R-Kleene: A High-Performance Divide-and-Conquer Algorithm for the All-Pair Shortest Path for Densely Connected Networks

Abstract

We propose a novel divide-and-conquer algorithm for the solution of the all-pair shortest-path problem for directed and dense graphs with no negative cycles. We propose R-Kleene, a compact and in-place recursive algorithm inspired by Kleene's algorithm. R-Kleene delivers a better performance than previous algorithms for randomly generated graphs represented by highly dense adjacency matrices, in which the matrix components can have any integer value. We show that R-Kleene, unchanged and without any machine tuning, yields consistently between 1 7 and 1 2 of the peak performance running on five very different uniprocessor systems. Introduction.The all-pair shortest-paths problem (APSP) is a well-studied and basic problem in graph theory but it is also a crucial and real problem in large networks such as sensor networks, switch networks or complex targeting systems.Consider the scenario where many thousands of nodes are located across a large area and every node has a processor with little memory space and computational power. In this scenario the computation of APSP is neither feasible nor practical by a single node, nonetheless it is a key feature for efficient data routing and broadcasting. Despite the node-processor computational/memory limitations, a node in the network is able to determine the locations and distances of its neighbors rather easily. Such local information can be coded, sent on the network and collected by an observer node such as a satellite, a global router or a computer cluster. Then the observer node may construct the adjacency matrix, compute the solution and send the result back on the network where each node will store the necessary local information.Any network is naturally represented by a directed graph and we formalize APSP as follows. Given a graph G = (V, E) where V is a set of nodes and E is a set of directed edges, we label every node in the graph by an integer ι ∈ [0, n − 1] where n = |V | (n = |V | is the cardinality of the set V ), and an edge in E is defined by a unique ordered pair of integers (i, j) with i, j ∈ [0, n − 1]. In fact, we assume that there is at most one directed edge connecting two nodes and, therefore, the graph has

Algorithms of All Pair Shortest Path Problem

“…In-place Parallel Recursive approach using kleene's algorithm [7] Kleene's algorithm is used for finding transitive closure that computes the path existence between every possible pair of vertices(i, j). Kleene's algorithm divides the nodes of the graph into n⁄√s zones.…”

The Cache-Oblivious Gaussian Elimination Paradigm: Theoretical Framework, Parallelization and Experimental Evaluation

“…A cache-oblivious algorithm for Floyd-Warshall's APSP algorithm is given in [27] (see also [13]). The algorithm runs in O(n 3 ) time and incurs O(…”

Solving path problems on the GPU

“…The correctness of the recursive algorithm has been formally proven in various ways before [21,22]. Here we present a simpler proof based on algebraic paths.…”

“…Recursive formulations of APSP have been presented by many researchers over the years [21,22,23]. The connection to semiring matrix multiplication was shown by Aho et al [12], but they did not present a complete algorithm.…”

“…The connection to semiring matrix multiplication was shown by Aho et al [12], but they did not present a complete algorithm. Ours is a modified version of the algorithm of Tiskin [23] and R-Kleene algorithm [21]. Especially, the in-place nature of the R-Kleene algorithm helped us avoid expensive global memory to global memory data copying.…”