Approximate matching for 2 families of trees

Abstract

We study approximate matching between h-ary trees (ordered trees whose nodes have exactly h sons) and ordered arbitrary trees, using a string representation of trees. For two h-ary trees P, T, the subtree distance is the number of subtrees to be inserted in P in place of empty nodes, or to be deleted from P, to obtain T. We consider the problem of finding all the occurrences of P in T, with bounded distance k. A known sequential solution requires O(h|P| + h|T| + k|T|) time. We show that the problem can be solved in O(log h + log|P| + log|T| + k) parallel time, in a CRCW-PRAM with O(h(|P| + |T|)) processors. For arbitrary ordered trees we solve a version of the classical tree pattern matching problem. We define the leaf distance between two trees P, T as the total number of subtrees to be inserted in P in place of its leaves, or to be deleted from P leaving leaves in their place, to obtain T. We show how all the occurrences of P as a subtree of T, with bounded distance k, can be determined in O(|P| + k|T|) sequential time, and in O(log|P| + log|T| + k) parallel time in a CRCW-PRAM with O(|P| + |T|) processors. We also discuss an extension of the above problems to labelled trees. © 1995 Academic Press, Inc.