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.