177 research outputs found
Efficient Parallel Implementation of the Ramalingam Decremental Algorithm for Updating the Shortest Paths Subgraph
We propose an efficient parallel implementation of the Ramalingam algorithm for dynamic updating the shortest paths subgraph of a directed weighted graph with a sink after deletion of an edge. To this end, a model of associative (content addressable) parallel systems with vertical processing (the STAR-machine) is used. On the STAR-machine, the Ramalingam decremental algorithm for dynamic updating the shortest paths subgraph is represented as the main procedure DeleteArc that uses a group of auxiliary procedures. We provide the DeleteArc procedure along with the auxiliary procedures, prove correctness of these procedures and evaluate the time complexity. We also consider an example of implementing the DeleteArc procedure on the STAR-machine
Shortest Paths Avoiding Forbidden Subpaths
In this paper we study a variant of the shortest path problem in graphs:
given a weighted graph G and vertices s and t, and given a set X of forbidden
paths in G, find a shortest s-t path P such that no path in X is a subpath of
P. Path P is allowed to repeat vertices and edges. We call each path in X an
exception, and our desired path a shortest exception-avoiding path. We
formulate a new version of the problem where the algorithm has no a priori
knowledge of X, and finds out about an exception x in X only when a path
containing x fails. This situation arises in computing shortest paths in
optical networks. We give an algorithm that finds a shortest exception avoiding
path in time polynomial in |G| and |X|. The main idea is to run Dijkstra's
algorithm incrementally after replicating vertices when an exception is
discovered.Comment: 12 pages, 2 figures. Fixed a few typos, rephrased a few sentences,
and used the STACS styl
Negative Cycle Detection in Dynamic Graphs
We examine the problem of detecting negative cycles in a
dynamic graph, which is a fundamental problem that arises in
electronic design automation and systems theory. Previous
approaches used for this have tried to modify Dijkstra's
algorithm since it is the fastest known Single-Source
Shortest Path algorithm.
We introduce the concept of {\em batch mode} negative cycle
detection, in which a graph changes over time, and negative
cycle detection needs to be done periodically. Such
scenarios arise, for example, during iterative design space
exploration for hardware and software synthesis. We present
an algorithm for this problem, based on the Bellman-Ford
algorithm, which outperforms previous approaches.
We also show that this technique leads to very fast
algorithms for the computation of the maximum-cycle mean
(MCM) of a graph, especially for a certain form of {\em
sparse graph}. Such sparseness often occurs in practice, as
demonstrated for example by the ISCAS 89/93 benchmarks.
We present experimental results that demonstrate the
advantages of our batch-processing techniques, and
illustrate their application to design-space exploration by
developing an automated local-search technique for
multiple-voltage scheduling of iterative data-flow graphs.
(Also cross-referenced as UMIACS-TR-99-59
New Algorithms and Hardness for Incremental Single-Source Shortest Paths in Directed Graphs
In the dynamic Single-Source Shortest Paths (SSSP) problem, we are given a
graph subject to edge insertions and deletions and a source vertex
, and the goal is to maintain the distance for all .
Fine-grained complexity has provided strong lower bounds for exact partially
dynamic SSSP and approximate fully dynamic SSSP [ESA'04, FOCS'14, STOC'15].
Thus much focus has been directed towards finding efficient partially dynamic
-approximate SSSP algorithms [STOC'14, ICALP'15, SODA'14,
FOCS'14, STOC'16, SODA'17, ICALP'17, ICALP'19, STOC'19, SODA'20, SODA'20].
Despite this rich literature, for directed graphs there are no known
deterministic algorithms for -approximate dynamic SSSP that
perform better than the classic ES-tree [JACM'81]. We present the first such
algorithm.
We present a \emph{deterministic} data structure for incremental SSSP in
weighted digraphs with total update time which is
near-optimal for very dense graphs; here is the ratio of the largest weight
in the graph to the smallest. Our algorithm also improves over the best known
partially dynamic \emph{randomized} algorithm for directed SSSP by Henzinger et
al. [STOC'14, ICALP'15] if .
We also provide improved conditional lower bounds. Henzinger et al. [STOC'15]
showed that under the OMv Hypothesis, the partially dynamic exact -
Shortest Path problem in undirected graphs requires amortized update or query
time , given polynomial preprocessing time. Under a hypothesis
about finding Cliques, we improve the update and query lower bound for
algorithms with polynomial preprocessing time to . Further,
under the -Cycle hypothesis, we show that any partially dynamic SSSP
algorithm with preprocessing time requires amortized update
or query time
Recent Advances in Fully Dynamic Graph Algorithms
In recent years, significant advances have been made in the design and
analysis of fully dynamic algorithms. However, these theoretical results have
received very little attention from the practical perspective. Few of the
algorithms are implemented and tested on real datasets, and their practical
potential is far from understood. Here, we present a quick reference guide to
recent engineering and theory results in the area of fully dynamic graph
algorithms
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 fĂŒr das KĂŒrzeste-Wege Problem (Single-Source Shortest-Paths, SSSP) auf Graphen mit n Knoten, M Kanten und nichtnegativen zufĂ€lligen Kantengewichten. Alle bisherigen SSSP Algorithmen benötigen auf gerichteten Graphen superlineare Zeit. Wir stellen den ersten SSSP Algorithmus vor, der auf beliebigen gerichteten Graphen mit zufĂ€lligen Kantengewichten eine beweisbar lineare average-case-KomplexitĂ€t
O(n+m)aufweist. Sind die Kantengewichte unabhĂ€ngig, so wird die lineare Zeitschranke auch mit hoher Wahrscheinlichkeit eingehalten. AuĂerdem impliziert unser Ergebnis verbesserte average-case-Schranken fĂŒr das All-Pairs Shortest-Paths (APSP) Problem auf dĂŒnnen Graphen und liefert die erste theoretische average-case-Analyse fĂŒr die "Approximate Bucket Implementierung" von Dijkstras SSSP Algorithmus (ABI-Dijkstra). Weiterhin fĂŒhren wir konstruktive Existenzbeweise fĂŒr Graphklassen mit zufĂ€lligen 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; fĂŒr umfangreiche Graphklassen mit zufĂ€lligen Kantengewichten wie z.B. dĂŒnne Zufallsgraphen oder Modelle fĂŒr das WWW, Telefonanrufe oder soziale Netzwerke stellen wir die derzeit schnellsten parallelen SSSP Algorithmen mit durchschnittlich linearer Arbeit vor
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