Ryan B. Hayward

Weakly triangulated graphs

Abstract A graph is triangulated if it has no chordless cycle with four or more vertices. It follows that the complement of a triangulated graph cannot contain a chordless cycle with five or more vertices. We introduce a class of graphs (namely, weakly triangulated graphs) which includes both triangulated graphs and complements of triangulated graphs (we define a graph as weakly triangulated if neither it nor its complement contains a chordless cycle with five or more vertices). Our main result is a structural theorem which leads to a proof that weakly triangulated graphs are perfect.

Monte Carlo Tree Search in Hex

Hex, the classic board game invented by Piet Hein in 1942 and independently by John Nash in 1948, has been a domain of AI research since Claude Shannons seminal work in the 1950s. Until the Monte Carlo Go revolution a few years ago, the best computer Hex players used knowledge-intensive alpha-beta search. Since that time, strong Monte Carlo Hex players have appeared that are on par with the best alpha-beta Hex players. In this paper, we describe MoHex, the Monte Carlo tree search Hex player that won gold at the 2009 Computer Olympiad. Our main contributions to Monte Carlo tree search include using inferior cell analysis and connection strategy computation to prune the search tree. In particular, we run our random game simulations not on the actual game position, but on a reduced equivalent board.

Optimizing weakly triangulated graphs

A graph is weakly triangulated if neither the graph nor its complement contains a chordless cycle with five or more vertices as an induced subgraph. We use a new characterization of weakly triangulated graphs to solve certain optimization problems for these graphs. Specifically, an algorithm which runs inO((n + e)n3) time is presented which solves the maximum clique and minimum colouring problems for weakly triangulated graphs; performing the algorithm on the complement gives a solution to the maximum stable set and minimum clique covering problems. Also, anO((n + e)n4) time algorithm is presented which solves the weighted versions of these problems.

Improved algorithms for weakly chordal graphs

We use a new structural theorem on the presence of two-pairs in weakly chordal graphs to develop improved algorithms. For the recognition problem, we reduce the time complexity from O(<i>mn</i>²) to O(<i>m</i>²) and the space complexity from O(<i>n</i>³) to O(<i>m</i> + <i>n</i>), and also produce a hole or antihole if the input graph is not weakly chordal. For the optimization problems, the complexity of the clique and coloring problems is reduced from O(<i>mn</i>²) to O(<i>n</i>³) and the complexity of the independent set and clique cover problems is improved from O(<i>n</i>⁴) to O(<i>mn</i>). The space complexity of our optimization algorithms is O(<i>m</i> + <i>n</i>).

Large Deviations for Quicksort

LetQnbe the random number of comparisons made by quicksort in sortingndistinct keys when we assume that alln! possible orderings are equally likely. Known results concerning moments forQndo not show how rare it is forQnto make large deviations from its mean. Here we give a good approximation to the probability of such a large deviation and find that this probability is quite small. As well as the basic quicksort, we consider the variant in which the partitioning key is chosen as the median of (2t+1) keys.

Meyniel weakly triangulated graphs—I: co-perfect orderability

Abstract We show that P5-free weakly triangulated graphs are perfectly orderable. Our proof is algorithmic, and relies on a notion concerning separating sets, a property of weakly triangulated graphs, and several properties of P5-free weakly triangulated graphs.

Solving hex: beyond humans

For the first time, automated Hex solvers have surpassed humans in their ability to solve Hex positions: they can now solve many 9×9 Hex openings. We summarize the methods that attained this milestone, and examine the future of Hex solvers.

Dead Cell Analysis in Hex and the Shannon Game

In 1981 Claude Berge asked about combinatorial properties that might be used to solve Hex puzzles. In response, we establish properties of dead, or negligible, cells in Hex and the Shannon game.

Hex and combinatorics

Inspired by Claude Berges interest in and writings on Hex, we discuss some results on the game.

Average case analysis of heap building by repeated insertion

We show that the average number of swaps required to construct a heap on n keys by Williams’ method of repeated insertion is (! + o(1))n, where the constant ! is about 1.3. Further, with high probability the number of swaps is close to this