y.algo
Class ShortestPaths

java.lang.Object
  y.algo.ShortestPaths

public class ShortestPaths
extends java.lang.Object

This class provides diverse algorithms and helper methods for solving the shortest path problem on weighted graphs.

Definitions

Given a weighted directed/undirected graph:

• The shortest path problem is the problem of finding a shortest path between a source node s and a target node t such that the sum of the edge costs is minimized.
• The k-shortest path problem is the problem of finding k shortest paths between a source node s and a target node t such that the sum of the edge costs is minimized.
• The single-source shortest path problem is the problem of finding shortest paths from a source node s to all other nodes such that the sum of the edge costs is minimized.
• The single-source single-sink shortest path problem is the problem of finding shortest paths from a source node s to a target node t such that the sum of the edge costs is minimized.
• The all-pairs shortest path problem is the problem of finding shortest paths between every pair of nodes such that the sum of the edge costs is minimized.

Method Summary

static boolean	acyclic(Graph graph, Node s, DataProvider cost, NodeMap dist, NodeMap pred) Like acyclic(Graph, Node, double[], double[]) but uses NodeMaps and DataProviders instead of arrays.
static boolean	acyclic(Graph graph, Node s, double[] cost, double[] dist) Solves the single-source shortest path problem for acyclic directed graphs.
static boolean	acyclic(Graph graph, Node s, double[] cost, double[] dist, Edge[] pred) Like acyclic(Graph, Node, double[], double[]) but, additionally, this method yields the path edges of each calculated shortest path.
static boolean	allPairs(Graph graph, boolean directed, double[] cost, double[][] dist) This method solves the all-pairs shortest path problem for graphs with arbitrary edge costs.
static EdgeList	aStar(Graph graph, Node s, Node t, boolean directed, DataProvider cost, DataProvider heuristicCost) Solves the single-source single-sink shortest path problem for arbitrary graphs using an implementation of the well-known a-star algorithm (A*).
static double	aStar(Graph graph, Node s, Node t, boolean directed, DataProvider cost, DataProvider heuristicCost, NodeMap pred) Solves the single-source single-sink shortest path problem for arbitrary graphs using an implementation of the well-known a-star algorithm (A*).
static EdgeList	aStar(Graph graph, Node s, Node t, boolean directed, double[] cost, double[] heuristicCost) Solves the single-source single-sink shortest path problem for arbitrary graphs using an implementation of the well-known a-star algorithm (A*).
static double	aStar(Graph graph, Node s, Node t, boolean directed, double[] cost, double[] heuristicCost, Edge[] pred) Solves the single-source single-sink shortest path problem for arbitrary graphs using an implementation of the well-known a-star algorithm (A*).
static boolean	bellmanFord(Graph graph, Node s, boolean directed, DataProvider cost, NodeMap dist, NodeMap pred) Like bellmanFord(Graph, Node, boolean, double[], double[], Edge[]) but uses NodeMaps and DataProviders instead of arrays.
static boolean	bellmanFord(Graph graph, Node s, boolean directed, double[] cost, double[] dist) Solves the single-source shortest path problem for arbitrary graphs.
static boolean	bellmanFord(Graph graph, Node s, boolean directed, double[] cost, double[] dist, Edge[] pred) Like bellmanFord(Graph, Node, boolean, double[], double[]) but, additionally, this method yields the path edges of each calculated shortest path.
static EdgeList	constructEdgePath(Node s, Node t, DataProvider pred) Like constructEdgePath(Node, Node, Edge[]) but the path edges are given by a DataProvider.
static EdgeList	constructEdgePath(Node s, Node t, Edge[] pred) Convenience method that constructs an explicit path of edges from the result returned by one of the shortest paths methods defined in this class.
static NodeList	constructNodePath(Node s, Node t, DataProvider pred) Like constructNodePath(Node, Node, Edge[]) but the path edges are given by a DataProvider.
static NodeList	constructNodePath(Node s, Node t, Edge[] pred) Convenience method that constructs an explicit path of nodes from the result returned by one of the shortest paths methods defined in this class.
static void	dijkstra(Graph graph, Node s, boolean directed, DataProvider cost, NodeMap dist, NodeMap pred) Like dijkstra(Graph, Node, boolean, double[], double[]) but uses NodeMaps and DataProviders instead of arrays.
static void	dijkstra(Graph graph, Node s, boolean directed, double[] cost, double[] dist) Solves the single-source shortest path problem for arbitrary graphs.
static void	dijkstra(Graph graph, Node s, boolean directed, double[] cost, double[] dist, Edge[] pred) Like dijkstra(Graph, Node, boolean, double[], double[]) but, additionally, this method yields the path edges of each calculated shortest path.
static void	findShortestUniformPaths(Graph graph, Node start, DataProvider targetMap, boolean directed, int maxLength, EdgeList pathEdges, NodeList pathNodes) Finds all nodes and edges that belong to a shortest path from a start node to a set of target nodes in the graph not farther away than a given distance.
static void	findShortestUniformPaths(Graph graph, Node start, Node end, boolean directed, EdgeMap pathMap) Marks all edges that belong to a shortest path from start node to target node.
static YList	kShortestPaths(Graph graph, DataProvider costDP, Node start, Node end, int k) This method finds the k shortest paths connecting a pair of nodes in a directed graph with non-negative edge costs.
static YCursor	kShortestPathsCursor(Graph graph, DataProvider costDP, Node start, Node end, int k) A variant of kShortestPaths(Graph, DataProvider, Node, Node, int) that returns the result as a special cursor that calculates the next path in the sequence only when needed.
static EdgeList[]	shortestPair(Graph graph, Node source, Node target, boolean directed, DataProvider costDP) Returns two edge-disjoint paths in a non-negatively weighted directed graph, such that both paths connect nodes s and t and have minimum total length.
static boolean	singleSource(Graph graph, Node s, boolean directed, DataProvider cost, NodeMap dist, NodeMap pred) Like singleSource(Graph, Node, boolean, double[], double[], Edge[]) but uses NodeMaps and DataProviders instead of arrays.
static boolean	singleSource(Graph graph, Node s, boolean directed, double[] cost, double[] dist) This method solves the single-source shortest path problem for arbitrary graphs.
static boolean	singleSource(Graph graph, Node s, boolean directed, double[] cost, double[] dist, Edge[] pred) Like singleSource(Graph, Node, boolean, double[], double[]) but, additionally, this method yields the path edges of each calculated shortest path.
static EdgeList	singleSourceSingleSink(Graph graph, Node s, Node t, boolean directed, DataProvider cost) Similar to singleSourceSingleSink(Graph, Node, Node, boolean, DataProvider, NodeMap) but instead of returning the shortest distance between the source and sink the actual shortest edge path between these nodes will be returned.
static double	singleSourceSingleSink(Graph graph, Node s, Node t, boolean directed, DataProvider cost, NodeMap pred) Like singleSourceSingleSink(Graph, Node, Node, boolean, double[], Edge[]) but uses NodeMaps and DataProviders instead of arrays.
static EdgeList	singleSourceSingleSink(Graph graph, Node s, Node t, boolean directed, double[] cost) Similar to singleSourceSingleSink(Graph, Node, Node, boolean, double[], Edge[]) but instead of returning the shortest distance between the source and sink the actual shortest edge path between these nodes will be returned.
static double	singleSourceSingleSink(Graph graph, Node s, Node t, boolean directed, double[] cost, Edge[] pred) This method solves the single-source single-sink shortest path problem for arbitrary graphs.
static void	uniform(Graph graph, Node s, boolean directed, double[] dist) Solves the single-source shortest path problem for arbitrary graphs in which each edge has a uniform cost of 1.0.
static void	uniform(Graph graph, Node s, boolean directed, double[] dist, Edge[] pred) Like uniform(Graph, Node, boolean, double[]) but, additionally, yields the edges of each calculated shortest path.
static void	uniform(Graph graph, Node s, boolean directed, NodeMap dist, NodeMap pred) Like uniform(Graph, Node, boolean, double[]) but uses NodeMaps instead of arrays.
static double[]	uniformCost(Graph graph) Convenience method that returns an array containing uniform edge costs of 1.0 for each edge of the given graph.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Detail

