1. All-Pairs SPs on DG Run Dijkstra;s algorithm for each vertex or
Use DP: Floyd-Warshall O(n3) algorithm.
The structure of SP:
Intermediate vertices of P={v1,v2,…,vk}:
{v2,v3,…,vk-1}
{1,2,…,k-1} notation for vertices.
For any i,j of V consider all paths with intermediate vertices in {1,2,…,k}.
Let P be min-weight path found this way.

2. All-Pairs SP (cont.)
Relation between P and SP(i,j) with intermediate vertices in {1,2,…,k-1}
K not in P: same
K in P: break P:
Recursive solution: dij(k) is weight of SP(i,j) with interm. vertices in {1,2,…,k}.
dij(k)=wij, if k=0;
dij(k)=min{dij(k-1), dik(k-1)+dkj(k-1)} if k >=1.
Matrix D(n)=dij(n) gives final solution.

3. SP Bottom-Up Computation
W: wij= 0, if i=j;
weight of edge (i,j) if exists;
“infinity”, otherwise.
D(0)=W;
For k=1 to n do
for i=1 to n do
for j=1 to n do
dij(k)=min{dij(k-1),dik(k-1)+dkj(k-1)}.
Return D(n).

4. Constructing a SP Compute predecessor matrix P(n) together with D(n).
Pij(k): predecesor of j on SP(i,j) with intermediate vertices in {1,2,…,k}.
Pij(0)= nil, if i=j or no edge (i,j).
i, if edge (i,j).
K >=1: ikj same predecessor as kj with intermediate vertices in {1,2,…,k-1}.
Pij(k)=Pij(k-1), if dij(k-1) <= dik(k-1)+dkj(k-1)
Pij(k)=Pkj(k-1), if dij(k-1) > dik(k-1)+dkj(k-1)