Modifying Dijkstra's Algorithm to Solve Many Instances of SSSP in Linear Time

We show that for graphs with positive edge weights the single-source shortest path planning problem (SSSP) can be solved using a novel partial ordering over nodes, instead of a full ordering, without sacrificing algorithmic correctness. The partial ordering we investigate is defined with respect to carefully chosen (but easy to calculate) "approximate" level-sets of shortest-path length. We present a family of easy-to-implement "approximate'' priority heaps, based on an array of linked-lists, that create this partial ordering automatically when used directly by Dijkstra's SSSP algorithm. For graphs G = (E,V) with positive edge lengths, and depending on which version of the heap is used, the resulting Dijkstra variant runs in either time O(|E| + |V | + K) with space O(|E| + |V | + l_max/l_min) or time O((|E| + |V|) log_w(l_max/l_min + 1)) with space O(|E| + log_w(l_max/l_min + 1)), where l_min and l_max are the minimum (non-zero) and maximum (finite) edge lengths, respectively, and w is the word length of the computer being used (e.g., 32 or 64 in most cases), and K is a function of G such that K = O(|E|) for many common types of graphs (e.g., K = 1 for graphs with unit edge lengths). We also describe a linear time pre-/post-processing procedure that extends these results to undirected graphs with non-negative edge weights. Thus, it possible to solve many instances of SSSP in O(|E| + |V |); for these instances our method ties the fastest known runtime for SSSP, while having smaller constant factor overhead than previous methods. This work can be viewed as an extension of Dial’s SSSP algorithm in order handle floating point edge weights, faster runtimes, and new theoretical results

Design and analysis of sequential and parallel single-source shortest-paths algorithms

We study the performance of algorithms for the Single-Source Shortest-Paths (SSSP) problem on graphs with n nodes and m edges with nonnegative random weights. All previously known SSSP algorithms for directed graphs required superlinear time. Wie give the first SSSP algorithms that provably achieve linear O(n-m)average-case execution time on arbitrary directed graphs with random edge weights. For independent edge weights, the linear-time bound holds with high probability, too. Additionally, our result implies improved average-case bounds for the All-Pairs Shortest-Paths (APSP) problem on sparse graphs, and it yields the first theoretical average-case analysis for the "Approximate Bucket Implementation" of Dijkstra\u27s SSSP algorithm (ABI-Dijkstra). Futhermore, we give constructive proofs for the existence of graph classes with random edge weights on which ABI-Dijkstra and several other well-known SSSP algorithms require superlinear average-case time. Besides the classical sequential (single processor) model of computation we also consider parallel computing: we give the currently fastest average-case linear-work parallel SSSP algorithms for large graph classes with random edge weights, e.g., sparse rondom graphs and graphs modeling the WWW, telephone calls or social networks.In dieser Arbeit untersuchen wir die Laufzeiten von Algorithmen für das Kürzeste-Wege Problem (Single-Source Shortest-Paths, SSSP) auf Graphen mit n Knoten, M Kanten und nichtnegativen zufälligen Kantengewichten. Alle bisherigen SSSP Algorithmen benötigen auf gerichteten Graphen superlineare Zeit. Wir stellen den ersten SSSP Algorithmus vor, der auf beliebigen gerichteten Graphen mit zufälligen Kantengewichten eine beweisbar lineare average-case-Komplexität O(n+m)aufweist. Sind die Kantengewichte unabhängig, so wird die lineare Zeitschranke auch mit hoher Wahrscheinlichkeit eingehalten. Außerdem impliziert unser Ergebnis verbesserte average-case-Schranken für das All-Pairs Shortest-Paths (APSP) Problem auf dünnen Graphen und liefert die erste theoretische average-case-Analyse für die "Approximate Bucket Implementierung" von Dijkstras SSSP Algorithmus (ABI-Dijkstra). Weiterhin führen wir konstruktive Existenzbeweise für Graphklassen mit zufälligen Kantengewichten, auf denen ABI-Dijkstra und mehrere andere bekannte SSSP Algorithmen durchschnittlich superlineare Zeit benötigen. Neben dem klassischen seriellen (Ein-Prozessor) Berechnungsmodell betrachten wir auch Parallelverarbeitung; für umfangreiche Graphklassen mit zufälligen Kantengewichten wie z.B. dünne Zufallsgraphen oder Modelle für das WWW, Telefonanrufe oder soziale Netzwerke stellen wir die derzeit schnellsten parallelen SSSP Algorithmen mit durchschnittlich linearer Arbeit vor

Evaluating multi-core graph algorithm frameworks

Multi-core and GPU-based systems offer unprecedented computational power. They are, however, challenging to utilize effectively, especially when processing irregular data such as graphs. Graphs are of great interest, as they are now used to model geographic-, social- andneural networks. Several interesting programming frameworks for graph processing have therefore been developed these past few years. In this work, we highlight the strengths and weaknesses of the Galois, GraphBLAST, Gunrock and Ligra graph frameworks through benchmarking their single source shortest path (SSSP) implementations using the SuiteSparse Matrix Collection. Tests were done on an Nvidia DGX2 system, except for Ligra, which only provides a multi-core framework. D-IrGL, built on Galois, also provided a multi-GPU option for SSSP. We also look at program size, documentation and overall ease of use. High performance generally comes at the price of high complexity. D-IrGL shows its strength on the very largest graphs, where it achieved the best run-time, while Gunrock processed most other large sets the fastest. However, GraphBLAST, with a relatively low-complexity interface, achieves the greatest median throughput across all our test cases. This despite that its SSSP implementation size is only 1/10th of Gunrock, which for our tests has the highest peak throughput and the fastest run-time in most cases. Ligra had less computational resources available, and consequently performed worse in most cases, but it is also a very compact and easy to use framework. Futher analyses and some suggestions for future work are also included