1. Shortest path algorithm
Zhiqi Chen
The slides are referenced from Dr. Parberry and Dr. Roden’s notes.

2. Single source shortest path problem
Given a labeled, directed (or undirected) graph, with nonnegative edge costs, G = (V, E) and a vertex, s Є V, find the shortest path from s to every other vertex in the graph.
Create a data structure that allows us to compute the vertices on a shortest path from s to w for each w Є V.
Vertex s is the source
let Δ(v, w) denote the cost of the shortest path from v to w
let C(v, w) be the cost of the edge from v to w (∞ if don’t
exist, 0 if v=w)

3. Dijkstra’s algorithm
Keep a set S of vertices whose minimum distance from the source s is known.
Initially S = {s}
keep adding vertices to S
eventually, S = V (V is all vertices in graph)
Maintain an array D (distance) with
If w Є S, then D[w] is the cost of the shortest path to w
If w Є S, then D[w] is the cost of the shortest path from s to w Δ(s, w)

4. The algorithm S = {s} for each v Є V D[v] = C(s, v) for i = 1 to n-1
choose w Є V – S with smallest D[w]
S = S υ {w}
for each vertex v Є V
D[v] = min { D[v], D[w] + C(w, v) }

5. Example A 3 10 B E 2 4 C 5 2 7 F 2 3 D

6. Step 1: S = {A} D[A] = 0 D[B] = 3 D[C] = ∞ D[D] = 5 D[E] = 10 D[F] = ∞
S = {s}
for each v Є V
D[v] = C(s, v)
D[A] = 0
D[B] = 3
D[C] = ∞
D[D] = 5
D[E] = 10
D[F] = ∞ A 3 10 B E 2 4 C 5 2 7 F 2 3 D

7. Step 2: S = {A} D[A] = 0 D[B] = 3 D[C] = ∞ D[D] = 5 D[E] = 10 D[F] = ∞
Pick
Smallest
D[w]
for i = 1 to n-1
choose w Є V – S with smallest D[w]
D[A] = 0
D[B] = 3
D[C] = ∞
D[D] = 5
D[E] = 10
D[F] = ∞ A 3 10 B E 2 4 C 5 2 7 F 2 3 D

8. Step 3: S = {A, B} S = S υ {w} for each vertex v Є V
D[v] = min { D[v], D[w] + C(w, v) }
W = B
D[A] = min {D[A], D[B] + C(B, A)}
D[B] = min {D[B], D[B] + C(B, B)}
D[C] = min {D[C], D[B] + C(B, C)}
D[D] = min {D[D], D[B] + C(B, D)}
D[E] = min {D[E], D[B] + C(B, E)}
D[F] = min {D[F], D[B] + C(B, F)} A 3 3 3 10 3 3 3 B E 2 5 ∞ 3 2 4 C 5 2 5 5 3 7 7 F 2 10 10 3 ∞ 3 D ∞ ∞ 3 ∞

9. Step 4: S = {A, B} D[A] = 0 D[B] = 3 D[C] = 5 D[D] = 5 D[E] = 10
Pick
Smallest
D[w]
for i = 1 to n-1
choose w Є V – S with smallest D[w]
D[A] = 0
D[B] = 3
D[C] = 5
D[D] = 5
D[E] = 10
D[F] = ∞ A 3 10 B E 2 4 C 5 2 7 F 2 3 D

10. Step 5: S = {A, B, C} S = S υ {w} for each vertex v Є V
D[v] = min { D[v], D[w] + C(w, v) }
W = C
D[A] = min {D[A], D[C] + C(C, A)}
D[B] = min {D[B], D[C] + C(C, B)}
D[C] = min {D[C], D[C] + C(C, C)}
D[D] = min {D[D], D[C] + C(C, D)}
D[E] = min {D[E], D[C] + C(C, E)}
D[F] = min {D[F], D[C] + C(C, F)} A 5 ∞ 3 10 3 3 5 2 B E 2 5 5 5 4 C 5 2 5 5 5 3 7 F 2 9 10 5 4 3 D 7 ∞ 5 2

11. Step 6: S = {A, B, C} D[A] = 0 D[B] = 3 D[C] = 5 D[D] = 5 D[E] = 9
Pick
Smallest
D[w]
for i = 1 to n-1
choose w Є V – S with smallest D[w]
D[A] = 0
D[B] = 3
D[C] = 5
D[D] = 5
D[E] = 9
D[F] = 7 A 3 10 B E 2 4 C 5 2 7 F 2 3 D

