ALG0183 Algorithms & Data Structures Lecture 22 n negative weights present (Bellman-Ford algorithm) 8/25/20091 ALG0183 Algorithms & Data Structures by

ALG0183 Algorithms & Data Structures by Dr Andy Brooks

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ALG0183 Algorithms & Data Structures

Lecture 22n negative weights present (Bellman-Ford algorithm)

8/25/2009

…as implemented in Graph.java by Weiss

Chapter 14 Weiss

“The shortest-path algorithms are all single-source algorithms, which begin at some starting point and compute the shortest paths from it to all vertices.” Weiss

...to all vertices.


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Java exception handling in Graph.java.

• In u, d, n, and a, an exception is thrown if the start vertex is not found. Vertex w = vertexMap.get( startName ); if( w == null ) throw new NoSuchElementException(“Start vertex not found" );

• In printPath(String), an exception is thrown if the destination vertex is not found. Vertex w = vertexMap.get( destName ); if( w == null ) throw new NoSuchElementException( "Destination vertex not found" );

• In d (Dijkstra), an exception is thrown if the graph has an negative edge. if( cvw < 0 ) throw new GraphException( "Graph has negative edges" );

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Java exception handling in Graph.java.

• In n (Bellman-Ford), an exception is thrown if a graph has an negative cycle. if( v.scratch++ > 2 * vertexMap.size( ) ) throw new GraphException( "Negative cycle detected" );

• In a (acyclic topological sort), an exception is thrown if a graph has a cycle. if( iterations != vertexMap.size( ) )

throw new GraphException( "Graph has a cycle!" );

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scratch

iterations


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Figure 14.28 A graph with a negative cost cycle (Weiss ©Addison-Wesley)

• The path from V3 to V4 has a cost of 2.

• The path V3V4V1V3V4, going through the destination V4 twice, has a cost of -3!

• We could stay in the cycle an arbitrary number of times, lowering the cost each cycle!

• “Thus the shortest path between these two points (V3 & V4) is undefined.” Weiss

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The problem with negative cost cycles.


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Figure 14.28 A graph with a negative cost cycle (Weiss ©Addison-Wesley)

• The shortest path from V2 to V5 is also undefined i.e. the problem with a negative cost cycle is not restricted to nodes within the cycle (V1, V3, V4).

• When a negative-cost cycle is present, many shortest paths in a graph can be undefined.

• “Negative-cost edges by themselves are not necessarily bad; it is the cycles that are.” Weiss

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The problem with negative cost cycles.


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Bellman-Ford Algorithmhttp://www.itl.nist.gov/div897/sqg/dads/HTML/bellmanford.html

Definition: An efficient algorithm to solve the single-source shortest-path problem. Weights may be negative. The algorithm initializes the distance to the source vertex to 0 and all other vertices to ∞.

It then does V-1 passes (V is the number of vertices) over all edges relaxing, or updating, the distance to the destination of each edge. Finally it checks each edge again to detect negative weight cycles, in which case it returns false. The time complexity is O(VE), where E is the number of edges.

The algorithm has to iterate over the edges several times to make sure that the effect of any negative edge is fully propagated.

http://www.itl.nist.gov/div897/sqg/dads/HTML/singleSourceShortestPath.html

http://www.itl.nist.gov/div897/sqg/dads/HTML/source.html

http://www.itl.nist.gov/div897/sqg/dads/HTML/edge.html

http://www.itl.nist.gov/div897/sqg/dads/HTML/cycle.html

http://www.itl.nist.gov/div897/sqg/dads/HTML/bigOnotation.html


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SUNY Binghamton CS480 mini-lectures -- Bellman-Fordhttp://www.youtube.com/watch?v=gzXSFdXqrH8

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|V| is the number of vertices|E| is the number of edges

if a better path has been found


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Pseudocode implementation (http://en.wikipedia.org/wiki/Bellman-Ford_algorithm)

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// Step 1: Initialize graph for each vertex v in vertices: if v is source then v.distance := 0 else v.distance := infinity v.predecessor := null // Step 2: relax edges repeatedly for i from 1 to size(vertices)-1: for each edge uv in edges: // uv is the edge from u to v u := uv.source v := uv.destination if u.distance + uv.weight < v.distance: if a better path has been found v.distance := u.distance + uv.weight v.predecessor := u // Step 3: check for negative-weight cycles for each edge uv in edges: u := uv.source v := uv.destination if u.distance + uv.weight < v.distance: error "Graph contains a negative-weight cycle"

2

3 -5

(2)

(-3)

zero-cost cycle


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Example from Digraphs: Theory, Algorithms and Applications, 1st Edition , Jørgen Bang-Jensen and Gregory Gutinhttp://www.cs.rhul.ac.uk/books/dbook/main.pdf

Initialize graph.

For assessment purposes, students should be able to produce the working shown in (a)-(f) for any small graph.


