Abstract. The efficiency of A * searching depends on the quality of the lower bound estimates of the solution cost. Pattern databases enumerate all possible subgoals required by any solution, subject to constraints on the subgoal size. Each subgoal in the database provides a tight lower bound on the cost of achieving it. For a given state in the search space, all possible subgoals are looked up, with the maximum cost over all lookups being the lower bound. For sliding tile puzzles, the database enumerates all possible patterns containing N tiles and, for each one, contains a lower bound on the distance to correctly move all N tiles into their correct final location. For the 15-Puzzle, iterative~deepening A * with pattern databases (N=8) reduces the total number of nodes searched on a standard problem set of 100 positions by over 1000-fold. 1

Retrograde Analysis (ra) is an AI search technique used to compute endgame databases, which contain optimal solutions for part of the search space of a game. ra has been applied successfully to several games, but its usefulness is restricted by the huge amount of cpu time and internal memory it requires. We present a parallel distributed algorithm for ra that addresses these problems. ra is hard to parallelize efficiently, because the communication overhead potentially is enormous. We show that the overhead can be reduced drastically using message combining. We implemented the algorithm on an Ethernet-based distributed system. For one example game (awari), we have computed a large database in 50 minutes on 64 processors, whereas one machine took 40 hours (a speedup of 48). An even larger database (computed in half a day) would have required 400 MByte of internal memory on a uniprocessor and would compute for weeks. Keywords: game-tree search, retrograde analysis, distribute...

Leading algorithms for Boolean satisfiability (SAT) are based on either a depth-first tree traversal of the search space (the DLL procedure [6]) or resolution (the DP procedure [7]). In this work we introduce a variant of BreadthFirst Search (BFS) based on the ability of Zero-Suppressed Binary Decision Diagrams (ZDDs) to compactly represent sparse or structured collections of subsets.

This paper discusses a new work-scheduling algorithm for parallel search of single-agent state spaces, called Transposition-Table-Driven Work Scheduling, that places the transposition table at the heart of the parallel work scheduling. The scheme results in less synchronization overhead, less processor idle time, and less redundant search effort. Measurements on a 128-processor parallel machine show that the scheme achieves close-to-linear speedups; for large problems the speedups are even superlinear due to better memory usage. On the same machine, the algorithm is 1.6 to 12.9 times faster than traditional work-stealing-based schemes.

This paper introduces a new scheduling algorithm for parallel single-agent search, transposition table driven work scheduling, that places the transposition table at the heart of the parallel work scheduling. The scheme results in less synchronization overhead, less processor idle time, and less redundant search effort. Measurements on a 128-processor parallel machine show that the scheme achieves nearly-optimal performance and scales well. The algorithm performs a factor of 2.0 to 13.7 times better than traditional work-stealing-based schemes.

For half a century since computers came into existence, the goal of finding elegant and efficient algorithms to solve "simple" (welldefined and well-structured) problems has dominated algorithm design. Over the same time period, both processing and storage capacity of computers have increased roughly by a factor of 10 6 . The next few decades may well give us a similar rate of growth in raw computing power, due to various factors such as continuing miniaturization, parallel and distributed computing. If a quantitative change of orders of magnitude leads to qualitative changes, where will the latter take place? Many problems exhibit no detectable regular structure to be exploited, they appear "chaotic ", and do not yield to efficient algorithms. Exhaustive search of large state spaces appears to be the only viable approach. We survey techniques for exhaustive search, typical combinatorial problems that have been solved, and present one case study in detail.

Abstract. For half a century since computers came into existence, the goal of finding elegant and efficient algorithms to solve “simple ” (welldefined and well-structured) problems has dominated algorithm design. Over the same time period, both processingand storage capacity of computers have increased by roughly a factor of a million. The next few decades may well give us a similar rate of growth in raw computing power, due to various factors such as continuingminiaturization, parallel and distributed computing. If a quantitative change of orders of magnitude leads to qualitative advances, where will the latter take place? Only empirical research can answer this question. Asymptotic complexity theory has emerged as a surprisingly effective tool for predictingrun times of polynomial-time algorithms. For NPhard problems, on the other hand, it yields overly pessimistic bounds. It asserts the non-existence of algorithms that are efficient across an entire problem class, but ignores the fact that many instances, perhaps

In this paper symbolic exploration with binary decision diagrams (BDDs) is applied to two-player games to improve main memory consumption for reachability analysis and game-theoretical classification, since BDDs provide a compact representation for large set of game positions. A number of examples are evaluated: Tic-Tac-Toe, Nim, Hex, and Four Connect. In Chess we restrict the considerations to the creation of endgame databases. The results are preliminary, but the study puts forth the idea that BDDs are widely applicable in game playing and provides a universal tool for people interested in quickly solving practical problems.

A graph of a puzzle is obtained by associating each possible position with a vertex and by inserting edges between vertices iff the corresponding positions can be obtained from each other in one move. Computational methods for finding the vertices at maximum distance ffi from a vertex associated with a goal position are presented. Solutions are given for small sliding block puzzles, and methods for obtaining upper and lower bounds on ffi for large puzzles are considered. Old results are surveyed, and a new upper bound for the 24-puzzle is obtained: ffi 210. 1. Introduction In the early 1980s, it was impossible to avoid hearing about Rubik's cube, a puzzle that became very popular all over the world. Very soon, mathematicians became interested in this puzzle, and several books have been written on the subject (for example, [4]). Another popular---and much older---puzzle is the 15-puzzle, which was invented by Sam Loyd in the 19th century. This puzzle, and its variants, will be consid...

Abstract — Over the years, an increasing number of twoplayer games has been added to the list of solved games. However, solving multi-player games has so far received little, if any, attention. One of the reasons is that while two-player games have a unique game-theoretical value, no unique gametheoretical value may exist in multi-player games as they can have many equilibrium points. We propose solving multi-player games under the paranoid condition. This is equivalent to find the optimal score that a player can achieve independent of the other players ’ strategies. We furthermore introduce and examine an algorithm, called Paranoid Proof-Number Search (PPNS), for solving multi-player games under the paranoid condition. PPNS is tested by solving the 4 × 4 and 6 × 6 variant of the multi-player game Rolit, a multi-player generalization of Reversi (Othello). Our results show that no player can win more than the analytical minimum score in 6 × 6 Rolit while on 4 × 4 Rolit the players are able to score higher. Moreover, the experiments show that for Rolit PPNS is taking advantage of the non-uniformity of the game tree. I.