1. Lecture 8 Shortest Path Shortest-path problems
Single-source shortest path
All-pair shortest path

2. Overview Shortest-path problems Single-source shortest path algorithms
Bellman-Ford algorithm
Dijkstra algorithm
All-Pair shortest path algorithms
Floyd-Warshall algorithm

3. Weighted graph Beijing Qingdao Lhasa Shanghai
Single-source shortest path
All-pair shortest path
Guangzhou

4. Shortest Path

5. Cycles. Can a shortest path contains cycles?
Optimal substructure:
Subpaths of shortest paths are shortest paths.
Cycles. Can a shortest path contains cycles?
Negative weights.

6. Where are we? Shortest-path problems
Single-source shortest path algorithms
Bellman-Ford algorithm
Dijkstra algorithm
All-Pair shortest path algorithms
Floyd-Warshall algorithm

7. Relaxing
The process of relaxing an edge (u,v) consists of testing whether we can improve the shortest path to v found so far by going through u.

8. Shortest-path properties
Notation: We fix the source to be s.
λ[v]: the length of path computed by our algorithms from s to v.
δ[v]: the length of the shortest path from s to v.
Triangle property
δ[v] <= δ[u] + w(u,v) for any edge (u,v).
Upper-bound property
δ[v] <= λ[v] for any vertex v.
Convergence property
If s⇒u→v is a shortest path, and if λ[u] = δ[u], then after relax(u,v), we have λ[v] = δ[v].

9. Bellman-Ford algorithm
E = {(t,x),(t,y),(t,z),(x,t),(y,x),(y,z),(z,x),(z,s),(s,t),(s,y)}
Θ(mn)

10. Negative cycle

11. Quiz Run Bellman-Ford algorithm for the following graph, provided by
E = {(S,A),(S,G),(A,E),(B,A),(B,C),(C,D),(D,E),(E,B),(F,A),(F,E),(G,F)}

12. Correctness Correctness of Bellman-Ford
If G contains no negative-weight cycles reachable from s, then the algorithm returns TRUE, and for all v, λ[v] = δ[v], otherwise the algorithm returns FALSE.
Proof.
No negative cycle. By the fact that the length of simple paths is bounded by |V| - 1.
After i-th iteration of relax, λ[vi] = δ[vi]. s v v1 Vj vi
Negative cycle. After i-th iteration of relax,
λ[vi] + w(vi,vj) < λ[vj].
The length of negative cycle is at most |V|.

13. Relax in topological order.
Observation 1
If there is no cycle (DAG), ...
Relax in topological order.
Θ(m) s v v1 vj vi

14. Weighted DAG application
seam

15. If (s, vi) is the lightest edge sourcing from s, then λ[vi] = δ[vi]
Observation 2
If there is no negative edge, ... vj vi v1 s
If (s, vi) is the lightest edge sourcing from s, then λ[vi] = δ[vi]

16. Dijkstra algorithm PV primitives Structural programming
Distributed algorithms
Edsger Wybe Dijkstra in 2002

17. Dijkstra algorithm

19. Correctness Assume u is chosen in Step 7, and 1. λ[u] > δ[u]
2. s ⇒ x → y ⇒ u is the shortest path
δ[u] = δ[x] + w(x,y) + w(p2)
= λ[x] + w(x,y) + w(p2)
≥ λ[x] + w(x,y)
≥ λ[y] ≥ λ[u]

20. Time complexity
Θ(n2)

21. What is the time complexity?
How about use d-heap?

22. Comparisons Array Binary heap d-ary heap Dijkstra O(n2) O(mlogn)
O(dnlogdn + mlogdn)

23. Dense graph dense: m = n1+ε, ε is not too small. d-heap, d = m/n
complexity: O(dnlogdn + mlogdn) = O(m)

24. Where are we? Shortest-path problems
Single-source shortest path algorithms
Bellman-Ford algorithm
Dijkstra algorithm
All-Pair shortest path algorithms
Floyd-Warshall algorithm

25. All-pairs shortest path
Define $d_{i,j}^{k}$ to be the length of a shortest path from $i$ to
$j$ that does not pass any vertex in $\{k+1,k+2,\ldots,n\}$. Clearly
\[d_{ij}^{k} = \left\{\begin{array}{ll}
l[i,j] & {\rm if}\ k=0 \\
min\{d_{i,j}^{k-1},d_{i,k}^{k-1}+d_{k,j}^{k-1}\} & {\rm if}\ 1\le
k\le n
\end{array}\right.\]

26. Floyd-Warshall algorithm
\left[
\begin{array}{ccccc}
0 & 10 & \infty & 5 & \infty\\
\infty & 0 & 1 & 2 & \infty\\
\infty & \infty & 0 & \infty & 4\\
\infty & 3 & 9 & 0 & 2\\
2 & \infty & 6 & \infty & 0
\end{array}
\right]

27. Floyd-Warshall algorithm
Θ(n3)

28. Quiz
Run Floyd-Warshall algorithm for the following graph:

29. Conclusion Dijkstra’s algorithm.
Nearly linear-time when weights are nonnegative.
Acyclic edge-weighted digraphs.
Faster than Dijkstra’s algorithm.
Negative weights are no problem.
Negative weights and negative cycles.
If no negative cycles, can find shortest paths via Bellman-Ford.
If negative cycles, can find one via Bellman-Ford.
All-pair shortest path.
can be solved via Floyd-Warshall
Floyd-Warshall can also compute the transitive closure of directed graph.

30. Which of the following statements are true for shortest path?
♠. If you run Dijkstra's algorithm on an edge-weighted DAG with positive weights, the order in which the vertices are relaxed is a topological order.
♥. Let P be a shortest path from some s to t in an edge- weighted digraph G. If the weight of each edge in G is increased by one, then P will still be a shortest path from s to t in the modified digraph G'.
♣. Let G be a digraph with positive edge weights. Suppose that you increase the length of an edge by x. Then, the length of the shortest path from s to t can increase by more than x.
♦. Bellman-Ford finds the shortest simple path from s to every other vertex, even if the edge weights are positive or negative integers, provided there are no negative cycles.
♣ ♦