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An externalmemory data structure for shortest path queries
 DIPLOMARBEIT, FRIEDRICHSCHILLERUNIVERSITIT JENA, NOV,1998
, 1998
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I/OEfficient Planar Separators and Applications
, 2001
"... We present a new algorithm to compute a subset S of vertices of a planar graph G whose removal partitions G into O(N/h) subgraphs of size O(h) and with boundary size O( p h) each. The size of S is O(N= p h). Computing S takes O(sort(N)) I/Os and linear space, provided that M 56hlog² B. Together with ..."
Abstract

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We present a new algorithm to compute a subset S of vertices of a planar graph G whose removal partitions G into O(N/h) subgraphs of size O(h) and with boundary size O( p h) each. The size of S is O(N= p h). Computing S takes O(sort(N)) I/Os and linear space, provided that M 56hlog² B. Together with recent reducibility results, this leads to O(sort(N)) I/O algorithms for breadthfirst search (BFS), depthfirst search (DFS), and single source shortest paths (SSSP) on undirected embedded planar graphs. Our separator algorithm does not need a BFS tree or an embedding of G to be given as part of the input. Instead we argue that "local embeddings" of subgraphs of G are enough.
I/Oefficient algorithms for planar graphs I: Separators
"... We present I/Oefficient algorithms for computing optimal separator partitions of planar graphs. Our main result shows that, given a planar graph G with N vertices and an integer r> 0, a vertex separator of size O (N / √ r) that partitions G into O(N/r) subgraphs of size at most r and boundary s ..."
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We present I/Oefficient algorithms for computing optimal separator partitions of planar graphs. Our main result shows that, given a planar graph G with N vertices and an integer r> 0, a vertex separator of size O (N / √ r) that partitions G into O(N/r) subgraphs of size at most r and boundary size O ( √ r) can be computed in O(sort(N)) I/Os, provided that M ≥ 56r log 2 B. Together with the planar embedding algorithm presented in the companion paper [27], this result is the basis for I/Oefficient solutions to many other fundamental problems on planar graphs, including breadthfirst search and shortest paths [5, 8], depthfirst search [6, 9], strong connectivity [9], and topological sorting [8]. Our second result shows that, given I/Oefficient solutions to these problems, a general separator algorithm for graphs with costs and weights on their vertices [3] can be made I/Oefficient. Many classical separator theorems are special cases of this result. In particular, our I/Oefficient version allows the computation of a separator as produced by our first separator algorithm, but without placing any constraints on r.