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Linear Algorithms for Partitioning Embedded Graphs of Bounded Genus
 SIAM Journal of Discrete Mathematics
, 1996
"... This paper develops new techniques for constructing separators for graphs embedded on surfaces of bounded genus. For any arbitrarily small positive " we show that any nvertex graph G of genus g can be divided in O(n + g) time into components whose sizes do not exceed "n by removing a set C of O( ..."
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Cited by 23 (5 self)
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This paper develops new techniques for constructing separators for graphs embedded on surfaces of bounded genus. For any arbitrarily small positive " we show that any nvertex graph G of genus g can be divided in O(n + g) time into components whose sizes do not exceed "n by removing a set C of O( p (g + 1=")n) vertices. Our result improves the best previous ones with respect to the size of C and the time complexity of the algorithm. Moreover, we show that one can cut off from G a piece of no more than (1 \Gamma ")n vertices by removing a set of O( p n"(g" + 1) vertices. Both results are optimal up to a constant factor. Keywords: graph separator, graph genus, algorithm, divideandconquer, topological graph theory AMS(MOS) subject classifications: 05C10, 05C85, 68R10 1 Bulgarian Academy of Sci., CICT, G.Bonchev 25A, 1113 Sofia, Bulgaria 2 Department of Comp.Sci.,Rice University, P.O.Box 1892, Houston, Texas 77251, USA 1 Introduction Let S be a class of graphs closed under t...
Towards overcoming the transitiveclosure bottleneck: efficient parallel algorithms for planar digraphs
 J. Comput. System Sci
, 1993
"... Abstract. Currently, there is a significant gap between the best sequential and parallel complexities of many fundamental problems related to digraph reachability. This complexity bottleneck essentially reflects a seemingly unavoidable reliance on transitive closure techniques in parallel algorithms ..."
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Cited by 11 (1 self)
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Abstract. Currently, there is a significant gap between the best sequential and parallel complexities of many fundamental problems related to digraph reachability. This complexity bottleneck essentially reflects a seemingly unavoidable reliance on transitive closure techniques in parallel algorithms for digraph reachability. To pinpoint the nature of the bottleneck, we de* velop a collection of polylogtime reductions among reachability problems. These reductions use only linear processors and work for general graphs. Furthermore, for planar digraphs, we give polylogtime algorithms for the following problems: (1) directed ear decomposition, (2) topological ordering, (3) digraph reachability, (4) descendent counting, and (5) depthfirst search. These algorithms use only linear processors and therefore reduce the complexity to within a polylog factor of optimal.
A Model Classifying Algorithms as Inherently Sequential with Applications to Graph Searching
, 1992
"... A model is proposed that can be used to classify algorithms as inherently sequential. The model captures the internal computations of algorithms. Previous work in complexity theory has focused on the solutions algorithms compute. Direct comparison of algorithms within the framework of the model is ..."
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Cited by 7 (3 self)
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A model is proposed that can be used to classify algorithms as inherently sequential. The model captures the internal computations of algorithms. Previous work in complexity theory has focused on the solutions algorithms compute. Direct comparison of algorithms within the framework of the model is possible. The model is useful for identifying hard to parallelize constructs that should be avoided by parallel programmers. The model's utility is demonstrated via applications to graph searching. A stack breadthfirst search (BFS) algorithm is analyzed and proved inherently sequential. The proof technique used in the reduction is a new one. The result for stack BFS sharply contrasts a result showing that a queue based BFS algorithm is in NC. An NC algorithm to compute greedy depthfirst search numbers in a dag is presented, and a result proving that a combination search strategy called breadthdepth search is inherently sequential is also given.
Planar Strong Connectivity Helps in Parallel DepthFirst Search
 SIAM Journal on Computing
, 1992
"... . This paper shows that for a strongly connected planar directed graph of size n, a depthfirst search tree rooted a specified vertex can be computed in O(log 5 n) time using n= log n processors. Previously, for planar directed graphs that may not be strongly connected, the best depthfirst searc ..."
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Cited by 3 (0 self)
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. This paper shows that for a strongly connected planar directed graph of size n, a depthfirst search tree rooted a specified vertex can be computed in O(log 5 n) time using n= log n processors. Previously, for planar directed graphs that may not be strongly connected, the best depthfirst search algorithm runs in O(log 10 n) time using n processors. Both algorithms run on a parallel random access machine that allows concurrent reads and concurrent writes in its shared memory, and in case of a write conflict, permits an arbitrary processor to succeed. Key words. linearprocessor NC algorithms, graph separators, depthfirst search, planar directed graphs, strong connectivity, bubble graphs, st graphs AMS(MOS) subject classification. 68Q10, 05C99 1. Introduction. Depthfirst search is one of the most useful tools in graph theory [32], [4]. The depthfirst search problem is the following: given a graph and a distinguished vertex, construct a tree that corresponds to performing de...
Efficient Parallel Algorithms for Planar stGraphs 1
, 2002
"... Abstract. Planar stgraphs find applications in a number of areas. In this paper we present efficient parallel algorithms for solving several fundamental problems on planar stgraphs. The problems we consider include allpairs shortest paths in weighted planar stgraphs, singlesource shortest paths ..."
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Cited by 1 (0 self)
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Abstract. Planar stgraphs find applications in a number of areas. In this paper we present efficient parallel algorithms for solving several fundamental problems on planar stgraphs. The problems we consider include allpairs shortest paths in weighted planar stgraphs, singlesource shortest paths in weighted planar layered digraphs (which can be reduced to singlesource shortest paths in certain special planar stgraphs), and depthfirst search in planar stgraphs. Our parallel shortest path techniques exploit the specific geometric and graphic structures of planar stgraphs, and involve schemes for partitioning planar stgraphs into subgraphs in a way that ensures that the resulting path length matrices have a monotonicity property [1], [2]. The parallel algorithms we obtain are a considerable improvement over the previously best known solutions (when they are applied to these stgraph problems), and are in fact relatively simple. The parallel computational models we use are the CREW PRAM and EREW PRAM. Key Words.