uniform

public static void uniform(Graph graph,
                           Node s,
                           boolean directed,
                           double[] dist)

Solves the single-source shortest path problem for arbitrary graphs in which each edge has a uniform cost of 1.0.

This method yields the shortest distance from a given node s to all other nodes.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Precondition:

dist.length == graph.N()

Complexity:

O(graph.N() + graph.E())

Parameters:

graph - the input graph

s - the node from which the shortest path search starts

directed - true if the graph should be considered as directed, false otherwise

dist - an array of values that will be filled during the execution and returns the shortest distance from node s to all other nodes. The distance from s to v is dist[v.index()]. If there is no path from s to v, then dist[v.index()] == Double.POSITIVE_INFINITY.

uniform

public static void uniform(Graph graph,
                           Node s,
                           boolean directed,
                           double[] dist,
                           Edge[] pred)

Like uniform(Graph, Node, boolean, double[]) but, additionally, yields the edges of each calculated shortest path.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Precondition:

pred.length == graph.N()

Complexity:

O(graph.N() + graph.E())

Parameters:

graph - the input graph

s - the node from which the shortest path search starts

directed - true if the graph should be considered as directed, false otherwise

dist - an array of values that will be filled during the execution and returns the shortest distance from node s to all other nodes. The distance from s to v is dist[v.index()]. If there is no path from s to v, then dist[v.index()] == Double.POSITIVE_INFINITY.

pred - an array of Edges that will be filled during the execution and returns for each node t the shortest path edge pred[t.index()] which is the last edge on the shortest path from s to t. If t == s or if there is no shortest path from s to t, then pred[t.index()] == null.

See Also:

uniform(Graph, Node, boolean, double[]), constructNodePath(Node, Node, Edge[]), constructEdgePath(Node, Node, Edge[])

uniform

public static void uniform(Graph graph,
                           Node s,
                           boolean directed,
                           NodeMap dist,
                           NodeMap pred)

Like uniform(Graph, Node, boolean, double[]) but uses NodeMaps instead of arrays.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Precondition:

pred.length == graph.N()

Complexity:

O(graph.N() + graph.E())

Parameters:

graph - the input graph

s - the node from which the shortest path search starts

directed - true if the graph should be considered as directed, false otherwise

dist - the NodeMap that will be filled during the execution and returns a double value (shortest distance) from node s to all other nodes or Double.POSITIVE_INFINITY if no such paths exist

pred - the NodeMap that will be filled during the execution and returns for each node t the last edge on the shortest path from s to t or null if t == s or no shortest path from s to t exists

See Also:

uniform(Graph, Node, boolean, double[]), constructNodePath(Node, Node, Edge[]), constructEdgePath(Node, Node, Edge[])

acyclic

public static boolean acyclic(Graph graph,
                              Node s,
                              double[] cost,
                              double[] dist)

Solves the single-source shortest path problem for acyclic directed graphs.

Each edge is associated with an arbitrary double value that represents the cost of that edge.

This method yields the shortest distance from a given node s to all other nodes.

Preconditions:

The graph must be acyclic.

cost.length == graph.E()

dist.length == graph.N()

Complexity:

O(graph.N() + graph.E())

Parameters:

graph - the input graph

s - the node from which the shortest path search starts

cost - an array of double values that returns the costs for traversing each edge; edge e has cost cost[e.index()]

dist - an array of values that will be filled during the execution and returns the shortest distance from node s to all other nodes. The distance from s to v is dist[v.index()]. If there is no path from s to v, then dist[v.index()] == Double.POSITIVE_INFINITY.

Returns:

true if the input graph is acyclic, false otherwise

See Also:

acyclic(Graph, Node, double[], double[], Edge[]), acyclic(Graph, Node, DataProvider, NodeMap, NodeMap)

acyclic

public static boolean acyclic(Graph graph,
                              Node s,
                              double[] cost,
                              double[] dist,
                              Edge[] pred)

Like acyclic(Graph, Node, double[], double[]) but, additionally, this method yields the path edges of each calculated shortest path.

