Pruning uplieate Nodes in th-First

Best-first search algorithms require exponential memory, while depth-first algorithms require only linear memory. On graphs with cycles, however, depth-first searches do not detect duplicate nodes, and hence may generate asymptotically more nodes than best-first searches. We present a technique for reducing the asymptotic complexity of depth-first search by eliminating the generation of duplicate nodes. The automatic discovery and application of a finite state machine (FSM) that enforces pruning rules in a depth-first search, has significantly extended the power of search in several domains. We have implemented and tested the technique on a grid, the Fifteen Puzzle, the Twenty-Four Puzzle, and two versions of Rubik’s Cube. In each case, the effective branching factor of the depth-first search is reduced, reducing the asymptotic time complexity. Introduction-The Problem Search techniques are fundamental to artificial intelligence. Best-first search algorithms such as breadthfirst search, Dijkstra’s algorithm [Dijkstra, 19591, and A* [Hart et al., 19681, all require enough memory to store all generated nodes. This results in exponential space complexity on many problems, making them impra:tical. I,1 contrast, depth-first searches run in space linear in the depth of the search. However, a major disadvantage of depth-first approaches is the generation of du$icate nodes in a graph with cycles [Nilsson, 19801. More than one combination of operators may produce the same node, but since depth-first search does not store the nodes already generated, it cannot detect the duplicates. As a result, the total number of nodes generated by a depth-first search may be asymptotically more than the number of nodes generated by a bestfirst search. *This research was partially supported by NSF Grant #KU-9119825, and a grant from Rockwell International. Figure 1: The grid search space, explored depth-first to depth 2, without pruning. To illustrate, consider a search of a grid with the operators: Up, Down, Left and Right, each moving one unit. A depth-first search to depth T would visit 4’ nodes (figure l), since 4 operators are applicable to each node. But only O(r2) distinct junctions are visited by a breadth-first search. Thus, a depth-first search has exponential complexity, while a breadthfirst search has only polynomial complexity. To reduce this effect, we would like to detect and prune duplicate nodes in a depth-first search. Unfortunately, there is no way to do this on an arbitrary graph without storing all the nodes. On a randomly connected explicit graph, for example, the only way to check for duplicate nodes is to maintain a list of all the nodes already generated. A partial solution to this problem is to compare new nodes against the current path from the root [Pearl, 19841. This detects duplicates in the case that the path has made a complete cycle. However, as we saw in the grid example, duplicates occur when the search explores two halves of a cycle, such as up-left and left-up. Only a fraction of duplicates can be found by comparing nodes on the current path [Dillenburg and Nelson, 19931. Other node caching schemes have been suggested [Ibaraki, 19781 [Sen and Bagchi, 19891 [Chakrabarti et a/., 19891 [Elkan, 19891, but their utility depends on the implementation (costs per node generation). 756 Taylor From: AAAI-93 Proceedings. Copyright © 1993, AAAI (www.aaai.org). All rights reserved.