An Expected-Cost Analysis of Backtracking and Non-Backtracking Algorithms

Consider an infinite binary search tree in which the branches have independent random costs. Suppose that we must find an optimal (cheapest) or nearly optimal path from the root to a node at depth n. Karp and Pearl [1983] show that a bounded-lookahead backtracking algorithm A2 usually finds a nearly optimal path in linear expected time (when the costs take only the values 0 or 1). From this successful performance one might conclude that similar heuristics should be of more general use. But we find here equal success for a simpler non-backtracking bounded-lookahead algorithm , so the search model cannot support this conclusion. If, however, the search tree is generated by a branching process so that there is a possibility of nodes having no sons (or branches having prohibitive costs), then the non-backtracking algorithm is hopeless while the backtracking algorithm still performs very well. These results suggest the general guideline that backtracking becomes attractive when there is the possibility of "dead-ends" or prohibitively costly outcomes. 1 INTRODUCTION Many algorithms considered in operations research, computer science and artificial intelligence may be represented as searches or partial searches through rooted trees. Such algorithms typically involve backtracking but try to minimize the time spent doing so (e.g. for some problems it may be best to avoid backtracking [de Kleer, 1984]. The paper extends work of [Karp and Pearl, 1983], and gives a probabilistic analysis of backtracking and non-backtracking search algorithms in certain trees with random branch costs. We thus cast some light on the question of when to backtrack: it seems that backtrack-ing is valuable just for problems with "dead-ends" (or outcomes with prohibitively high costs). Let us review briefly the model and results of Karp and Pearl [1983]. They consider an infinite search tree in which each node has exactly two sons. The branches have independent (0, l)-valued random costs X, with p = P(X = 0). 1 The problem is to find an optimal (cheapest) or nearly optimal path from the root to a node at depth n. The problem changes nature depending on whether the expected number 2p of zero-cost branches leaving a node is > 1, = 1 or < 1. When 2p > 1 a simple uniform cost breadth-first search algorithm Al finds an optimal solution in expected time O(n); and when 2p = 1 this algorithm takes expected time 0(n 2). When 2p < 1 any algorithm …