Constructing trees in parallel

Abstract

An O(log ~ 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.

Fast Distance Multiplication of Unit-Monge Matrices

“…for each sign combination taken consistently on both sides of the inequality, and, by (6), (7), (8)) Thus, in this case P C (î,k) = 1 is equivalent to P C,lo (î,k) = 1. Note that this also implies δ(î − ,k − ) < 0, since otherwise we would have δ(î ± ,k ± ) = 0 for all four sign combinations, and hence, by symmetry, also P C,hi (î,k) = 1.…”

“…The fastest currently known algorithm is by Chan [15], running in time O( n 3 (log log n) 3 log 2 n ). For Monge matrices, distance multiplication can easily be performed in time O(n 2 ), using the standard row-minima searching technique of Aggarwal et al [1] to perform matrix-vector multiplication in linear time (see also [6,57]). Alternatively, an algorithm with the same quadratic running time can be obtained directly by the divide-and-conquer technique (see e.g.…”

Parallel Searching in Generalized Monge Arrays

“…In the parallel case, APSP can be solved by repeated squaring in O(lg 2 n) time using n 3 /lg n processors on a CREW PRAM. Atallah et al [15] show how to solve APSP in O(lg 2 n) time using n 3 /lg n processors on a CREW PRAM (this solution follows from their O(lg 2 n)-time (n 2 /lg n)-processor solution to the single source shortest paths problem on such a graph). In Section 4.1 we give the algorithm of Aggarwal et al [2] which runs in O(lg 2 n) time using n 2 CREW-PRAM processors for the special case of the APSP problem when the graph is acyclic and the edge weights satisfy the quadrangle inequality.…”

“…In the sequential domain, Huffman in [25] showed how to construct Huffman codes greedily in O(n) time (once the character frequencies are in sorted order). In [15], Atallah et al reduced Huffman coding to O(lg n) tube minimization problems on Monge-composite arrays, thereby obtaining parallel algorithms for Huffman coding that run in O(lg 2 n) time using n 2 /lg n processors on a CREW PRAM and in O(lg n(lg lg n) 2 ) time using n 2 /(lg lg n) 2 processors on a CRCW PRAM. Larmore and Przytycka in [30] reduce Huffman coding to the Concave Least Weight Subsequence (CLWS) problem (defined in Section 4.2) and then show how to solve CLWS, and thereby Huffman coding, in O( √ n lg n) time using n processors on a CREW PRAM.…”

“…(See also [12].) Similarly, Atallah et al [15] have used Monge-composite arrays to construct Huffman and other such codes on CRCW and CREW PRAMs. Larmore and Przytycka in [30] used Monge arrays to solve the Concave Least Weight Subsequence (CLWS) problem (defined in Section 4.2).…”