Each edge is associated with an arbitrary double value that represents the cost of that edge.

This method yields the shortest distance from a given node s to all other nodes.

Precondition:

pred.length == graph.N()

Parameters:

graph - the input graph

s - the node from which the shortest path search starts

cost - an array of double values that returns the costs for traversing each edge; edge e has cost cost[e.index()]

dist - an array of values that will be filled during the execution and returns the shortest distance from node s to all other nodes. The distance from s to v is dist[v.index()]. If there is no path from s to v, then dist[v.index()] == Double.POSITIVE_INFINITY.

pred - an array of Edges that will be filled during the execution and returns for each node t the shortest path edge pred[t.index()] which is the last edge on the shortest path from s to t. If t == s or if there is no shortest path from s to t, then pred[t.index()] == null.

Returns:

true if the input graph is acyclic, false otherwise

See Also:

acyclic(Graph, Node, double[], double[]), acyclic(Graph, Node, double[], double[], Edge[]), constructNodePath(Node, Node, Edge[]), constructEdgePath(Node, Node, Edge[])

acyclic

public static boolean acyclic(Graph graph,
                              Node s,
                              DataProvider cost,
                              NodeMap dist,
                              NodeMap pred)

Like acyclic(Graph, Node, double[], double[]) but uses NodeMaps and DataProviders instead of arrays.

Each edge is associated with an arbitrary double value that represents the cost of that edge.

This method yields the shortest distance from a given node s to all other nodes.

Precondition:

pred.length == graph.N()

Parameters:

graph - the input graph

s - the node from which the shortest path search starts

cost - the DataProvider that returns the double value (cost) for traversing each edge

dist - the NodeMap that will be filled during the execution and returns a double value (shortest distance) from node s to all other nodes or Double.POSITIVE_INFINITY if no such paths exist

pred - the NodeMap that will be filled during the execution and returns for each node t the last edge on the shortest path from s to t or null if t == s or no shortest path from s to t exists

Returns:

true if the input graph is acyclic, false otherwise

See Also:

acyclic(Graph, Node, double[], double[]), acyclic(Graph, Node, double[], double[], Edge[])

aStar

public static EdgeList aStar(Graph graph,
                             Node s,
                             Node t,
                             boolean directed,
                             double[] cost,
                             double[] heuristicCost)

Solves the single-source single-sink shortest path problem for arbitrary graphs using an implementation of the well-known a-star algorithm (A*).

Each edge is associated with a non-negative double value that represents the cost of the edge. The costs should be non-negative.

Furthermore, the algorithm uses so-called heuristic costs for each node. The heuristic cost for a node n is the estimated cost to reach the sink node t from n. The heuristic has a crucial impact on the performance of the shortest path search:

• If the heuristic is 'good', i.e., heuristic costs are close to the actual costs, then the algorithm is able to avoid unprofitable search paths and move faster towards the target. Using a good heuristic, this algorithm outperforms other shortest-path algorithms.
• The heuristic must be admissible, which means that it must never overestimate the actual cost to reach the target node. Otherwise, it is not guaranteed that the algorithm actually finds the shortest path. Use dijkstra(Graph, Node, Node, boolean, double[], Edge[]) if you have no proper problem-specific heuristic.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Preconditions:

cost.length == graph.E()

heuristicCost.length == graph.N()

Parameters:

graph - the input graph

s - the source node

t - the sink node

directed - true if the graph should be considered as directed, false otherwise

cost - an array of double values that represent the costs for traversing each edge; edge e has cost cost[e.index()]

heuristicCost - an array of double values that represent an estimate to reach the sink node t. The value of heuristicCost[n.index()] is the estimated heuristic cost to reach node t from node n

Returns:

the list of edges constituting the shortest path from the source node s to the sink node t or an empty list if there is no such path

See Also:

aStar(Graph, Node, Node, boolean, DataProvider, DataProvider), aStar(Graph, Node, Node, boolean, double[], double[], Edge[]), aStar(Graph, Node, Node, boolean, DataProvider, DataProvider, NodeMap)

aStar

public static EdgeList aStar(Graph graph,
                             Node s,
                             Node t,
                             boolean directed,
                             DataProvider cost,
                             DataProvider heuristicCost)

Solves the single-source single-sink shortest path problem for arbitrary graphs using an implementation of the well-known a-star algorithm (A*).

This method is like aStar(Graph, Node, Node, boolean, double[], double[]), with the only difference that it takes DataProviders as parameters instead of arrays.

Each edge is associated with a non-negative double value that represents the cost of the edge. The costs should be non-negative.

Furthermore, the algorithm uses so-called heuristic costs for each node. The heuristic cost for a node n is the estimated cost to reach the sink node t from n. The heuristic has a crucial impact on the performance of the shortest path search:

• If the heuristic is 'good', i.e., heuristic costs are close to the actual costs, then the algorithm is able to avoid unprofitable search paths and move faster towards the target. Using a good heuristic, this algorithm outperforms other shortest-path algorithms.
• The heuristic must be admissible, which means that it must never overestimate the actual cost to reach the target node. Otherwise, it is not guaranteed that the algorithm actually finds the shortest path. Use dijkstra(Graph, Node, boolean, double[], double[], Edge[]) if you have no proper problem-specific heuristic.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Parameters:

graph - the input graph

s - the source node

t - the sink node

directed - true if the graph should be considered as directed, false otherwise

cost - the DataProvider that returns the cost for traversing an edge

heuristicCost - the DataProvider that yields for each node the heuristic cost estimate to reach the sink node t starting from the node

Returns:

the list of edges constituting the shortest path from the source node s to the sink node t or an empty list if there is no such path

See Also:

aStar(Graph, Node, Node, boolean, double[], double[]), aStar(Graph, Node, Node, boolean, double[], double[], Edge[]), aStar(Graph, Node, Node, boolean, DataProvider, DataProvider, NodeMap)

aStar

public static double aStar(Graph graph,
                           Node s,
                           Node t,
                           boolean directed,
                           double[] cost,
                           double[] heuristicCost,
                           Edge[] pred)

Solves the single-source single-sink shortest path problem for arbitrary graphs using an implementation of the well-known a-star algorithm (A*).

