Algorithm to count number of disjoint paths
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The document outlines an algorithm to find all edge-disjoint paths between a source and target in a graph, utilizing Dijkstra's shortest-path algorithm. It includes pseudo-code for the algorithm, which counts the disjoint paths by repeatedly finding and deleting paths until none remain. The algorithm is applicable to both unweighted and weighted graphs, and can be modified to work for vertex-disjoint paths as well.
![Pseudo-Code
disjoint-path(G[n][n],X,Y){
path=path_find(X,Y);
while(path!=NULL){
count++;
delete_path(G,path);
path=path_find(X,Y);
}
return count;
}](https://image.slidesharecdn.com/algorithmtocountnumberofdisjointpaths-121213204452-phpapp01/85/Algorithm-to-count-number-of-disjoint-paths-3-320.jpg)
![path_find(G,X,Y){
for each vertex V in G:
dist[v]:=infinity;
previous[v]:= undefined;
dist[x]:=0;
Q:= set of all nodes in G;
while Q is not empty
u:=vertex in Q with smallest distance in
dist[];](https://image.slidesharecdn.com/algorithmtocountnumberofdisjointpaths-121213204452-phpapp01/85/Algorithm-to-count-number-of-disjoint-paths-5-320.jpg)
![If dist[u]=infinity:
break;
remove u from Q;
for each neighbour v of u:
alt:= dist[u]+1;
if alt<dist[v]:
dist[v]:=alt;
previous[v]=u;](https://image.slidesharecdn.com/algorithmtocountnumberofdisjointpaths-121213204452-phpapp01/85/Algorithm-to-count-number-of-disjoint-paths-6-320.jpg)
![path:=empty sequence;
k=Y;
while (previous[k]!=NULL)
insert k at the end of path;
k:=previous[k];
return path;
}](https://image.slidesharecdn.com/algorithmtocountnumberofdisjointpaths-121213204452-phpapp01/85/Algorithm-to-count-number-of-disjoint-paths-7-320.jpg)
Algorithm to count number of disjoint paths
- 1. Algorithm To Count Number of Disjoint Paths Between A Source and Target, In A Given Graph Sujith Nair
- 2. Introduction • This algorithm finds all the edge-disjoint paths between a source S ∈ V and a target T ∈ V, in a given graph G(E,V). • The algorithm uses Dijkstra’s Single-Source- Shortest-Path algorithm as a sub-routine.
- 3. Pseudo-Code disjoint-path(G[n][n],X,Y){ path=path_find(X,Y); while(path!=NULL){ count++; delete_path(G,path); path=path_find(X,Y); } return count; }
- 4. delete_path(G,path){ for each edge in path G:=G-edge; }
- 5. path_find(G,X,Y){ for each vertex V in G: dist[v]:=infinity; previous[v]:= undefined; dist[x]:=0; Q:= set of all nodes in G; while Q is not empty u:=vertex in Q with smallest distance in dist[];
- 6. If dist[u]=infinity: break; remove u from Q; for each neighbour v of u: alt:= dist[u]+1; if alt<dist[v]: dist[v]:=alt; previous[v]=u;
- 7. path:=empty sequence; k=Y; while (previous[k]!=NULL) insert k at the end of path; k:=previous[k]; return path; }
- 8. Example Source Vertex: a Target Vertex: f
- 9. Example Continued. List Of Edge- Disjoint Paths: 1. a-b-f
- 10. Example Continued List Of Edge-Disjoint Paths: 1.a-b-f 2.a-c-f
- 11. Example Continued List Of Edge- Disjoint Paths: 1.a-b-f 2.a-c-f 3.a-d-c-f
- 12. Endnotes • This algorithm will also work in the case of weighted graphs. In the case of weighted graphs, the algorithm returns a list of paths with the lowest weights. • This algorithm, with minor modifications, will also work for the case of vertex-disjoint paths.
- 13. Thank You