Computing connected components on parallel computers

Abstract

motivated in part by practical considerations. Among the many areas treated in the recent literature are sorting [2,3,7,12,15], the evaluation of arithmetic expressions, linear recurrences and polynomials [4,8,10], matrix algorithms [5,6,13], and graph theory [9,14,15]. In this paper we present a parallel algorithm CONNECT which determines the connected components of an undirected graph with n vertices in time O(log2n) using n 2 processors. Next, we modify the algorithm to demonstrate an observation due to F.P. Preparata and R.L. Probert, viz., n[n/lg n] processors 1 suffice to achieve a time bound of O(log2n). Finally, we show that CONNECT can be modified to compute the transitive closure of an nxn symmetric Boolean matrix in time O(log2n) using n[n/lg n] processors. We use the single instruction stream-multiple data stream (SIMD) model of parallel processors. It is assumed that the processors have access to a common memory, and that simultaneous access to the same location is permitted for fetch instructions but not for store instructions. The Algorithm ConnectWe present a parallel algorithm which uses n 2 processors to find the connected components of an undirected graph with n vertices in time O(log2n). An O0og2n) time bound also can be achieved using only n[n/[log2n]] processors. The algorithm can he used to find the transitive closure of a symmetric Boolean matrix. We assume that the processors have access to a common memory. Simultaneous access to the same location is permitted for fetch instructions but not for store instructions.

Hypercube computing: Connected components

“…[Carlson 1987, Gopalakrishnan et al 1985, Hirschberg et al 1979, Huang 1985, Nassimi and Sahni 1980, and Shiloach and Vishkin 1982 are some examples of such research. None of these algorithms provides a suitable starting point for our work.…”

Structural parallel algorithmics

“…First randomized, and later deterministic, opti inal parallel algorithms for list ranking were given Two logarithmic time connectivity algorithms were given: (1) a deterministic one which is optimal on all except very sparse graphs [CV86aJ; (2) a randomized optimal one [Gaz86]. For Figure 1, the deterministic algorithms builds on a restricted union find problem, a scheduling problem, dubbed duration unknown task scheduling, and the Euler tour technique, as well as ideas from two previous connectivity algorithms [11CS79] and [SV82aJ. It should be pointed out that the logarithmic time version of the deterministic connectivity algorithm requires the use of expander graphs and thus is highly impractical at present; however, a slightly less parallel version involves much smaller constants.…”

Data Movement Techniques for the Pyramid Computer

“…(If the column has no black pixel then set the coordinates to o.) Similarly, for each row j we form records (3, j, lc (j), 11 (j)), (4, j, rc (j), rl (j)) for the leftmost and rightmost black pixels in the row. These are the records needed for the next stage.…”

“…Each problem involves a graph G (V, E) where V is the set of vertices and E is the set of edges. The [9], [11], [13], [18] The initial value of forest_level is [log4(n/v)J and forest_level increases every 2 iterations, so the total time of the algorithm is…”