This method is like aStar(Graph, Node, Node, boolean, double[], double[]), with the difference that it does not return the shortest path but the cost of the shortest path. To construct the actual shortest path use constructEdgePath(Node, Node, Edge[]) or constructNodePath(Node, Node, Edge[]).

Each edge is associated with a non-negative double value that represents the cost of the edge. The costs should be non-negative.

Furthermore, the algorithm uses so-called heuristic costs for each node. The heuristic cost for a node n is the estimated cost to reach the sink node t from n. The heuristic has a crucial impact on the performance of the shortest path search:

• If the heuristic is 'good', i.e., heuristic costs are close to the actual costs, then the algorithm is able to avoid unprofitable search paths and move faster towards the target. Using a good heuristic, this algorithm outperforms other shortest-path algorithms.
• The heuristic must be admissible, which means that it must never overestimate the actual cost to reach the target node. Otherwise, it is not guaranteed that the algorithm actually finds the shortest path. Use dijkstra(Graph, Node, boolean, double[], double[], Edge[]) if you have no proper problem-specific heuristic.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Preconditions:

cost.length == graph.E()

heuristicCost.length == graph.N()

pred.length == graph.N()

Parameters:

graph - the input graph

s - the source node

t - the sink node

directed - true if the graph should be considered as directed, false otherwise

cost - an array of double values that represent the costs for traversing each edge; edge e has cost cost[e.index()]

heuristicCost - an array of double values that represent an estimate to reach the sink node t. The value of heuristicCost[n.index()] is the estimated heuristic cost to reach node t from node n

pred - an array of Edges that will be filled during the execution and holds for each node t the shortest path edge pred[t.index()] which is the last edge on the shortest path from s to t. If t == s or if there is no shortest path from s to t, then pred[t.index()] == null.

Returns:

the shortest distance (cost) between s and t if a path between these two nodes exists or Double.POSITIVE_INFINITY otherwise

See Also:

aStar(Graph, Node, Node, boolean, double[], double[]), aStar(Graph, Node, Node, boolean, DataProvider, DataProvider), aStar(Graph, Node, Node, boolean, DataProvider, DataProvider, NodeMap)

aStar

public static double aStar(Graph graph,
                           Node s,
                           Node t,
                           boolean directed,
                           DataProvider cost,
                           DataProvider heuristicCost,
                           NodeMap pred)

Solves the single-source single-sink shortest path problem for arbitrary graphs using an implementation of the well-known a-star algorithm (A*).

This method is like aStar(Graph, Node, Node, boolean, double[], double[], Edge[]), with the only difference that it takes DataProviders and a NodeMap as input instead of arrays.

Each edge is associated with a non-negative double value that represents the cost of the edge. The costs should be non-negative.

Furthermore, the algorithm uses so-called heuristic costs for each node. The heuristic cost for a node n is the estimated cost to reach the sink node t from n. The heuristic has a crucial impact on the performance of the shortest path search:

• If the heuristic is 'good', i.e., heuristic costs are close to the actual costs, then the algorithm is able to avoid unprofitable search paths and move faster towards the target. Using a good heuristic, this algorithm outperforms other shortest-path algorithms.
• The heuristic must be admissible, which means that it must never overestimate the actual cost to reach the target node. Otherwise, it is not guaranteed that the algorithm actually finds the shortest path. Use dijkstra(Graph, Node, boolean, double[], double[], Edge[]) if you have no proper problem-specific heuristic.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Parameters:

graph - the input graph

s - the source node

t - the sink node

directed - true if the graph should be considered as directed, false otherwise

cost - the DataProvider that returns the cost for traversing an edge

heuristicCost - the DataProvider that yields for each node the heuristic cost estimate to reach the sink node t starting from the node

pred - the map that will be filled during the execution and returns for each node t the last edge on the shortest path from s to t or null if t == s or no shortest path from s to t exists

Returns:

the shortest distance (cost) between s and t if a path between these two nodes exists or Double.POSITIVE_INFINITY otherwise

See Also:

aStar(Graph, Node, Node, boolean, double[], double[]), aStar(Graph, Node, Node, boolean, DataProvider, DataProvider), aStar(Graph, Node, Node, boolean, double[], double[], Edge[])

dijkstra

public static void dijkstra(Graph graph,
                            Node s,
                            boolean directed,
                            double[] cost,
                            double[] dist)

Solves the single-source shortest path problem for arbitrary graphs.

Each edge is associated with a non-negative double value that represents the cost of the edge.

This method yields the shortest distance from a given node s to all other nodes.

The costs should be non-negative.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Preconditions:

cost.length == graph.E()

dist.length == graph.N()

Complexity:

O(graph.E() + graph.N() * log(graph.N())

Parameters:

graph - the input graph

s - the node from which the shortest path search starts

directed - true if the graph should be considered as directed, false otherwise

cost - an array of double values that returns the costs for traversing each edge; edge e has cost cost[e.index()]

dist - an array of values that will be filled during the execution and returns the shortest distance from node s to all other nodes. The distance from s to v is dist[v.index()]. If there is no path from s to v, then dist[v.index()] == Double.POSITIVE_INFINITY.

See Also:

dijkstra(Graph, Node, boolean, DataProvider, NodeMap, NodeMap), dijkstra(Graph, Node, boolean, double[], double[], Edge[])

dijkstra

public static void dijkstra(Graph graph,
                            Node s,
                            boolean directed,
                            double[] cost,
                            double[] dist,
                            Edge[] pred)

Like dijkstra(Graph, Node, boolean, double[], double[]) but, additionally, this method yields the path edges of each calculated shortest path.

