6 research outputs found
Planar Reachability in Linear Space and Constant Time
We show how to represent a planar digraph in linear space so that distance
queries can be answered in constant time. The data structure can be constructed
in linear time. This representation of reachability is thus optimal in both
time and space, and has optimal construction time. The previous best solution
used space for constant query time [Thorup FOCS'01].Comment: 20 pages, 5 figures, submitted to FoC
On-line and Dynamic Shortest Paths through Graph Decompositions (Preliminary Version)
We describe algorithms for finding shortest paths and distances in a planar digraph which exploit the particular topology of the input graph. We give both sequential and parallel algorithms that work on a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. For outerplanar digraphs, for instance, the data structures can be updated after any such change in only time, where is the number of vertices of the digraph. The parallel algorithms presented here are the first known ones for solving this problem. Our results can be extended to hold for digraphs of genus
On-line and dynamic algorithms for shortest path problems
We describe algorithms for finding shortest paths and distances in a planar digraph which exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. For outerplanar digraphs, for instance, the data structures can be updated after any such change in only time, where is the number of vertices of the digraph. We also describe the first parallel algorithms for solving the dynamic version of the shortest path problem. Our results can be extended to hold for digraphs of genus
Hammock-on-ears decomposition: a technique for the efficient parallel solution of shortest paths and other problems
We show how to decompose efficiently in parallel {\em any} graph into a number, , of outerplanar subgraphs (called {\em hammocks}) satisfying certain separator properties. Our work combines and extends the sequential hammock decomposition technique introduced by G.~Frederickson and the parallel ear decomposition technique, thus we call it the {\em hammock-on-ears decomposition}. We mention that hammock-on-ears decomposition also draws from techniques in computational geometry and that an embedding of the graph does not need to be provided with the input. We achieve this decomposition in time using CREW PRAM processors, for an -vertex, -edge graph or digraph. The hammock-on-ears decomposition implies a general framework for solving graph problems efficiently. Its value is demonstrated by a variety of applications on a significant class of (di)graphs, namely that of {\em sparse (di)graphs}. This class consists of all (di)graphs which have a between and , and includes planar graphs and graphs with genus . We improve previous bounds for certain instances of shortest paths and related problems, in this class of graphs. These problems include all pairs shortest paths, all pairs reachability
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学位の種別: 課程博士審査委員会委員 : (主査)国立情報学研究所教授 佐藤 真一, 東京大学教授 佐藤 洋一, 東京大学教授 相澤 清晴, 東京大学准教授 山崎 俊彦, 東京大学准教授 大石 岳史University of Tokyo(東京大学