12. Step 7: S = {A, B, C, D} S = S υ {w} for each vertex v Є V
D[v] = min { D[v], D[w] + C(w, v) }
W = D
D[A] = min {D[A], D[D]+ C(D, A)}
D[B] = min {D[B], D[D] + C(D, B)}
D[C] = min {D[C], D[D] + C(D, C)}
D[D] = min {D[D], D[D] + C(D, D)}
D[E] = min {D[E], D[D] + C(D, E)}
D[F] = min {D[F], D[D] + C(D, F)} A 5 5 3 10 3 3 5 7 B E 2 5 5 5 3 4 C 5 2 5 5 5 7 F 2 9 9 5 ∞ 3 D 7 7 5 ∞

13. Step 8: S = {A, B, C, D} D[A] = 0 D[B] = 3 D[C] = 5 D[D] = 5 D[E] = 9
Pick
Smallest
D[w]
for i = 1 to n-1
choose w Є V – S with smallest D[w]
D[A] = 0
D[B] = 3
D[C] = 5
D[D] = 5
D[E] = 9
D[F] = 7 A 3 10 B E 2 4 C 5 2 7 F 2 3 D

14. Step 9: S = {A, B, C, D, F} D[A] = min {D[A], D[F]+ C(F, A)}
S = S υ {w}
for each vertex v Є V
D[v] = min { D[v], D[w] + C(w, v) }
W = F
D[A] = min {D[A], D[F]+ C(F, A)}
D[B] = min {D[B], D[F] + C(F, B)}
D[C] = min {D[C], D[F] + C(F, C)}
D[D] = min {D[D], D[F] + C(F, D)}
D[E] = min {D[E], D[F] + C(F, E)}
D[F] = min {D[F], D[F] + C(F, F)} A 7 ∞ 3 10 3 3 7 ∞ B E 2 5 5 7 2 4 C 5 2 5 5 7 ∞ 7 F 2 9 9 7 2 3 D 7 7 7

15. Step 10: S = {A, B, C, D} D[A] = 0 D[B] = 3 D[C] = 5 D[D] = 5 D[E] = 9
Pick
Smallest
D[w]
for i = 1 to n-1
choose w Є V – S with smallest D[w]
D[A] = 0
D[B] = 3
D[C] = 5
D[D] = 5
D[E] = 9
D[F] = 7 A 3 10 B E 2 4 C 5 2 7 F 2 3 D

16. Step 11: S = {A, B, C, D, F, E} S = S υ {w} for each vertex v Є V
D[v] = min { D[v], D[w] + C(w, v) }
W = E
D[A] = min {D[A], D[E] + C(E, A)}
D[B] = min {D[B], D[E] + C(E, B)}
D[C] = min {D[C], D[E] + C(E, C)}
D[D] = min {D[D], D[E] + C(E, D)}
D[E] = min {D[E], D[E] + C(E, E)}
D[F] = min {D[F], D[E] + C(E, F)} A 9 10 3 10 3 3 9 ∞ B E 2 5 5 9 4 4 C 5 2 5 5 9 ∞ 7 F 2 9 9 9 3 D 7 7 9 2

17. S = {A, B, C, D, F, E} D[A] = 0 D[B] = 3 D[C] = 5 D[D] = 5 D[E] = 910 B E 2 4 C 5 2 7 F 2 3 D

18. Implementation using heap
Store S’ = V – S as a (minimum) heap with minimum D value, vertex at top of heap (where each entry in the heap is a vertex and its corresponding D value)

19. Implementation using heap
makeNull (S’) // S’ = V - s
For each v Є V except s
D[v] = C(s, v)
insert (S’, v) // build the min heap
for i = 1 to n-1
w = S’.deleteMin() // always pick the smallest D
for each v connected by an edge to w
D[v] = min { D[v], D[w] + C(w, v) }
move v up the heap

20. Exercise A 2 5 B C 1 3 10 D

21. Answer: S = {A, B, C, D} D[A] = 0 D[B] = 2 D[C] = 3 D[D] = 6 3 10 A 25 B C 1 3 10 D