Each edge is associated with a non-negative double value that represents the cost of the edge.

This method yields the shortest distance from a given node s to all other nodes.

The costs should be non-negative.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Precondition:

pred.length == graph.N()

Parameters:

graph - the input graph

s - the node from which the shortest path search starts

directed - true if the graph should be considered as directed, false otherwise

cost - an array of double values that returns the costs for traversing each edge; edge e has cost cost[e.index()]

dist - an array of values that will be filled during the execution and returns the shortest distance from node s to all other nodes. The distance from s to v is dist[v.index()]. If there is no path from s to v, then dist[v.index()] == Double.POSITIVE_INFINITY.

pred - an array of Edges that will be filled during the execution and returns for each node t the shortest path edge pred[t.index()] which is the last edge on the shortest path from s to t. If t == s or if there is no shortest path from s to t, then pred[t.index()] == null.

See Also:

constructNodePath(Node, Node, Edge[]), constructEdgePath(Node, Node, Edge[]), dijkstra(Graph, Node, boolean, double[], double[]), dijkstra(Graph, Node, boolean, DataProvider, NodeMap, NodeMap)

dijkstra

public static void dijkstra(Graph graph,
                            Node s,
                            boolean directed,
                            DataProvider cost,
                            NodeMap dist,
                            NodeMap pred)

Like dijkstra(Graph, Node, boolean, double[], double[]) but uses NodeMaps and DataProviders instead of arrays.

Each edge is associated with a non-negative double value that represents the cost of the edge.

This method yields the shortest distance from a given node s to all other nodes.

The costs should be non-negative.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Parameters:

graph - the input graph

s - the node from which the shortest path search starts

directed - true if the graph should be considered as directed, false otherwise

cost - the DataProvider that returns the double value (cost) for traversing each edge

dist - the NodeMap that will be filled during the execution and returns a double value (shortest distance) from node s to all other nodes or Double.POSITIVE_INFINITY if no such paths exist

pred - the NodeMap that will be filled during the execution and returns for each node t the last edge on the shortest path from s to t or null if t == s or no shortest path from s to t exists

See Also:

constructNodePath(Node, Node, Edge[]), constructEdgePath(Node, Node, Edge[]), dijkstra(Graph, Node, boolean, double[], double[]), dijkstra(Graph, Node, boolean, double[], double[], Edge[])

singleSourceSingleSink

public static double singleSourceSingleSink(Graph graph,
                                            Node s,
                                            Node t,
                                            boolean directed,
                                            double[] cost,
                                            Edge[] pred)

This method solves the single-source single-sink shortest path problem for arbitrary graphs.

Each edge is associated with a non-negative double value that represents the cost of the edge.

This method returns the shortest distance from node s to node t. It also returns information to construct the actual path between these two nodes.

The costs should be non-negative.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Preconditions:

cost.length == graph.E()

pred.length == graph.N()

Complexity:

O(graph.E() + graph.N() * log(graph.N())

Parameters:

graph - the input graph

s - the source node

t - the sink node

directed - true if the graph should be considered as directed, false otherwise

cost - an array of double values that returns the costs for traversing each edge; edge e has cost cost[e.index()]

pred - an array of Edges that will be filled during the execution and returns for each node t the shortest path edge pred[t.index()] which is the last edge on the shortest path from s to t. If t == s or if there is no shortest path from s to t, then pred[t.index()] == null.

Returns:

the distance between sand t if a path between these two nodes exists or Double.POSITIVE_INFINITY otherwise

See Also:

constructNodePath(Node, Node, Edge[]), constructEdgePath(Node, Node, Edge[]), singleSourceSingleSink(Graph, Node, Node, boolean, DataProvider), singleSourceSingleSink(Graph, Node, Node, boolean, DataProvider, NodeMap), singleSourceSingleSink(Graph, Node, Node, boolean, double[])

singleSourceSingleSink

public static EdgeList singleSourceSingleSink(Graph graph,
                                              Node s,
                                              Node t,
                                              boolean directed,
                                              double[] cost)

Similar to singleSourceSingleSink(Graph, Node, Node, boolean, double[], Edge[]) but instead of returning the shortest distance between the source and sink the actual shortest edge path between these nodes will be returned.

Each edge is associated with a non-negative double value that represents the cost of the edge.

If the returned path is empty, then there is no path between the nodes.

The costs should be non-negative.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Parameters:

graph - the input graph

s - the source node

t - the sink node

directed - true if the graph should be considered as directed, false otherwise

cost - an array of double values that returns the costs for traversing each edge; edge e has cost cost[e.index()]

Returns:

a shortest path of edges between source and sink

See Also:

singleSourceSingleSink(Graph, Node, Node, boolean, double[], Edge[]), singleSourceSingleSink(Graph, Node, Node, boolean, DataProvider), singleSourceSingleSink(Graph, Node, Node, boolean, DataProvider, NodeMap)

singleSourceSingleSink

public static EdgeList singleSourceSingleSink(Graph graph,
                                              Node s,
                                              Node t,
                                              boolean directed,
                                              DataProvider cost)

Similar to singleSourceSingleSink(Graph, Node, Node, boolean, DataProvider, NodeMap) but instead of returning the shortest distance between the source and sink the actual shortest edge path between these nodes will be returned.

Each edge is associated with a non-negative double value that represents the cost of the edge.

If the returned path is empty, then there is no path between the nodes.

The costs should be non-negative.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Parameters:

graph - the input graph

s - the source node

t - the sink node

directed - true if the graph should be considered as directed, false otherwise

cost - the DataProvider that returns the double value (cost) for traversing each edge

Returns:

a shortest path of edges between source and sink

See Also:

singleSourceSingleSink(Graph, Node, Node, boolean, double[], Edge[]), singleSourceSingleSink(Graph, Node, Node, boolean, DataProvider, NodeMap), singleSourceSingleSink(Graph, Node, Node, boolean, double[])

singleSourceSingleSink

public static double singleSourceSingleSink(Graph graph,
                                            Node s,
                                            Node t,
                                            boolean directed,
                                            DataProvider cost,
                                            NodeMap pred)

Like singleSourceSingleSink(Graph, Node, Node, boolean, double[], Edge[]) but uses NodeMaps and DataProviders instead of arrays.

