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 nodes and edges with nonnegative random weights. All previously known SSSP algorithms for directed graphs required superlinear time. We give the first SSSP algorithms that provably achieve linear 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’s SSSP algorithm (ABI–Dijkstra). Furthermore, 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 averagecase 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 random graphs and graphs modeling the WWW, telephone calls or social networks. Kurzzusammenfassung. In dieser Arbeit untersuchen wir die Laufzeiten von Algorithmen fur das Kurzeste-Wege Problem (Single-Source Shortest-Paths, SSSP) auf Graphen mit Knoten, Kanten und nichtnegativen zufalligen Kantengewichten. Alle bisherigen SSSP Algorithmen benotigten auf gerichteten Graphen superlineare Zeit. Wir stellen den ersten SSSP Algorithmus vor, der auf beliebigen gerichteten Graphen mit zufalligen Kantengewichten eine beweisbar lineare average-case-Komplexitat aufweist. Sind die Kantengewichte unabhangig, so wird die lineare Zeitschranke auch mit hoher Wahrscheinlichkeit eingehalten. Auserdem impliziert unser Ergebnis verbesserte averagecase-Schranken fur das All-Pairs Shortest-Paths (APSP) Problem auf dunnen Graphen und liefert die erste theoretische average-case-Analyse fur die “Approximate Bucket Implementierung” von Dijkstras SSSP Algorithmus (ABI-Dijkstra). Weiterhin fuhren wir konstruktive Existenzbeweise fur Graphklassen mit zufalligen Kantengewichten, auf denen ABIDijkstra und mehrere andere bekannte SSSP Algorithmen durchschnittlich superlineare Zeit benotigen. Neben dem klassischen seriellen (Ein-Prozessor) Berechnungsmodell betrachten wir auch Parallelverarbeitung; fur umfangreiche Graphklassen mit zufalligen Kantengewichten wie z.B. dunne Zufallsgraphen oder Modelle fur das WWW, Telefonanrufe oder soziale Netzwerke stellen wir die derzeit schnellsten parallelen SSSP Algorithmen mit durchschnittlich linearer Arbeit vor.

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