Question

Consider a graph that has non-negative edge weights. Suppose that I want to find the distance of the shortest path from a start node s to every other node v in the graph, subject to the constraint that no path may have more than e edges. Denote the distance between any two nodes u and v by d(u,v). Denote the minimum distance from node s to some node u with no more than e edges by dist(u,e). Then dist(s,0)=0. Write a recurrence for the distance dist (v,e) as the minimum over a set of shorter distances.

Answer 1

The distance dist(v, e) from the start node s to node v with no more than e edges can be calculated using a recurrence relation. If e = 0, the distance is 0 if v = s, and infinity otherwise. If e > 0, the distance is obtained by taking the minimum of the sum of the distance from s to a neighbor u of v and the distance from u to v with e-1 edges.

Step-by-step explanation:

The distance dist(v, e) from the start node s to node v with no more than e edges can be calculated using a recurrence relation. If e = 0, the distance is 0 if v = s, and infinity otherwise. If e > 0, the distance is obtained by taking the minimum of the sum of the distance from s to a neighbor u of v and the distance from u to v with e-1 edges.

The distance dist(v, e) from the start node s to node v with no more than e edges can be calculated using a recurrence relation:

1. If e = 0, then dist(v, e) = 0 if v = s, and ∞ otherwise.
2. If e > 0, then dist(v, e) = min(d(v, u) + dist(u, e-1)) for all u ∈ neighbors(v)

This recurrence relation states that if e = 0, the distance from s to any node v is 0 if v = s, and ∞ (infinity) otherwise. If e > 0, the distance is obtained by taking the minimum of the sum of the distance from s to a neighbor u of v and the distance from u to v with e-1 edges.