Each edge is associated with a non-negative double value that represents the cost of the edge.

The costs should be non-negative.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Parameters:

graph - the input graph

s - the source node

t - the sink node

directed - true if the graph should be considered as directed, false otherwise

cost - the DataProvider that returns the double value (cost) for traversing each edge

pred - the NodeMap that will be filled during the execution and returns for each node t the last edge on the shortest path from s to t or null if t == s or no shortest path from s to t exists

Returns:

the distance between sand t if a path between these two nodes exists or Double.POSITIVE_INFINITY otherwise

See Also:

singleSourceSingleSink(Graph, Node, Node, boolean, double[], Edge[]), singleSourceSingleSink(Graph, Node, Node, boolean, DataProvider), singleSourceSingleSink(Graph, Node, Node, boolean, double[])

bellmanFord

public static boolean bellmanFord(Graph graph,
                                  Node s,
                                  boolean directed,
                                  double[] cost,
                                  double[] dist)

Solves the single-source shortest path problem for arbitrary graphs.

Each edge is associated with an arbitrary double value that represents the cost of this edge.

In case the given weighted graph contains no negative cost cycles, this method will yield the shortest distance from a given node s to all other nodes. If, on the other hand, the given graph contains negative-cost cycles, this method will yield no reasonable result which will be indicated by the return value false.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

If the graph is considered undirected, each edge with negative costs induces a negative-cost cycle and, thus, the algorithm doesn't return a valid result. Hence, for undirected graphs, all costs must not be negative.

Preconditions:

cost.length == graph.E()

dist.length == graph.N()

Complexity:

O(graph.E() * min(D, graph.N())), where D is the maximum number of edges in any shortest path

Parameters:

graph - the input graph

s - the node from which the shortest path search starts

directed - true if the graph should be considered as directed, false otherwise

cost - an array of double values that returns the costs for traversing each edge; edge e has cost cost[e.index()]

dist - an array of values that will be filled during the execution and returns the shortest distance from node s to all other nodes. The distance from s to v is dist[v.index()]. If there is no path from s to v, then dist[v.index()] == Double.POSITIVE_INFINITY.

Returns:

true if the algorithm found a valid result, false otherwise (i.e., graph contains negative-cost cycles)

See Also:

bellmanFord(Graph, Node, boolean, DataProvider, NodeMap, NodeMap), bellmanFord(Graph, Node, boolean, double[], double[], Edge[])

bellmanFord

public static boolean bellmanFord(Graph graph,
                                  Node s,
                                  boolean directed,
                                  double[] cost,
                                  double[] dist,
                                  Edge[] pred)

Like bellmanFord(Graph, Node, boolean, double[], double[]) but, additionally, this method yields the path edges of each calculated shortest path.

Each edge is associated with an arbitrary double value that represents the cost of this edge.

In case the given weighted graph contains no negative cost cycles, this method will yield the shortest distance from a given node s to all other nodes. If, on the other hand, the given graph contains negative-cost cycles, this method will yield no reasonable result which will be indicated by the return value false.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

If the graph is considered undirected, each edge with negative costs induces a negative-cost cycle and, thus, the algorithm doesn't return a valid result. Hence, for undirected graphs, all costs must not be negative.

Precondition:

pred.length == graph.N()

Parameters:

graph - the input graph

s - the node from which the shortest path search starts

directed - true if the graph should be considered as directed, false otherwise

cost - an array of double values that returns the costs for traversing each edge; edge e has cost cost[e.index()]

dist - an array of values that will be filled during the execution and returns the shortest distance from node s to all other nodes. The distance from s to v is dist[v.index()]. If there is no path from s to v, then dist[v.index()] == Double.POSITIVE_INFINITY.

pred - an array of Edges that will be filled during the execution and returns for each node t the shortest path edge pred[t.index()] which is the last edge on the shortest path from s to t. If t == s or if there is no shortest path from s to t, then pred[t.index()] == null.

Returns:

true if the algorithm found a valid result, false otherwise (i.e., graph contains negative-cost cycles)

See Also:

constructNodePath(Node, Node, Edge[]), constructEdgePath(Node, Node, Edge[]), bellmanFord(Graph, Node, boolean, double[], double[]), bellmanFord(Graph, Node, boolean, DataProvider, NodeMap, NodeMap)

bellmanFord

public static boolean bellmanFord(Graph graph,
                                  Node s,
                                  boolean directed,
                                  DataProvider cost,
                                  NodeMap dist,
                                  NodeMap pred)

Like bellmanFord(Graph, Node, boolean, double[], double[], Edge[]) but uses NodeMaps and DataProviders instead of arrays.

Each edge is associated with an arbitrary double value that represents the cost of this edge.

In case the given weighted graph contains no negative-cost cycles, this method will yield the shortest distance from a given node s to all other nodes. If, on the other hand, the given graph contains negative-cost cycles, this method will yield no reasonable result which will be indicated by the return value false.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

If the graph is considered undirected, each edge with negative costs induces a negative-cost cycle and, thus, the algorithm doesn't return a valid result. Hence, for undirected graphs, all costs must not be negative.

Parameters:

graph - the input graph

s - the source node

directed - true if the graph should be considered as directed, false otherwise

cost - the DataProvider that returns the double value (cost) for traversing each edge

dist - the NodeMap that will be filled during the execution and returns a double value (shortest distance) from node s to all other nodes or Double.POSITIVE_INFINITY if no such paths exist

pred - the NodeMap that will be filled during the execution and returns for each node t the last edge on the shortest path from s to t or null if t == s or no shortest path from s to t exists

Returns:

true if the algorithm found a valid result, false otherwise (i.e., graph contains negative-cost cycles)

See Also:

bellmanFord(Graph, Node, boolean, double[], double[]), bellmanFord(Graph, Node, boolean, double[], double[], Edge[])

singleSource

public static boolean singleSource(Graph graph,
                                   Node s,
                                   boolean directed,
                                   double[] cost,
                                   double[] dist)

This method solves the single-source shortest path problem for arbitrary graphs.

