first social variant is User-centered Knowledge, to contrast Collective Knowledge. It improves tag recommendation by grouping historic tag data according to friend relationships and interests. The second variant is dubbed 'social batched personomy' and attempts to address both quality and scalability issues by processing queries in batches instead of individually, such as done in a conventional personomy approach. For the social group of interest, 'community batched personomy' is proposed to provide better accuracy groups of recommendation systems in contrast also to Collective Knowledge. By taking social information into account can enhance the performance of recommendation systems.Comment: 6 page
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