1. ALG0183 Algorithms & Data Structures
“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.
ALG0183 Algorithms & Data Structures
Lecture 22 n negative weights present (Bellman-Ford algorithm)
…as implemented in Graph.java by Weiss
Chapter 14 Weiss
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

2. 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" );
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

3. 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!" );
scratch
iterations
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

4. The problem with negative cost cycles.
Figure 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
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

5. The problem with negative cost cycles.
Figure 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
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

6. ALG0183 Algorithms & Data Structures by Dr Andy Brooks
Bellman-Ford Algorithm
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.
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

7. ALG0183 Algorithms & Data Structures by Dr Andy Brooks
SUNY Binghamton CS480 mini-lectures -- Bellman-Ford
if a better path has been found
|V| is the number of vertices |E| is the number of edges
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

8. ALG0183 Algorithms & Data Structures by Dr Andy Brooks
Pseudocode implementation (
// 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:
if u.distance + uv.weight < v.distance:
error "Graph contains a negative-weight cycle"
(-3) 3 -5
zero-cost cycle
(2) 2
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

9. ALG0183 Algorithms & Data Structures by Dr Andy Brooks
Example from Digraphs: Theory, Algorithms and Applications, 1st Edition , Jørgen Bang-Jensen and Gregory Gutin
Initialize graph.
For assessment purposes, students should be able to produce the working shown in (a)-(f) for any small graph.
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

10. ALG0183 Algorithms & Data Structures by Dr Andy Brooks
* Infinity is represented by a large finite number in code, so a distance cost to a node might be reduced from “infinity”. d a
ab, b still infinite
ac, c still infinite
* ba, a still infinite
bc, c still infinite
* cb, b still infinite
da, a still infinite
* dc, c still infinite
ec, c still infinite
*ed, d still infinite
sd, d->4
se, e->8
“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.”
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

11. ALG0183 Algorithms & Data Structures by Dr Andy Brooks
* Infinity is represented by a large finite number in code, so a distance cost to a node might be reduced from “infinity”. d a
ab, b still infinite
ac, c still infinite
* ba, a still infinite
bc, c still infinite
* cb, b still infinite
da, a ->8
dc, c->2
ec, c no change
ed, d->3
sd, d no change
se, e no change
“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.”
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

12. ALG0183 Algorithms & Data Structures by Dr Andy Brooks
ab, b->17
ac, c no change
ba, a no change
bc, c no change
cb, b->0
da, a ->7
dc, c->1
ec, c no change
ed, d no change
sd, d no change
se, e no change
“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.”
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

13. ALG0183 Algorithms & Data Structures by Dr Andy Brooks
ab, b no change
ac, c no change
ba, a->-3
bc, c no change
cb, b->-1
da, a no change
dc, c no change
ec, c no change
ed, d no change
sd, d no change
se, e no change
“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.”
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

14. ALG0183 Algorithms & Data Structures by Dr Andy Brooks
6 vertices
5th iteration
ab, b no change
ac, c no change
ba, a->-4
bc, c no change
cb, b no change
da, a no change
dc, c no change
ec, c no change
ed, d no change
sd, d no change
se, e no change
“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.”
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

15. 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).
simple, not prioritised queue
a 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!
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

16. 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
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

17. ALG0183 Algorithms & Data Structures by Dr Andy Brooks
startName from user
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
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.
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

18. ALG0183 Algorithms & Data Structures by Dr Andy Brooks
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.
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

19. ALG0183 Algorithms & Data Structures by Dr Andy Brooks
Note this graph also contains a positive-cost cycle.
NegGraphTest1.txt
V2 V0 4
V2 V5 5
V0 V1 2
V0 V3 1
V3 V2 2
V3 V5 8
V3 V6 4
V6 V5 1
V3 V4 2
V1 V3 3
V4 V1 -10
V4 V6 6
running algorithm n
File read...
7 vertices
Enter start node:V2
Enter destination node:V4
Enter algorithm (u, d, n, a ): n
gr.GraphException: Negative cycle detected
Enter start node:
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks

20. ALG0183 Algorithms & Data Structures by Dr Andy Brooks
NegGraphTest1.txt
V2 V0 4
V2 V5 5
V0 V1 2
V0 V3 1
V3 V2 2
V3 V5 8
V3 V6 4
V6 V5 1
V3 V4 2
V4 V1 -10
V4 V6 6 X
V1 V3 3
running algorithm n
Skipping ill-formatted line
File read...
7 vertices
Enter start node:V2
Enter destination node:V1
Enter algorithm (u, d, n, a ): n
(Cost is: -3.0) V2 to V0 to V3 to V4 to V1
Enter start node:
8/25/2009
ALG0183 Algorithms & Data Structures by Dr Andy Brooks