Depending on the structure of the given graph and the values of the given edge costs it delegates its work to the algorithm with the theoretically best running time. Please note that theory does not necessarily reflect practice.

Preconditions:

cost.length == graph.E()

dist.length == graph.N()

Parameters:

graph - the input graph

s - the node from which the shortest path search starts

directed - true if the graph should be considered as directed, false otherwise

cost - an array of double values that returns the costs for traversing each edge; edge e has cost cost[e.index()]

dist - an array of values that will be filled during the execution and returns the shortest distance from node s to all other nodes. The distance from s to v is dist[v.index()]. If there is no path from s to v, then dist[v.index()] == Double.POSITIVE_INFINITY.

Returns:

true if the weighted graph does not contain a negative-cost cycle, false otherwise

See Also:

singleSource(Graph, Node, boolean, DataProvider, NodeMap, NodeMap), singleSource(Graph, Node, boolean, double[], double[], Edge[])

singleSource

public static boolean singleSource(Graph graph,
                                   Node s,
                                   boolean directed,
                                   double[] cost,
                                   double[] dist,
                                   Edge[] pred)

Like singleSource(Graph, Node, boolean, double[], double[]) but, additionally, this method yields the path edges of each calculated shortest path.

Depending on the structure of the given graph and the values of the given edge costs it delegates its work to the algorithm with the theoretically best running time. Please note that theory does not necessarily reflect practice.

Precondition:

pred.length == graph.N()

Parameters:

graph - the input graph

s - the node from which the shortest path search starts

directed - true if the graph should be considered as directed, false otherwise

cost - an array of double values that returns the costs for traversing each edge; edge e has cost cost[e.index()]

dist - an array of values that will be filled during the execution and returns the shortest distance from node s to all other nodes. The distance from s to v is dist[v.index()]. If there is no path from s to v, then dist[v.index()] == Double.POSITIVE_INFINITY.

pred - an array of Edges that will be filled during the execution and returns for each node t the shortest path edge pred[t.index()] which is the last edge on the shortest path from s to t. If t == s or if there is no shortest path from s to t, then pred[t.index()] == null.

Returns:

true if the weighted graph does not contain a negative-cost cycle, false otherwise

See Also:

constructNodePath(Node, Node, Edge[]), constructEdgePath(Node, Node, Edge[]), singleSource(Graph, Node, boolean, double[], double[]), singleSource(Graph, Node, boolean, DataProvider, NodeMap, NodeMap)

singleSource

public static boolean singleSource(Graph graph,
                                   Node s,
                                   boolean directed,
                                   DataProvider cost,
                                   NodeMap dist,
                                   NodeMap pred)

Like singleSource(Graph, Node, boolean, double[], double[], Edge[]) but uses NodeMaps and DataProviders instead of arrays.

Depending on the structure of the given graph and the values of the given edge costs it delegates its work to the algorithm with the theoretically best running time. Please note that theory does not necessarily reflect practice.

Parameters:

graph - the input graph

s - the source node

directed - true if the graph should be considered as directed, false otherwise

cost - the DataProvider that returns the double value (cost) for traversing each edge

dist - the NodeMap that will be filled during the execution and returns a double value (shortest distance) from node s to all other nodes or Double.POSITIVE_INFINITY if no such paths exist

pred - the NodeMap that will be filled during the execution and returns for each node t the last edge on the shortest path from s to t or null if t == s or no shortest path from s to t exists

Returns:

true if the weighted graph does not contain a negative-cost cycle, false otherwise

See Also:

constructNodePath(Node, Node, Edge[]), constructEdgePath(Node, Node, Edge[]), singleSource(Graph, Node, boolean, double[], double[]), singleSource(Graph, Node, boolean, double[], double[], Edge[])

allPairs

public static boolean allPairs(Graph graph,
                               boolean directed,
                               double[] cost,
                               double[][] dist)

This method solves the all-pairs shortest path problem for graphs with arbitrary edge costs.

If the given graph contains a negative-cost cycle, then false is returned and the values returned in dist are left unspecified.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Preconditions:

cost.length == graph.E()

Dimension of dist: [graph.N()] [graph.N()]]

Complexity:

O(graph.N()) * O(singleSource)

Parameters:

graph - the input graph

directed - true if the graph should be considered as directed, false otherwise

cost - an array of double values that returns the costs for traversing each edge; edge e has cost cost[e.index()]

dist - an array of values that will be filled during the execution and returns the shortest path distances from all pairs of nodes s and t in the graph. The distance from s to t is dist[s.index()][t.index()]. If there is no path from s to t, then dist[s.index()][t.index()] == Double.POSITIVE_INFINITY.

Returns:

true if the given graph does not contain a negative-cost cycle, false otherwise

findShortestUniformPaths

public static void findShortestUniformPaths(Graph graph,
                                            Node start,
                                            Node end,
                                            boolean directed,
                                            EdgeMap pathMap)

Marks all edges that belong to a shortest path from start node to target node.

This method assumes that each edge of the input graph has a cost of 1.0.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Complexity:

O(graph.N() + graph.E())

Parameters:

graph - the input graph

start - the start node

end - the target node

directed - true if the graph should be considered as directed, false otherwise

pathMap - the EdgeMap that will be filled during the execution and returns a boolean value indicating whether or not the edge belongs to a shortest path connecting the two nodes

kShortestPaths

public static YList kShortestPaths(Graph graph,
                                   DataProvider costDP,
                                   Node start,
                                   Node end,
                                   int k)

This method finds the k shortest paths connecting a pair of nodes in a directed graph with non-negative edge costs.

The result will be returned as a list of EdgeList objects.

The cost should be non-negative.

The returned paths are not required to be simple, i.e. they may contain a node or an edge multiple times.

