1. All-Pairs Shortest Path: Floyd-Warshall Algorithm
J.-S. Roger Jang (張智星)
MIR Lab, CSIE Dept.
National Taiwan University
2019/5/21

2. Shortest-Paths Algorithms
Several variants of shortest-path algorithms
Single-source all-destination
Dijkstra’s algorithm
Bellman-Ford algorithm
All-pairs shortest paths
Floyd-Warshall algorithm

3. Floyd-Warshall Algorithm
2019/5/21
Characteristics
All-pairs shortest paths
Based on DP
Supports negative edge-weights, but not negative-weight cycles
O(n3)

4. Three-step DP Formula for Floyd-Warshall Algorithm
2019/5/21
Optimum-value function: 𝑑 𝑖,𝑗 𝑘 is the shortest path from nodes 𝑖 to 𝑗 using nodes only from {1,2,…,𝑘} (𝑘≤𝑛) as intermediate points
Recurrent equation:
𝑑 𝑖,𝑗 𝑘 =𝑚𝑖𝑛 𝑑 𝑖,𝑗 𝑘−1 𝑑 𝑖,𝑘 𝑘−1 + 𝑑 𝑘,𝑗 𝑘−1 , with 𝑑 𝑖,𝑗 0 =𝑤 𝑖,𝑗 .
Final answer: 𝑑 𝑖,𝑗 𝑛
Weight between 𝑖 and 𝑗

5. An Illustrative Example

6. Example 1 (1/5) Distance: {} Predecessor 𝐷 0 𝜋 0 Quiz! B A B C D 2 6 82 6 8 ∞ -2 3 4 1 A B C D - 2 3 -2 A D 8 4 1 6 C

7. Example 1 (2/5) Distance: {A} Predecessor 𝐷 1 𝜋 1 B 3 A B C D 2 6 8 ∞2 6 8 ∞ -2 3 4 1 A B C D - 2 -2 A D 8 4 1 6 C

8. Example 1 (3/5) Distance: {A,B} Predecessor 𝐷 2 𝜋 2 B 3 A B C D 2 5 ∞2 5 ∞ -2 3 4 6 1 A B C D - 2 -2 A D 8 4 1 6 C

9. Example 1 (4/5) Distance: {A,B,C} Predecessor 𝐷 3 𝜋 3 B 3 A B C D 2 12 1 -2 -1 4 6 ∞ A B C D - 2 -2 A D 8 4 1 6 C

10. Example 1 (5/5) Distance: {A,B,C,D} Predecessor 𝐷 4 𝜋 4 B 3 A B C D 22 1 -2 -1 4 6 ∞ A B C D - 2 -2 A D 8 4 1 6 C

11. Example 2 Distance: {} Predecessor 𝐷 0 𝜋 0 Quiz! B 4 A B C D ∞ 3 -2 1∞ 3 -2 1 5 4 A B C D - -2 1 A D 5 3 C
Solution

12. Example 3 Distance: {} Predecessor 𝐷 0 𝜋 0 A B C D Quiz! A B C D ∞ -2∞ -2 4 3 2 -1 A B C D - B C D
Solution

13. Reference Youtube tutorials Web resources Abdul Bari: Basics
Joe James: Basics plus predecessors
Erik Demaine: Comprehensive
Web resources
Chiu CC (in Chinese): Detailed descriptions