Class EppsteinShortestPathIterator<V,​E>

  • Type Parameters:
    V - the graph vertex type
    E - the graph edge type
    All Implemented Interfaces:
    java.util.Iterator<GraphPath<V,​E>>

    public class EppsteinShortestPathIterator<V,​E>
    extends java.lang.Object
    implements java.util.Iterator<GraphPath<V,​E>>
    Iterator over the shortest paths (not required to be simple) between two vertices in a graph sorted by weight.

    This implementation can only be used for directed simple graphs. Also for this iterator to work correctly the graph must not be modified during iteration. Currently there are no means to ensure that, nor to fail-fast. The results of such modifications are undefined.

    First the shortest paths tree in the edge reversed graph starting at sink is built. Thus we get distances $d(v)$ from every vertex $v$ to sink. We then define a sidetrack edge to be an edge, which is not in the shortest paths tree. The key observation is that every path between the source and the sink can be solely determined by a sub-sequence of its edges which are sidetracks.

    Let $d(v)$ be the distance from $v$ to sink and $w()$ be the weight function for edges in graph. If $e$ connects a pair of vertices $(u, w)$, the $\delta(e)$ is defined as $w(e)+d(w)-d(u)$. Intuitively, $\delta(e)$ measures how much distance is lost by being “sidetracked” along $e$ instead of taking a shortest path to sink.

    The idea of the algorithm is to build a heap of sidetracks. This heap can be then traversed with breadth-first search in order to retrieve the implicit representations of the paths between source and sink.

    This implementation has several improvements in comparison to the original description in the article:

    1. An outgoing edge of vertex $v$ is inserted in the paths graph iff it is reachable from the source.
    2. The cross edges in the paths graph are added only for those vertices which are reachable from the root vertex.
    3. Weights of the edges in the paths graph are mot maintained explicitly, because they are computed during its traversal.
    • Field Detail

      • graph

        private final Graph<V,​E> graph
        Underlying graph.
      • source

        private final V source
        Source vertex.
      • sink

        private final V sink
        Sink vertex.
      • distanceAndPredecessorMap

        private java.util.Map<V,​Pair<java.lang.Double,​E>> distanceAndPredecessorMap
        Shortest paths tree in the edge reversed graph graph rooted at sink.
    • Constructor Detail

      • EppsteinShortestPathIterator

        public EppsteinShortestPathIterator​(Graph<V,​E> graph,
                                            V source,
                                            V sink)
        Constructs an instance of the algorithm for the given graph, source and sink.
        Parameters:
        graph - graph
        source - source vertex
        sink - sink vertex
    • Method Detail

      • hasNext

        public boolean hasNext()
        Specified by:
        hasNext in interface java.util.Iterator<V>
      • next

        public GraphPath<V,​E> next()
        Specified by:
        next in interface java.util.Iterator<V>
      • buildPathsGraph

        private void buildPathsGraph()
        Guides the building process of the paths graph. The process is divided into three stages. First the D(g) is constructed, then cross edges are added and finally the root vertex is created.
      • buildDGraph

        private void buildDGraph()
        If the graph is denoted by $G$, then for every vertex $v$ reachable from source in $G$ $D(G)$ contains balanced heaps of all outroots, which corresponds to vertices on the path from $v$ to sink. If there are no sidetracks on the path from $v$ to sink, the value $null$ is stored. An outroot is connected to its rest heap if the corresponding vertex has more than one sidetrack.
      • addCrossEdges

        private void addCrossEdges()
        Adds cross edges for every vertex $v$ reachable from the root of balanced heap of source in the paths graph. If a sidetrack, which corresponds to $v$ connects some pair of vertices $(u,w)$, a cross edge from $v$ to the root of the balanced heap of $w$ is added.
      • addPathGraphRoot

        private void addPathGraphRoot()
        Creates the root vertex $r$ of the paths graph and connects it to the root of the balanced heap of source.
      • insertVertex

        private void insertVertex​(V v,
                                  EppsteinShortestPathIterator.PathsGraphVertex predecessorHeap)
        Guides the process of adding the sidetracks of v to the paths graph. First receives the outroot and root of the rest heap of v by calling getOutrootAndRestHeapRoot(Object). If the outroot if $null$ maps $v$ to predecessorHeap in hMapping. Otherwise inserts outroot of $v$ in the balanced heap rooted at predecessorHeap and links it to the received rest heap root.
        Parameters:
        v - vertex
        predecessorHeap - balanced heap root
      • delta

        private double delta​(E e)
        Calculates the $\delta(e)$ value for a given edge e.
        Parameters:
        e - edge
        Returns:
        value of $\delta(e)$