Package org.jgrapht.alg.shortestpath

Class DeltaSteppingShortestPath<V,​E>

  • Type Parameters:
    V - the graph vertex type
    E - the graph edge type
    All Implemented Interfaces:
    ShortestPathAlgorithm<V,​E>
    public class DeltaSteppingShortestPath<V,​E>
extends Object
    Parallel implementation of a single-source shortest path algorithm: the delta-stepping algorithm. The algorithm computes single source shortest paths in a graphs with non-negative edge weights. When using multiple threads, this implementation typically outperforms DijkstraShortestPath and BellmanFordShortestPath.

    The delta-stepping algorithm is described in the paper: U. Meyer, P. Sanders, $\Delta$-stepping: a parallelizable shortest path algorithm, Journal of Algorithms, Volume 49, Issue 1, 2003, Pages 114-152, ISSN 0196-6774.

    The $\Delta$-stepping algorithm takes as input a weighted graph $G(V,E)$, a source node $s$ and a parameter $\Delta > 0$. Let $tent[v]$ be the best known shortest distance from $s$ to vertex $v\in V$. At the start of the algorithm, $tent[s]=0$, $tent[v]=\infty$ for $v\in V\setminus \{s\}$. The algorithm partitions vertices in a series of buckets $B=(B_0, B_1, B_2, \dots)$, where a vertex $v\in V$ is placed in bucket $B_{\lfloor\frac{tent[v]}{\Delta}\rfloor}$. During the execution of the algorithm, vertices in bucket $B_i$, for $i=0,1,2,\dots$, are removed one-by-one. For each removed vertex $v$, and for all its outgoing edges $(v,w)$, the algorithm checks whether $tent[v]+c(v,w) < tent[w]$. If so, $w$ is removed from its current bucket, $tent[w]$ is updated ($tent[w]=tent[v]+c(v,w)$), and $w$ is placed into bucket $B_{\lfloor\frac{tent[w]}{\Delta}\rfloor}$. Parallelism is achieved by processing all vertices belonging to the same bucket concurrently. The algorithm terminates when all buckets are empty. At this stage the array $tent$ contains the minimal cost from $s$ to every vertex $v \in V$. For a more detailed description of the algorithm, refer to the aforementioned paper.

    For a given graph $G(V,E)$ and parameter $\Delta$, let a $\Delta$-path be a path of total weight at most $\Delta$ with no repeated edges. The time complexity of the algorithm is $O(\frac{(|V| + |E| + n_{\Delta} + m_{\Delta})}{p} + \frac{L}{\Delta}\cdot d\cdot l_{\Delta}\cdot \log n)$, where

    • $n_{\Delta}$ - number of vertex pairs $(u,v)$, where $u$ and $v$ are connected by some $\Delta$-path.
    • $m_{\Delta}$ - number of vertex triples $(u,v^{\prime},v)$, where $u$ and $v^{\prime}$ are connected by some $\Delta$-path and edge $(v^{\prime},v)$ has weight at most $\Delta$.
    • $L$ - maximum weight of a shortest path from selected source to any sink.
    • $d$ - maximum vertex degree.
    • $l_{\Delta}$ - maximum number of edges in a $\Delta$-path $+1$.

    For parallelization, this implementation relies on the ExecutorService.

    Since:
    January 2018
    Author:
    Semen Chudakov

    • Field Detail

      • GRAPH_CONTAINS_A_NEGATIVE_WEIGHT_CYCLE

        protected static final String GRAPH_CONTAINS_A_NEGATIVE_WEIGHT_CYCLE
        Error message for reporting the existence of a negative-weight cycle.
        See Also:
        Constant Field Values

      • GRAPH_MUST_CONTAIN_THE_SOURCE_VERTEX

        protected static final String GRAPH_MUST_CONTAIN_THE_SOURCE_VERTEX
        Error message for reporting that a source vertex is missing.
        See Also:
        Constant Field Values

      • GRAPH_MUST_CONTAIN_THE_SINK_VERTEX

        protected static final String GRAPH_MUST_CONTAIN_THE_SINK_VERTEX
        Error message for reporting that a sink vertex is missing.
        See Also:
        Constant Field Values

      • graph

        protected final Graph<V,​E> graph
        The underlying graph.

    • Constructor Detail

      • DeltaSteppingShortestPath

        public DeltaSteppingShortestPath​(Graph<V,​E> graph)
        Constructs a new instance of the algorithm for a given graph.
        Parameters:
        graph - graph

      • DeltaSteppingShortestPath

        public DeltaSteppingShortestPath​(Graph<V,​E> graph,
                                 double delta)
        Constructs a new instance of the algorithm for a given graph and delta.
        Parameters:
        graph - the graph
        delta - bucket width

      • DeltaSteppingShortestPath

        public DeltaSteppingShortestPath​(Graph<V,​E> graph,
                                 int parallelism)
        Constructs a new instance of the algorithm for a given graph and parallelism.
        Parameters:
        graph - the graph
        parallelism - maximum number of threads used in the computations

      • DeltaSteppingShortestPath

        public DeltaSteppingShortestPath​(Graph<V,​E> graph,
                                 double delta,
                                 int parallelism)
        Constructs a new instance of the algorithm for a given graph, delta, parallelism. If delta is $0.0$ it will be computed during the algorithm execution. In general if the value of $\frac{maximum edge weight}{maximum outdegree}$ is known beforehand, it is preferable to specify it via this constructor, because processing the whole graph to compute this value may significantly slow down the algorithm.
        Parameters:
        graph - the graph
        delta - bucket width
        parallelism - maximum number of threads used in the computations

    • Method Detail

      • getPath

        public GraphPath<V,​E> getPath​(V source,
                                    V sink)
        Get a shortest path from a source vertex to a sink vertex.
        Parameters:
        source - the source vertex
        sink - the target vertex
        Returns:
        a shortest path or null if no path exists

      • getPathWeight

        public double getPathWeight​(V source,
                            V sink)
        Get the weight of the shortest path from a source vertex to a sink vertex. Returns Double.POSITIVE_INFINITY if no path exists.
        Specified by:
        getPathWeight in interface ShortestPathAlgorithm<V,​E>
        Parameters:
        source - the source vertex
        sink - the sink vertex
        Returns:
        the weight of the shortest path from a source vertex to a sink vertex, or Double.POSITIVE_INFINITY if no path exists

      • createEmptyPath

        protected final GraphPath<V,​E> createEmptyPath​(V source,
                                                     V sink)
        Create an empty path. Returns null if the source vertex is different than the target vertex.
        Parameters:
        source - the source vertex
        sink - the sink vertex
        Returns:
        an empty path or null null if the source vertex is different than the target vertex