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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 Shannon's 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(More)
Let Q n be the random number of comparisons made by quicksort in sorting n distinct keys, when we assume that all n! possible orderings are equally likely. Known results concerning moments for Q n do not show how rare it is for Q n to make large deviations from its mean. Here we give a good approximation to the probability of such a large deviation, and(More)
Many tasks require evaluating a specified Boolean expression over a set of probabilistic tests whose costs and success probabilities are each known. A strategy specifies when to perform which test, towards determining the overall outcome of. We are interested in finding the strategy with the minimum expected cost. As this task is typically NP-hard — for(More)
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. A cell is dead if the colour of any stone placed there is irrelevant to the theoretical outcome of the game. We show that dead cell recognition is NP-complete for(More)
Consider a drawing in the plane of K~, the complete graph on n vertices. If all edges are restricted to be straight line segments, the drawing is called rectilinear. Consider a Hamiltonian cycle in a drawing of K,. If no pair of the edges of the cycle cross, it is called a crossing-free Hamiltonian cycle (cfhc). Let ~(n) represent the maximum number of(More)