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“In the inner loop of the second step of the algorithm the arcs are considered in the lexicographic order: ab, ac, ba, bc, cb, da, dc, ec, ed, sd, se.”

ab, b still infiniteac, c still infinite* ba, a still infinitebc, c still infinite* cb, b still infiniteda, a still infinite* dc, c still infiniteec, c still infinite*ed, d still infinitesd, d->4se, e->8

a

b

c

d

e

* Infinity is represented by a large finite number in code, so a distance cost to a node might be reduced from “infinity”.


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ab, b still infiniteac, c still infinite* ba, a still infinitebc, c still infinite* cb, b still infiniteda, a ->8dc, c->2ec, c no changeed, d->3sd, d no changese, e no change

a

b

c

d

e

* Infinity is represented by a large finite number in code, so a distance cost to a node might be reduced from “infinity”.


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a

b

c

d

e

ab, b->17 ac, c no changeba, a no changebc, c no changecb, b->0da, a ->7dc, c->1ec, c no changeed, d no changesd, d no changese, e no change


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a

b

c

d

e

ab, b no changeac, c no changeba, a->-3bc, c no changecb, b->-1da, a no changedc, c no changeec, c no changeed, d no changesd, d no changese, e no change


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a

b

c

d

e

ab, b no changeac, c no changeba, a->-4bc, c no changecb, b no changeda, a no changedc, c no changeec, c no changeed, d no changesd, d no changese, e no change

6 vertices5th iteration


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Features of algorithm n by Weiss.• “Whenever a vertex has its distance lowered, it

must be placed on a queue. This may happen repeatedly for each vertex.”

• “Consequently, we use the queue as in the unweighted algorithm, but we use Dv + cv,w as the distance measure (as in Dijkstra´s algorithm).

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simple, not prioritised queuea vertex can be enqueued several times

Note: In Weiss´s implementation, the outer loop is controlled by a test on the queue.Weiss´s implementation might require less computation than an implementation which is based on a fixed number of iterations through all the edges: we need an experiment!


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Features of algorithm n by Weiss.

• The variable scratch is incremented when a vertex is put on the queue (“enqueued”).

• The variable scratch is also incremented when a vertex is taken off the queue (“dequeued”).

• If the vertex is on the queue, scratch is odd.• The number of times a vertex has been

dequeued is scratch/2.• “We do not enqueue a vertex if it is already on

the queue.”8/25/2009


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public void negative( String startName ) { clearAll( ); call to the reset method for each vertex

Vertex start = vertexMap.get( startName ); get start vertex from the map if( start == null ) throw new NoSuchElementException( "Start vertex not found" );

Queue<Vertex> q = new LinkedList<Vertex>( ); make queue q.add( start ); add the start to the queue start.dist = 0; start to itself has distance zero start.scratch++; start is enqueued, so increment scratch

startName from user

The increment operator in Java is ++. The increment operator adds one.If a postfix increment is used (e.g. a++) then the value of a is used in the expression in which a resides, and then a is incremented by 1.


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while( !q.isEmpty( ) ) { Vertex v = q.remove( ); remove the head of the queue if( v.scratch++ > 2 * vertexMap.size( ) ) scratch/2 is number times dequeued throw new GraphException( "Negative cycle detected" );

for( Edge e : v.adj ) { iterate over a vertex´s edges Vertex w = e.dest; double cvw = e.cost; if( w.dist > v.dist + cvw ) { if a better path has been found w.dist = v.dist + cvw; w.prev = v; note of previous vertext (v) on the current shortest path // Enqueue only if not already on the queue if( w.scratch++ % 2 == 0 ) if w.scratch is odd, w is already on the queue q.add( w ); add to queue else w.scratch--; // undo the enqueue increment w not added to queue } } }}

When better paths are no longer found, vertices are no longer added to the queue, and the algorithm terminates when the queue is empty.


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V2 V0 4V2 V5 5V0 V1 2V0 V3 1V3 V2 2V3 V5 8V3 V6 4V6 V5 1V3 V4 2V1 V3 3V4 V1 -10V4 V6 6

NegGraphTest1.txt

File read...7 verticesEnter start node:V2Enter destination node:V4 Enter algorithm (u, d, n, a ): ngr.GraphException: Negative cycle detectedEnter start node:

running algorithm n

Note this graph also contains a positive-cost cycle.


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V2 V0 4V2 V5 5V0 V1 2V0 V3 1V3 V2 2V3 V5 8V3 V6 4V6 V5 1V3 V4 2

V4 V1 -10V4 V6 6

NegGraphTest1.txt

Skipping ill-formatted line File read...7 verticesEnter start node:V2Enter destination node:V1 Enter algorithm (u, d, n, a ): n(Cost is: -3.0) V2 to V0 to V3 to V4 to V1Enter start node:

running algorithm nV1 V3 3

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ALG0183 Algorithms & Data Structures Lecture 22 n negative weights present (Bellman-Ford algorithm) 8/25/20091 ALG0183 Algorithms & Data Structures by