scholarly journals Optimal Sequential and Parallel Algorithms for Cut Vertices and Bridges on Trapezoid Graphs

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Hon-Chan Chen
Keyword(s):  
Parallel Algorithms ◽  
Computational Model ◽  
Connected Subgraph ◽  
Number Of Components ◽  
Erew Pram ◽  
Cut Vertex ◽  
International Audience ◽  
Trapezoid Graph ◽  
Trapezoid Graphs

International audience Let G be a graph. A component of G is a maximal connected subgraph in G. A vertex v is a cut vertex of G if k(G-v) > k(G), where k(G) is the number of components in G. Similarly, an edge e is a bridge of G if k(G-e) > k(G). In this paper, we will propose new O(n) algorithms for finding cut vertices and bridges of a trapezoid graph, assuming the trapezoid diagram is given. Our algorithms can be easily parallelized on the EREW PRAM computational model so that cut vertices and bridges can be found in O(log n) time by using O(n / log n) processors.

FAST AND EFFICIENT OPERATIONS ON PARALLEL PRIORITY QUEUES

1996 ◽  
Vol 06 (04) ◽  
pp. 451-467
Author(s):  
DANNY Z. CHEN ◽  
XIAOBO SHARON HU
Keyword(s):  
Data Structure ◽  
Parallel Algorithms ◽  
Computational Model ◽  
Data Structures ◽  
Priority Queue ◽  
Minimal Path ◽  
Erew Pram ◽  
New Ideas ◽  
The Right ◽  
Path Partition

The Parallel Priority Queue (PPQ) data structure supports parallel operations for manipulating data items with keys, such as inserting n new items, deleting n items with the first n smallest keys, creating a new PPQ that contains a set of items, and melding two PPQ’s into one. In this paper, we present fast and efficient parallel algorithms for performing operations on the PPQ’s that maintain data items with real-valued keys. The data structures that we use for implementing the PPQ’s are the unmeldable and meldable parallel heaps. Our algorithms have considerably less time and/or work bounds than the previously best known algorithms, and use a less powerful parallel computational model (EREW PRAM). The new ideas that make our improvement possible are two partition schemes dynamically maintained on the parallel heap structures: the minimal- path partition and the right-path partition. These partition schemes could be of interest in their own right. Our results also lead to optimal parallel algorithms for implementing sequential operations on several commonly-used heap structures.

ON OPTIMAL OROW-PRAM ALGORITHMS FOR COMPUTING RECURSIVELY DEFINED FUNCTIONS

1995 ◽  
Vol 05 (02) ◽  
pp. 299-309
Author(s):  
ROLF NIEDERMEIER ◽  
PETER ROSSMANITH
Keyword(s):  
Parallel Algorithms ◽  
Computational Model ◽  
Random Access ◽  
Erew Pram

We investigate parallel algorithms to compute recursively defined functions. Our computational model are parallel random access machines (PRAM's). We preferably make use of the OROW-PRAM (owner read, owner write), a model supposed to be even weaker and more realistic than the EREW-PRAM (exclusive read, exclusive write) and that still provides the opportunities of a completely connected processor network. For OROW-PRAM's we show that our parallel algorithms are work-optimal.

SUB-LOGARITHMIC ALGORITHMS FOR THE LARGEST EMPTY RECTANGLE PROBLEM

1993 ◽  
Vol 03 (01) ◽  
pp. 79-85
Author(s):  
STEPHAN OLARIU ◽  
WENHUI SHEN ◽  
LARRY WILSON
Keyword(s):  
Parallel Algorithms ◽  
Erew Pram ◽  
Pram Model ◽  
The Common ◽  
Crcw Pram ◽  
Natural Way

We show that the Largest Empty Rectangle problem can be solved by reducing it, in a natural way, to the All Nearest Smaller Values problem. We provide two classes of algorithms: the first one assumes that the input points are available sorted by x (resp. y) coordinate. Our algorithm corresponding to this case runs in O(log log n) time using [Formula: see text] processors in the Common-CRCW-PRAM model. For unsorted input, we present algorithms that run in [Formula: see text] time using [Formula: see text] processors in the Common-CRCW-PRAM, or in O( log n) time using [Formula: see text] processors in the EREW-PRAM model. No sub-logarithmic time parallel algorithms have been previously reported for this problem.

