Optimal path-finding algorithms*

Path-finding search occurs in the presence of various sources of knowledge such as heuristic evaluation functions, subgoals, macro-operators, and abstractions. For a given source of knowledge, we explore optimal algorithms in terms of time, space, and cost of solution path, and quantify their performance. In the absence of any knowledge, a depth-first iterative-deepening algorithm is asymptotically optimal in time and space among minimal-cost exponential tree searches. A heuristic version of the algorithm, called iterative-deepening-A*, is shown to be optimal in the same sense over heuristic searches. For solving large problems, more powerful knowledge sources, such as subgoals, macro-operators, and abstraction are required, usually at the cost of suboptimal solutions. Subgoals often drastically reduce problem solving complexity, depending upon the subgoal distance, a new measure of problem difficulty. Macro-operators store solutions to previously solved subproblems in order to speed-up solutions to new problems, and are subject to a multiplicative time-space tradeoff. Finally, an analysis of abstraction concludes that abstraction hierarchies can reduce exponential problems to linear complexity.