“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
ALG0183 Algorithms & Data Structures
Lecture 22
n negative weights present (Bellman-Ford algorithm)
Chapter 14 Weiss
8/25/2009
ALG0183 Algorithms & Data Structures by
Dr Andy Brooks
1
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
2
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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ALG0183 Algorithms & Data Structures by
Dr Andy Brooks
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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
8/25/2009
ALG0183 Algorithms & Data Structures by
Dr Andy Brooks
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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
8/25/2009
ALG0183 Algorithms & Data Structures by
Dr Andy Brooks
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Bellman-Ford Algorithm
http://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.
8/25/2009
ALG0183 Algorithms & Data Structures by
Dr Andy Brooks
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SUNY Binghamton CS480 mini-lectures -- Bellman-Ford
http://www.youtube.com/watch?v=gzXSFdXqrH8
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
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Pseudocode implementation (http://en.wikipedia.org/wiki/Bellman-Ford_algorithm)
// 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
(-3)
for each edge uv in edges:
u := uv.source
3
-5
zero-cost cycle
v := uv.destination
(2)
if u.distance + uv.weight < v.distance:
2
error "Graph contains a negative-weight cycle"
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ALG0183 Algorithms & Data Structures by
Dr Andy Brooks
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Example from Digraphs: Theory, Algorithms and Applications, 1st Edition ,
Jørgen Bang-Jensen and Gregory Gutin
http://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. ALG0183 Algorithms & Data Structures by
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Dr Andy Brooks
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e
c
b
d
a
* Infinity is represented by
a large finite number in
code, so a distance cost to
a node might be reduced
from “infinity”.
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
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e
c
b
d
a
* Infinity is represented by
a large finite number in
code, so a distance cost to
a node might be reduced
from “infinity”.
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.”
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ALG0183 Algorithms & Data Structures by
Dr Andy Brooks
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e
c
b
d
a
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
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e
c
b
d
a
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
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e
c
b
d
a
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
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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).
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
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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.”
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ALG0183 Algorithms & Data Structures by
Dr Andy Brooks
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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
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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ALG0183 Algorithms & Data Structures by
Dr Andy Brooks
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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 ) {
Vertex w = e.dest;
double cvw = e.cost;
iterate over a vertex´s edges
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
}
}
}
}
8/25/2009
When better paths are no longer found, vertices are no longer added to
the queue, and the algorithm terminates when the queue is empty.
ALG0183 Algorithms & Data Structures by
Dr Andy Brooks
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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
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:
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ALG0183 Algorithms & Data Structures by
Dr Andy Brooks
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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
X
V1 V3 3
V4 V1 -10
V4 V6 6
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: ALG0183 Algorithms & Data Structures by
8/25/2009
Dr Andy Brooks
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