Jean R. S. Blair

An Introduction to Chordal Graphs and Clique Trees

Clique trees and chordal graphs have carved out a niche for themselves in recent work on sparse matrix algorithms, due primarily to research questions associated with advanced computer architectures. This paper is a unified and elementary introduction to the standard characterizations of chordal graphs and clique trees. The pace is leisurely, as detailed proofs of all results are included. We also briefly discuss applications of chordal graphs and clique trees in sparse matrix computations.

Puzzles and games: addressing different learning styles in teaching operating systems concepts

Because students have different learning styles, its important to incorporate multiple teaching techniques into the classroom experience. One such technique is the use of puzzles and games in the classroom to reinforce the learning objectives. Many topics in Computer Science are well suited for coverage in such a game. Several in-class puzzles and games have been used in the Computer Science program at this institution in recent years. In basic and advanced courses, simple crossword puzzles reinforce terminology and Jeopardy!®-style games help students master material with short answers. In the most recent iteration of the Operating Systems course, a BattleThreads game and a Process State Transition game helped students appreciate different approaches to process and thread management. The latter two games have been assessed for their effectiveness, providing several insights into what makes a good in-class game for teaching operating systems concepts, and how the existing games can be improved.

Maximum Cardinality Search for Computing Minimal Triangulations of Graphs

Abstract We present a new algorithm, called MCS-M, for computing minimal triangulations of graphs. Lex-BFS, a seminal algorithm for recognizing chordal graphs, was the genesis for two other classical algorithms: LEX M and MCS. LEX M extends the fundamental concept used in Lex-BFS, resulting in an algorithm that not only recognizes chordality, but also computes a minimal triangulation of an arbitrary graph. MCS simplifies the fundamental concept used in Lex-BFS, resulting in a simpler algorithm for recognizing chordal graphs. The new algorithm MCS-M combines the extension of LEX M with the simplification of MCS, achieving all the results of LEX M in the same time complexity.

A practical algorithm for making filled graphs minimal

For an arbitrary filled graph G+ of a given original graph G, we consider the problem of removing fill edges from G+ in order to obtain a graph M that is both a minimal filled graph of G and a subgraph of G+. For G+ with f fill edges and e original edges, we give a simple O(f(e+f)) algorithm which solves the problem and computes a corresponding minimal elimination ordering of G. We report on experiments with an implementation of our algorithm, where we test graphs G corresponding to some real sparse matrix applications and apply well-known and widely used ordering heuristics to find G+. Our findings show the amount of fill that is commonly removed by a minimalization for each of these heuristics, and also indicate that the runtime of our algorithm on these practical graphs is better than the presented worst-case bound.

Efficient self-stabilizing algorithms for tree networks

Many proposed self-stabilizing algorithms require an exponential number of moves before stabilizing on a global solution, including some rooting algorithms for tree networks [1, 2, 3]. These results are vastly improved upon in [6] with tree rooting algorithms that require only O(n/sup 3/ + n/sup 2//spl middot/c/sub h/) moves, where n is the number of nodes in the network and c/sub h/ is the highest initial value of a variable. In the current paper, we describe a new set of tree rooting algorithms that brings the complexity down to O(n/sup 2/) moves. This not only reduces the first term by an order of magnitude, but also reduces the second term by an unbounded factor We further show a generic mapping that can be used to instantiate an efficient self-stabilizing tree algorithm from any traditional sequential tree algorithm that makes a single bottom-up pass through a rooted tree. The new generic mapping improves on the complexity of the technique presented in [8].

Maximum Cardinality Search for Computing Minimal Triangulations

We present a new algorithm, called MCS-M, for computing minimal triangulations of graphs. Lex-BFS, a seminal algorithm for recognizing chordal graphs, was the genesis for two other classical algorithms: Lex-M and MCS. Lex-M extends the fundamental concept used in Lex-BFS, resulting in an algorithm that also computes a minimal triangulation of an arbitrary graph. MCS simplified the fundamental concept used in Lex-BFS, resulting in a simpler algorithm for recognizing chordal graphs. The new simpler algorithm MCS-M combines the extension of Lex-M with the simplification of MCS, achieving all the results of Lex-M in the same time complexity.

Experiments on union-find algorithms for the disjoint-set data structure

The disjoint-set data structure is used to maintain a collection of non-overlapping sets of elements from a finite universe. Algorithms that operate on this data structure are often referred to as Union-Find algorithms. They are used in numerous practical applications and are also available in several software libraries. This paper presents an extensive experimental study comparing the time required to execute 55 variations of Union-Find algorithms. The study includes all the classical algorithms, several recently suggested enhancements, and also different combinations and optimizations of these. Our results clearly show that a somewhat forgotten simple algorithm developed by Rem in 1976 is the fastest, in spite of the fact that its worst-case time complexity is inferior to that of the commonly accepted “best” algorithms.

On finding minimum-diameter clique trees

A clique-tree representation of a chordal graph often reduces the size of the data structure needed to store the graph, permitting the use of extremely efficient algorithms that take advantage of the compactness of the representation. Since some chordal graphs have many distinct clique-tree representations, it is interesting to consider which one is most desirable under various circumstances. A clique tree of minimum diameter (or height) is sometimes a natural candidate when choosing clique trees to be processed in a parallel-computing environment. This paper introduces a linear-time algorithm for computing a minimum-diameter clique tree.

PERFECT RECALL AND PRUNING IN GAMES WITH IMPERFECT INFORMATION

Games with imperfect information are an interesting and important class of games. They include most card games (e.g., bridge and poker) as well as many economic and political models. Here we investigate algorithms for findi ng the simplest form of a solution (a pure‐strategy equilibrium point) to imperfect information games expressed in their extensive (game tree) form. We introduce to the artificial intelligence community a classic algorithm, due to Wilson, that solves one‐player games with perfect recall. Wilsons algorithm, which we call iMP‐minimax, runs in time linear in the size of the game‐tree searched. In contrast to Wilsons result, Koller and Meggido have shown that finding a pure‐strategy equilibrium point in one‐player games without perfect recall is NP‐hard. Here, we provide another contrast to Wilsons result–we show that in games with perfect recall but more than one player, finding a pure‐strategy equilibrium point, given that such an equilibrium point exists, is NP‐hard.

Graph Extremities Defined by Search Algorithms

Graph search algorithms have exploited graph extremities, such as the leaves of a tree and the simplicial vertices of a chordal graph. Recently, several well-known graph search algorithms have been collectively expressed as two generic algorithms called MLS and MLSM. In this paper, we investigate the properties of the vertex that is numbered 1 by MLS on a chordal graph and by MLSM on an arbitrary graph. We explain how this vertex is an extremity of the graph. Moreover, we show the remarkable property that the minimal separators included in the neighborhood of this vertex are totally ordered by inclusion.