Fast Sequential and Parallel Vertex Relabelings of Km,m

Given an undirected, connected, simple graph G = (V,E), two vertex labelings LV and L'V of the vertices of G, and a label flip operation that interchanges a pair of labels on adjacent vertices, the Vertex Relabeling Problem is to transform G from LV into L'V using the flip operation. Agnarsson et al. showed solving the Vertex Relabeling Problem on arbitrary graphs can be done in θ(n2), where n is the number of vertices in G. In this article we study the Vertex Relabeling Problem on graphs Km,m and introduce the concept of parity and precise labelings. We show that, when we consider the parity labeling, the problem on graphs Km,m can be solved quickly in O(log m) time using m processors on an EREW PRAM. Additionally, we also show that the number of processors can be further reduced to [Formula: see text] in this case while the time complexity does not change. When the labeling is precise, the parallel time complexity increases by a factor of log m while the processor complexities remain m and [Formula: see text]. We also show that, when graphs are restricted to Km,m, this problem can be solved optimally in O(m) time when the labeling is parity, and can be solved in O(m log m) time when the labeling is precise, thereby improving the result in Agnarsson et al. for this specific case. Moreover, we generalize the result in the case of precise labeling to the cases when LV and L'V can be any configuration. In the end we give a conclusion and a list of some interesting open problems.

AN EFFICIENT NC ALGORITHM FOR APPROXIMATE 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.

A Cost Optimal Parallel Algorithm for Patience Sorting

In this paper, we consider a parallel algorithm for the patience sorting. The problem is not known to be in the class NC or P-complete. We propose two algorithms for the patience sorting of n distinct integers. The first algorithm runs in [Formula: see text] time using p processors on the EREW PRAM, where m is the number of decreasing subsequences in a solution of the patience sorting. The second algorithm runs in [Formula: see text] time using p processors on the EREW PRAM. If [Formula: see text] is satisfied, the second algorithm becomes cost optimal.

ARITHMETIC CODING IN PARALLEL

We present an EREW PRAM cost optimal parallel algorithm for arithmetic coding computation. We solve the problem in [Formula: see text] time using n/log n processors. Each part of the algorithm as well as a well-known parallel prefix computation forming a basis of the algorithm are clarified on simple examples.