Lecture 03-22-17

All pairs shortest path

• For every pair of vertices x and y, we want to know the shortest path from x to y
• Djikstra’s
  • n * O(n²)
  • non-negative weight edges
  • greedy algorithm
    • doesn’t work because longer edge can take you to negative weight edge later
• Floyd’s
  • Deals with negative weigths & negative weight cycles
  • O(n³)

Frequent flyers story

• How to find a negative weight edges & especially cycles

Floyd’s Algorithm

• Negative weights & cycles
• Will work unless there is a negative weight cycle
  • But will be able to detect
• Dynamic programming
• Applies to automata: convert FA to RE
• Standard assumption in graphs: vertices are numbered
• S_ij^k
  • ij: coordinates
  • k: cost of shortest path from i to j which goes through no vertex numbered greater than k
  • Example: 5 -> 12 -> 9 -> 4 -> 8 -> 3
    • Goes through 12, 9, 4, 8
    • Does not include start and end
    • Into and out of vertex to go through
• Going to calculate all
• Steps
  1. Solve all S_ij⁰
  2. Solve all S_ij¹
  3. Solve all S_ij² (vertices 1 and 2)
• Create a matrix M⁰
  • Represents shortest path from i to j without going through edge greater than k
  • If i to j, then cost is cost
  • If i = j, then 0
  • Otherwise, infinity
• Create a matrix M^k
  • Adding a new edge at each step, using the previous matrix
  • Row and column containing i is going to be the previous values
  • Then compare with and without new vertex k
• min(M_ij^{k - 1}, M_ik^{k - 1} + M_kj^{k - 1})
• How will we determine if there is a negative cycle
  • iff proof
  • If you find a negative value along the diagonal, you will have a negative cycle
  • If negative cycle, then Mx, there will be a negative along the diagonal
• Keep track of intermediate vertex if better
  • This will give the ability to reconstrut the path
• Can do it all in one matrix
  • Slightly mindblown