Constructing trees in parallel

Abstract

An O(log 2 n) time, n2/logn processor as well as an O(log n) time, n3/log n processor CREW deterministic parallel algorithms are presented for constructing Huffman codes from a given list of frequences. The time can be reduced to O(log n(loglog n) 2) on an CRCW model, using only n2/(log log n) 2 processors. Also presented is an optimal O(log n) time, O(n/log n) processor EREW parallel algorithm for constructing a tree given a list of leaf depths when the depths are monotonic. An O(log 2 n) time, n processor parallel algorithm is given for the general tree construction problem. We also give an O(log 2 n) time n2/log2n processor algorithm which finds a nearly optimal binary search tree. An O(log 2 n) time n 2 36 processor algorithm for recognizing linear context free languages is given. A crucial ingredient in achieving those bounds is a formulation of these problems as multiplications of special matrices which we call concave matrices. The structure of these matrices makes their parallel multiplication dramatically more efficient than that of arbitrary matrices.