Finding the Shortest Path with Vertex Constraint over Large Graphs

Algorithm 1

PERMUTATION-EXPANDING ().

Input:.

Output:.

// Input: : an undirected weighted graph

// : a vertex subset of

// , : starting vertex and ending vertex respectively

// Output: : the shortest path between and with vertex

// constraint of

1: Let be a min priority queue with entries in the form , sorted in ascending order of ;

2: for  each    do

3: Enqueue an entry into ;

4: Dequeue the first entry from and let be the last vertex of ;

5: while    do

6: if    then

7: for  each    do

8: Enqueue an entry into ;

9: else

10: Enqueue an entry into ;

11: Dequeue the first entry   from    and let be the last vertex of ;

12: Generate the shortest path    between    and under a permutation ;

13: return  ;