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Simplified external memory algorithms for planar DAGs
 Proc. 9th SWAT, LNCS 3111
, 2004
"... Abstract. In recent years a large number I/Oefficient algorithms have been developed for fundamental planar graph problems. Most of these algorithms rely on the existence of small planar separators as well as an O(sort(N)) I/O algorithm for computing a partition of a planar graph based on such sepa ..."
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Abstract. In recent years a large number I/Oefficient algorithms have been developed for fundamental planar graph problems. Most of these algorithms rely on the existence of small planar separators as well as an O(sort(N)) I/O algorithm for computing a partition of a planar graph based on such separators, where O(sort(N)) is the number of I/Os needed to sort N elements. In this paper we simplify and unify several of the known planar graph results by developing linear I/O algorithms for the fundamental singlesource shortest path, breadthfirst search and topological sorting problems on planar directed acyclic graphs, provided that a partition is given; thus our results give O(sort(N)) I/Os algorithms for the three problems. While algorithms for all these problems were already known, the previous algorithms are all considerably more complicated than our algorithms and use Θ(sort(N)) I/Os even if a partition is known. Unlike the previous algorithm, our topological sorting algorithm is simple enough to be of practical interest. 1
Cacheoblivious planar shortest paths
 In Proc. 32nd International Colloquium on Automata, Languages, and Programming. LNCS
, 2005
"... Abstract. We present an efficient cacheoblivious implementation of the shortestpath algorithm for planar graphs by Klein et al., and prove that it incurs no more than O ` N B1/2−ɛ + N B log N ´ block transfers on a graph with N vertices. This is the first cacheoblivious algorithm for this problem ..."
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Abstract. We present an efficient cacheoblivious implementation of the shortestpath algorithm for planar graphs by Klein et al., and prove that it incurs no more than O ` N B1/2−ɛ + N B log N ´ block transfers on a graph with N vertices. This is the first cacheoblivious algorithm for this problem that incurs o(N) block transfers. 1
External Data Structures for Shortest Path Queries on Planar Digraphs
"... Abstract. In this paper we present spacequery tradeoffs for external memory data structures that answer shortest path queries on planar directed graphs. For any S = Ω(N 1+ɛ)andS = O(N 2 /B), our main result is a family of structures that use S space and answer queries in O ( N2 SB) I/Os, thus obta ..."
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Abstract. In this paper we present spacequery tradeoffs for external memory data structures that answer shortest path queries on planar directed graphs. For any S = Ω(N 1+ɛ)andS = O(N 2 /B), our main result is a family of structures that use S space and answer queries in O ( N2 SB) I/Os, thus obtaining optimal spacequery product O(N2 /B). An S space structure can be constructed in O ( √ S · sort(N)) I/Os, where sort(N) is the number of I/Os needed to sort N elements, B is the disk block size, and N is the size of the graph. 1
I/OOptimal Algorithms for Outerplanar Graphs
, 2001
"... We present linearI/O algorithms for fundamental graph problems on embedded outerplanar graphs. We show that breadthfirst search, depthfirst search, singlesource shortest paths, triangulation, and computing an ɛseparator of size O(1/ɛ) takeO(scan(N)) I/Os on embedded outerplanar graphs. We also s ..."
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We present linearI/O algorithms for fundamental graph problems on embedded outerplanar graphs. We show that breadthfirst search, depthfirst search, singlesource shortest paths, triangulation, and computing an ɛseparator of size O(1/ɛ) takeO(scan(N)) I/Os on embedded outerplanar graphs. We also show that it takes O(sort(N)) I/Os to test whether a given graph is outerplanar and to compute an outerplanar embedding of an outerplanar graph, thereby providing O(sort(N))I/O algorithms for the above problems if no embedding of the graph is given. As all these problems have lineartime algorithms in internal memory, a simple simulation technique can be used to improve the I/Ocomplexity of our algorithms from O(sort(N)) to O(perm(N)). We prove matching lower bounds for embedding, breadthfirst search, depthfirst search, and singlesource shortest paths if no embedding is given. Our algorithms for the above problems use a simple linearI/O timeforward processing algorithm for rooted trees whose vertices are stored in preorder.
I/OEfficient Algorithms on NearPlanar Graphs
, 2011
"... Obtaining I/Oefficient algorithms for basic graph problems on sparse directed graphs has been a longstanding open problem. The best known algorithms for most basic problems on such graphs still require Ω(V) I/Os in the worst case, where V is the number of vertices in the graph. Nevertheless optima ..."
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Obtaining I/Oefficient algorithms for basic graph problems on sparse directed graphs has been a longstanding open problem. The best known algorithms for most basic problems on such graphs still require Ω(V) I/Os in the worst case, where V is the number of vertices in the graph. Nevertheless optimal O(sort(V)) I/O algorithms are known for special classes of sparse graphs, like planar graphs and grid graphs. It is hard to accept that a problem becomes difficult as soon as the graph contains a few deviations from planarity. In this paper we extend the class of graphs on which basic graph problems can be solved I/Oefficiently. We discuss several ways to transform graphs that are almost planar into planar graphs (given a suitable drawing), and based on those transformations we obtain the first I/Oefficient algorithms for directed graphs that are almost planar. Let G be a directed graph that is given as a planar subgraph (V, E) and a set of additional edges EC. Our main result is a singlesourceshortestpaths algorithm that runs in O(EC + sort(V + EC)) I/Os. When EC is small our algorithm is a significant improvement over the best previously known algorithms, which required Ω(V) I/Os. Alternatively, when G is given with a drawing with T crossings, we can compute singlesource shortest paths in O(sort(V + T)) I/Os. We obtain similar bounds for computing (strongly) connected components, breadthfirst and depthfirst traversals and topological ordering. Submitted:
Abstract
"... 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. 1