We describe efficient deterministic techniques for breaking symmetry in parallel. These techniques work well on rooted trees and graphs of constant degree or genus. OUf primary technique allows us to 3-color a rooted tree in O(lg*n) time on an EREW PRAM using a linear number of processors. We use these techniques to construct fast linear processor algorithms for several problems, including the problem of (.6. + 1)-coloring constant-degree graphs and 5-coloring planar graphs. We also prove lower bounds for 2-coloring directed lists and for finding maximal independent sets in arbitrary graphs.

We describe efficient deterministic techniques for breaking symmetry in parallel. These techniques work well on rooted trees and graphs of constant degree or genus. OUf primary technique allows us to 3-color a rooted tree in O(lg*n) time on an EREW PRAM using a linear number of processors. We use these techniques to construct fast linear processor algorithms for several problems, including the problem of (.6. + 1)coloring constant-degree graphs and 5-coloring planar graphs. We also prove lower bounds for 2-coloring directed lists and for finding maximal independent sets in arbitrary graphs.

We describe efficient deterministic techniques for breaking symmetry in parallel. These techniques work well on rooted trees and graphs of constant degree or genus. OUf primary technique allows us to 3-color a rooted tree in O(lg*n) time on an EREW PRAM using a linear number of processors. We use these techniques to construct fast linear processor algorithms for several problems, including the problem of (.6. + 1)-coloring constant-degree graphs and 5-coloring planar graphs. We also prove lower bounds for 2-coloring directed lists and for finding maximal independent sets in arbitrary graphs.

This paper introduces the LINK system as a flexible tool for the creation, manipulation, and drawing of graphs and hypergraphs. We describe the basic architecture of the system and illustrate its flexibility with several examples. LINK is distinguished from existing software for discrete mathematics by its layered interface, including a graphical user interface tied into an object-oriented Scheme language interface with access to Tk, and an extensible underlying set of C++ libraries. We conclude by briefly discussing roles LINK has played in research and education. 1 Background Over the past several years, there have been several efforts to construct software systems for discrete mathematics, and in particular, for the manipulation of graphs. None, however, has resulted in a product with influence comparable to the familiar symbolic mathematics packages. Some notable existing systems for discrete mathematics are Combinatorica [14], Steven Skiena's extension package for Mathematica, NETPAD [11] due to Nathaniel Dean and others at Bellcore, SetPlayer [1], due to Mark Goldberg and his students at Rensselaer Polytechnic Institute, and Gregory Shannon et. al.'s GraphLab [13]. For various reasons, none of these systems has the potential to be a widely-useful environment for both graph manipulation and computation. The authors of these systems recognized this and proposed the development of LINK, which was to be a freely-available and portable software system for discrete mathematics overcoming the various shortcomings of existing systems. After three years of development, the system is now freely available from the LINK web site: http://dimacs.rutgers.edu/Projects/LINK.html. LINK features a 200 page on-line manual and an on-line tutorial. LINK's design philosophy placed flexibility as the highest priority, and this led to the selection of STk, Erick Gallesio's object-oriented Scheme language interface to John Ousterhout's portable, interpretive Tk graphics system. [12][6]. Tk enables involved graphics programming without any knowledge of the X-window system, and offers the advantages of interpretation and portability at the cost of speed. This means that the system is not appropriate for viewing massive data sets. Graph views with

This paper gives an O(lg3 n)-time and n/lg n processor algorithm for solving the mat& chain ordering problem and forfinding optimal triangulations of a convez polygon on the Common CRCW PRAM model. This algorithm works by finding shortest paths in ape-cia1 digraphs modeling dynamic programming tables. Also, a key part of the algorithm ia improved by computing row minima of a totally monotone matriz, thirleta the algorithm run in O(1g'n) time with n procesaors on the EREW PRAM or even O(1g'nlglgn) time with n/lg lg n processors an the CRCW PRAM. 0-8186-5602-6/94 0 1994 IEEE