Module org.jgrapht.core
Package org.jgrapht.alg.shortestpath

Class DeltaSteppingShortestPath<V,E>

java.lang.Object
org.jgrapht.alg.shortestpath.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 ThreadPoolExecutor which is supplied to this algorithm from outside.

Since:
January 2018
Author:
Semen Chudakov

  • Field Details

    • 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:

    • 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:

    • 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:

    • graph

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

  • Constructor Details

    • DeltaSteppingShortestPath

      public DeltaSteppingShortestPath(Graph<V,E> graph, ThreadPoolExecutor executor)
      Constructs a new instance of the algorithm for a given graph and executor. It is up to a user of this algorithm to handle the creation and termination of the provided executor. For utility methods to manage a ThreadPoolExecutor see ConcurrencyUtil.
      Parameters:
      graph - graph
      executor - executor which will be used for parallelization

    • DeltaSteppingShortestPath

      public DeltaSteppingShortestPath(Graph<V,E> graph, ThreadPoolExecutor executor, Comparator<V> vertexComparator)
      Constructs a new instance of the algorithm for a given graph, executor and vertexComparator. It is up to a user of this algorithm to handle the creation and termination of the provided executor. For utility methods to manage a ThreadPoolExecutor see ConcurrencyUtil. vertexComparator provided via this constructor is used to create instances of ConcurrentSkipListSet for the individual buckets. This gives a gives a small performance benefit for shortest paths computation.
      Parameters:
      graph - graph
      executor - executor which will be used for parallelization
      vertexComparator - comparator for vertices of the graph

    • DeltaSteppingShortestPath

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

    • DeltaSteppingShortestPath

      public DeltaSteppingShortestPath(Graph<V,E> graph, double delta, ThreadPoolExecutor executor)
      Constructs a new instance of the algorithm for a given graph, delta and executor. It is up to a user of this algorithm to handle the creation and termination of the provided executor. For utility methods to manage a ThreadPoolExecutor see ConcurrencyUtil.
      Parameters:
      graph - the graph
      delta - bucket width
      executor - executor which will be used for parallelization

    • DeltaSteppingShortestPath

      public DeltaSteppingShortestPath(Graph<V,E> graph, double delta, ThreadPoolExecutor executor, Comparator<V> vertexComparator)
      Constructs a new instance of the algorithm for a given graph, delta, executor and vertexComparator. It is up to a user of this algorithm to handle the creation and termination of the provided executor. For utility methods to manage a ThreadPoolExecutor see ConcurrencyUtil. vertexComparator provided via this constructor is used to create instances of ConcurrentSkipListSet for the individual buckets. This gives a gives a small performance benefit for shortest paths computation.
      Parameters:
      graph - the graph
      delta - bucket width
      executor - executor which will be used for parallelization
      vertexComparator - comparator for vertices of the graph

    • DeltaSteppingShortestPath

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

    • DeltaSteppingShortestPath

      @Deprecated public DeltaSteppingShortestPath(Graph<V,E> graph, double delta, int parallelism)
      Deprecated.
      replaced with DeltaSteppingShortestPath(Graph, double, ThreadPoolExecutor)
      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 Details

    • 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

    • getPaths

      public ShortestPathAlgorithm.SingleSourcePaths<V,E> getPaths(V source)
      Compute all shortest paths starting from a single source vertex.
      Specified by:
      getPaths in interface ShortestPathAlgorithm<V,E>
      Parameters:
      source - the source vertex
      Returns:
      the shortest paths

    • 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