PARALLEL ALGORITHMS FOR SOME DOMINANCE PROBLEMS BASED ON THE PRAM MODEL

1993 ◽  
Vol 03 (04) ◽  
pp. 367-382
Author(s):  
I.W. CHAN ◽  
D.K. FRIESEN
Keyword(s):  
Computational Model ◽  
Single Point ◽  
Random Access ◽  
Geometric Algorithms ◽  
Counting Problem ◽  
Erew Pram ◽  
Pram Model ◽  
Input Size ◽  
Parallel Random Access Machine ◽  
Direct Dominance

Two parallel geometric algorithms based on the idea of point domination are presented. The first algorithm solves the d-dimensional isothetic rectangles intersection counting problem of input size N/2d, where d>1 and N is a multiple of 2d, in O( log d−1 N) time and O(N log N) space. The second algorithm solves the direct dominance reporting problem for a set of N points in the plane in O( log N+J) time and O(N log N) space, where J denotes the maximum of the number of direct dominances reported by any single point in the set. Both algorithms make use of the EREW PRAM (Exclusive Read Exclusive Write Parallel Random Access Machine) consisting of O(N) processors as the computational model.

scholarly journals Simple and Work-Efficient Parallel Algorithms for the Minimum Spanning Tree Problem

1997 ◽  
Vol 07 (01) ◽  
pp. 25-37 ◽  
Author(s):  
Christos D. Zaroliagis
Keyword(s):  
Parallel Algorithms ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Erew Pram ◽  
Minimum Spanning Tree Problem ◽  
Crcw Pram

Two Simple and work-efficient parallel algorithms for the minimum spanning tree problem are presented. Both algorithms perform O(m log n) work. The first algorithm runs in O( log 2 n) time on an EREW PRAM, while the second algorithm runs in O( log n) time on a COMMON CRCW PRAM.

PARALLEL ALGORITHMS FOR FINDING MAXIMAL k-DEPENDENT SETS AND MAXIMAL f-MATCHINGS

1993 ◽  
Vol 04 (02) ◽  
pp. 179-192 ◽  
Author(s):  
KRZYSZTOF DIKS ◽  
OSCAR GARRIDO ◽  
ANDRZEJ LINGAS
Keyword(s):  
Positive Integer ◽  
Parallel Algorithms ◽  
Parallel Algorithm ◽  
Erew Pram

Let k be a positive integer, a subset Q of the set of vertices of a graph G is k-dependent in G if each vertex of Q has no more than k neighbours in Q. We present a parallel algorithm which computes a maximal k-dependent set in a graph on n nodes in time O( log 4 n) on an EREW PRAM with O(n2) processors. In this way, we establish the membership of the problem of constructing a maximal k-dependent set in the class NC. Our algorithm can be easily adapted to compute a maximal k-dependent set in a graph of bounded valence in time O( log * n) using only O(n) EREW PRAM processors. Let f be a positive integer function defined on the set V of vertices of a graph G. A subset F of the set of edges of G is said to be an f-matching if every vertex vɛV is adjacent to at most f(v) edges in-F. We present the first NC algorithm for constructing a maximal f-matching. For a graph on n nodes and m edges the algorithm runs in time O( log 4 n) and uses O(n+m) EREW PRAM processors. For graphs of constantly bounded valence, we can construct a maximal f-matching in O( log * n) time on an EREW PRAM with O(n) processors.

THREE TECHNIQUES FOR PARALLEL MAINTENANCE OF A MINIMUM SPANNING TREE UNDER BATCH OF UPDATES

1996 ◽  
Vol 06 (02) ◽  
pp. 213-222 ◽  
Author(s):  
PAOLO FERRAGINA ◽  
FABRIZIO LUCCIO
Keyword(s):  
Data Structure ◽  
Parallel Algorithms ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Undirected Graph ◽  
Batch Size ◽  
Insertions And Deletions ◽  
Erew Pram ◽  
Pram Model ◽  
Proper Values

In this paper we provide three simple techniques to maintain in parallel the minimum spanning tree of an undirected graph under single or batch of edge updates (i.e., insertions and deletions). Our results extend the use of the sparsification data structure to the EREW PRAM model. For proper values of the batch size, our algorithms require less time and work than the best known dynamic parallel algorithms.

DIAMONDS ARE NOT A MINIMUM WEIGHT TRIANGULATION'S BEST FRIEND

2002 ◽  
Vol 12 (06) ◽  
pp. 445-453 ◽  
Author(s):  
PROSENJIT BOSE ◽  
LUC DEVROYE ◽  
WILLIAM EVANS
Keyword(s):  
Polynomial Time ◽  
Minimum Weight ◽  
Connected Subgraph ◽  
Constant Number ◽  
Point Sets ◽  
Expected Number ◽  
Number Of Components ◽  
Best Friend ◽  
Point Set ◽  
Minimum Weight Triangulation

Two recent methods have increased hopes of finding a polynomial time solution to the problem of computing the minimum weight triangulation of a set S of n points in the plane. Both involve computing what was believed to be a connected or nearly connected subgraph of the minimum weight triangulation, and then completing the triangulation optimally. The first method uses the light graph of S as its initial subgraph. The second method uses the LMT-skeleton of S. Both methods rely, for their polynomial time bound, on the initial subgraphs having only a constant number of components. Experiments performed by the authors of these methods seemed to confirm that randomly chosen point sets displayed this desired property. We show that there exist point sets where the number of components is linear in n. In fact, the expected number of components in either graph on a randomly chosen point set is linear in n, and the probability of the number of components exceeding some constant times n tends to one.

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