Complexity:

O(graph.E() + graph.N() * log(graph.N()) + k)

Parameters:

graph - the input graph

costDP - the DataProvider that returns a double value (cost) for traversing each edge

start - the given start node

end - the given target node

k - a non-negative integer value

Returns:

a list of EdgeList objects each of which represents a path from start node to target node. The i-th path in the list contains the i-th shortest path between the start and target node.

kShortestPathsCursor

public static YCursor kShortestPathsCursor(Graph graph,
                                           DataProvider costDP,
                                           Node start,
                                           Node end,
                                           int k)

A variant of kShortestPaths(Graph, DataProvider, Node, Node, int) that returns the result as a special cursor that calculates the next path in the sequence only when needed.

The returned cursor only supports the operation YCursor.ok(), YCursor.current() and YCursor.next().

The cost should be non-negative.

The returned paths are not required to be simple, i.e. they may contain a node or an edge multiple times.

Complexity:

O(graph.E() + graph.N() * log(graph.N()) + k)

Parameters:

graph - the input graph

costDP - the DataProvider that returns a double value (cost) for traversing each edge

start - the given start node

end - the given target node

k - a non-negative integer value

Returns:

a cursor that calculates the next path in the sequence only when needed

uniformCost

public static double[] uniformCost(Graph graph)

Convenience method that returns an array containing uniform edge costs of 1.0 for each edge of the given graph.

Parameters:

graph - the input graph

Returns:

an array that contains uniform edge costs of value 1.0 (i.e., for each edge e: cost[e.index()] == 1.0)

constructNodePath

public static NodeList constructNodePath(Node s,
                                         Node t,
                                         Edge[] pred)

Convenience method that constructs an explicit path of nodes from the result returned by one of the shortest paths methods defined in this class.

If there is no path from node s to t, then an empty list is returned.

Parameters:

s - the start node of the shortest path which must be the same start node that was specified when pred was calculated

t - the target node of the path

pred - an array of Edges that will be filled during the execution and returns for each node t the shortest path edge pred[t.index()] which is the last edge on the shortest path from s to t. If t == s or if there is no shortest path from s to t, then pred[t.index()] == null.

Returns:

a list containing the nodes on the shortest path from s to t in the correct order

See Also:

constructNodePath(Node, Node, DataProvider)

constructNodePath

public static NodeList constructNodePath(Node s,
                                         Node t,
                                         DataProvider pred)

Like constructNodePath(Node, Node, Edge[]) but the path edges are given by a DataProvider.

If there is no path from node s to t, then an empty list is returned.

Parameters:

s - the start node of the shortest path which must be the same start node that was specified when pred was calculated

t - the target node of the path

pred - the NodeMap that will be filled during the execution and returns for each node t the last edge on the shortest path from s to t or null if t == s or no shortest path from s to t exists

Returns:

a list containing the nodes on the shortest path from s to t in the correct order

See Also:

constructNodePath(Node, Node, Edge[])

constructEdgePath

public static EdgeList constructEdgePath(Node s,
                                         Node t,
                                         Edge[] pred)

Convenience method that constructs an explicit path of edges from the result returned by one of the shortest paths methods defined in this class.

If there is no path from node s to t, then an empty list is returned.

Parameters:

s - the start node of the shortest path which must be the same start node that was specified when pred was calculated

t - the target node of the path

pred - an array of Edges that will be filled during the execution and returns for each node t the shortest path edge pred[t.index()] which is the last edge on the shortest path from s to t. If t == s or if there is no shortest path from s to t, then pred[t.index()] == null.

Returns:

a list containing the edges on the shortest path from s to t in the correct order

See Also:

constructEdgePath(Node, Node, DataProvider)

constructEdgePath

public static EdgeList constructEdgePath(Node s,
                                         Node t,
                                         DataProvider pred)

Like constructEdgePath(Node, Node, Edge[]) but the path edges are given by a DataProvider.

If there is no path from node s to t, then an empty list is returned.

Parameters:

s - the start node of the shortest path which must be the same start node that was specified when pred was calculated

t - the target node of the path

pred - the NodeMap that will be filled during the execution and returns for each node t the last edge on the shortest path from s to t or null if t == s or no shortest path from s to t exists

Returns:

a list containing the edges on the shortest path from s to t in the correct order

See Also:

constructEdgePath(Node, Node, Edge[])

findShortestUniformPaths

public static void findShortestUniformPaths(Graph graph,
                                            Node start,
                                            DataProvider targetMap,
                                            boolean directed,
                                            int maxLength,
                                            EdgeList pathEdges,
                                            NodeList pathNodes)

Finds all nodes and edges that belong to a shortest path from a start node to a set of target nodes in the graph not farther away than a given distance.

This method assumes that each edge of the input graph has a cost of 1.0.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Shortest paths longer than maxLength will not be considered.

Complexity:

O(graph.N() + graph.E())

Parameters:

graph - the input graph

start - the start node

targetMap - the DataProvider that returns a boolean value indicating whether or not a node belongs to the set of target nodes

directed - true if the graph should be considered as directed, false otherwise

maxLength - the maximum edge length of the shortest paths

pathEdges - a list that will be filled during the execution and returns the edges on the shortest path from s to t in the correct order

pathNodes - a list that will be filled during the execution and returns the nodes on the shortest path from s to t in the correct order

shortestPair

public static final EdgeList[] shortestPair(Graph graph,
                                            Node source,
                                            Node target,
                                            boolean directed,
                                            DataProvider costDP)

Returns two edge-disjoint paths in a non-negatively weighted directed graph, such that both paths connect nodes s and t and have minimum total length.

If the graph is considered undirected, each edge can be traversed in both directions and the returned shortest paths can, thus, be undirected.

Parameters:

graph - the input graph

source - the source node of the shortest pair

target - the target node of the shortest pair

directed - true if the graph should be considered as directed, false otherwise

costDP - the DataProvider that returns a double value (cost) for traversing each edge

Returns:

a two-dimensional array of EdgeLists holding the resulting edge-disjoint paths or null if no such edge-disjoint paths exist