AN Ω(log n−k log k) TIME LINEAR COST LOWER BOUND FOR THE k FUNCTIONS COARSEST PARTITION PROBLEM

1996 ◽  
Vol 06 (02) ◽  
pp. 195-202
Author(s):  
CLIVE N. GALLEY
Keyword(s):  
Lower Bound ◽  
Partition Problem ◽  
Model Of Computation ◽  
Pram Model ◽  
Linear Cost ◽  
Time Linear ◽  
Crcw Pram

We consider the k functions coarsest partition problem for a set S, where |S|=n, and k functions from S to S. We present an Ω(log n−k log k) time, linear work, lower bound for this problem on the CRCW PRAM model of computation.

A SIMPLE PARALLEL ALGORITHM FOR THE SINGLE FUNCTION COARSEST PARTITION PROBLEM

1994 ◽  
Vol 04 (04) ◽  
pp. 437-445 ◽  
Author(s):  
CLIVE N. GALLEY ◽  
COSTAS S. ILIOPOULOS
Keyword(s):  
Parallel Algorithm ◽  
Parallel Computation ◽  
Simple Algorithm ◽  
Partition Problem ◽  
Pram Model ◽  
Single Function ◽  
Crcw Pram

This paper shows a simple algorithm for solving the single function coarsest partition problem on the CRCW PRAM model of parallel computation using O(n) processors in O( log n) time with O(n1+ε) space.

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.

scholarly journals SEQUENTIAL AND PARALLEL ALGORITHMS FOR THE k CLOSEST PAIRS PROBLEM

1995 ◽  
Vol 05 (03) ◽  
pp. 273-288 ◽  
Author(s):  
HANS-PETER LENHOF ◽  
MICHIEL SMID
Keyword(s):  
Decision Tree ◽  
Parallel Algorithms ◽  
Dimensional Space ◽  
Decision Tree Model ◽  
Sequential Algorithm ◽  
Tree Model ◽  
Parallel Time ◽  
Parallel Version ◽  
Pram Model ◽  
Crcw Pram

Let S be a set of n points in D-dimensional space, where D is a constant, and let k be an integer between 1 and [Formula: see text]. A new and simpler proof is given of Salowe’s theorem, i.e., a sequential algorithm is given that computes the k closest pairs in the set S in O(n log n+k) time, using O(n+k) space. The algorithm fits in the algebraic decision tree model and is, therefore, optimal. Salowe’s algorithm seems difficult to parallelize. A parallel version of our algorithm is given for the CRCW-PRAM model. This version runs in O((log n)2 log log n) expected parallel time and has an O(n log n log log n+k) time-processor product. Finally, actual running times are given of an implementation of our sequential algorithm.

PARALLEL VERTEX COLOURING OF INTERVAL GRAPHS

1999 ◽  
Vol 10 (01) ◽  
pp. 19-31 ◽  
Author(s):  
G. SAJITH ◽  
SANJEEV SAXENA
Keyword(s):  
Chromatic Number ◽  
Interval Graph ◽  
Interval Graphs ◽  
Chromatic Numbers ◽  
Erew Pram ◽  
Pram Model ◽  
Interval Representation ◽  
Left And Right ◽  
Crcw Pram ◽  
Vertex Colouring

Evidence is given to suggest that minimally vertex colouring an interval graph may not be in NC 1. This is done by showing that 3-colouring a linked list is NC 1-reducible to minimally colouring an interval graph. However, it is shown that an interval graph with a known interval representation and an O(1) chromatic number can be minimally coloured in NC 1. For the CRCW PRAM model, an o( log n) time, polynomial processors algorithm is obtained for minimally colouring an interval graph with o( log n) chromatic number and a known interval representation. In particular, when the chromatic number is O(( log n)1-ε), 0<ε<1, the algorithm runs in O( log n/ log log n) time. Also, an O( log n) time, O(n) cost, EREW PRAM algorithm is found for interval graphs of arbitrary chromatic numbers. The following lower bound result is also obtained: even when the left and right endpoints of the interval are separately sorted, minimally colouring an interval graph needs Ω( log n/ log log n) time, on a CRCW PRAM, with a polynomial number of processors.

Synchronizing Almost-Group Automata

2020 ◽  
pp. 1-22
Author(s):  
Mikhail V. Berlinkov ◽  
Cyril Nicaud
Keyword(s):  
Lower Bound ◽  
Efficient Algorithm ◽  
High Probability ◽  
Worst Case ◽  
Average Case ◽  
Model Of Computation ◽  
Letter Alphabet ◽  
Strongly Connected ◽  
Small Change ◽  
Random Automata

In this paper we address the question of synchronizing random automata in the critical settings of almost-group automata. Group automata are automata where all letters act as permutations on the set of states, and they are not synchronizing (unless they have one state). In almost-group automata, one of the letters acts as a permutation on [Formula: see text] states, and the others as permutations. We prove that this small change is enough for automata to become synchronizing with high probability. More precisely, we establish that the probability that a strongly-connected almost-group automaton is not synchronizing is [Formula: see text], for a [Formula: see text]-letter alphabet. We also present an efficient algorithm that decides whether a strongly-connected almost-group automaton is synchronizing. For a natural model of computation, we establish a [Formula: see text] worst-case lower bound for this problem ([Formula: see text] for the average case), which is almost matched by our algorithm.

Optimization in Cost of Sorting by CRCW PRAM Model

2020 ◽  
Author(s):  
Sandeep Kumar Gill ◽  
Anju Sharma
Keyword(s):  
Pram Model ◽  
Crcw Pram

Parallel Algorithms for Recognizing P5-free and ${\bar P}_5$-free Weakly Chordal Graphs

2004 ◽  
Vol 14 (01) ◽  
pp. 119-129
Author(s):  
Stavros D. Nikolopoulos ◽  
Leonidas Palios
Keyword(s):  
Parallel Algorithms ◽  
Chordal Graphs ◽  
Input Graph ◽  
Model Of Computation ◽  
Erew Pram ◽  
Recognition Algorithms ◽  
Pram Model

We prove algorithmic characterizations of weakly chordal graphs, which lead to efficient parallel algorithms for recognizing P5-free and [Formula: see text]-free weakly chordal graphs. For an input graph on n vertices and m edges, our algorithms run in O( log 2n) time and require O(m2/ log n) processors on the EREW PRAM model of computation. The proposed recognition algorithms efficiently detect P5 s and [Formula: see text] in weakly chordal graphs in O( log n) time with O(m2/ log n) processors on the EREW PRAM. Additionally, we show how the algorithms can be augmented to provide a certificate for the existence of a P5 (or a [Formula: see text]) in case the input graph is not P5-free (respectively, [Formula: see text]-free) weakly chordal.

AN EFFICIENT NC ALGORITHM FOR APPROXIMATE MAXIMUM WEIGHT MATCHING

2013 ◽  
Vol 05 (03) ◽  
pp. 1350013
Author(s):  
SATYAJIT BANERJEE
Keyword(s):  
Approximation Algorithm ◽  
Execution Time ◽  
Maximum Weight ◽  
Model Of Computation ◽  
Erew Pram ◽  
Pram Model ◽  
Maximum Weight Matching

We present an alternative implementation of the (1 – ϵ) factor NC approximation algorithm for the maximum weight matching by Hougardy et al. [1]. Our implementation, on the EREW PRAM model of computation, achieves an [Formula: see text] factor improvement on both the execution time and the number of processors.

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