Lecture 02-15-17

Shortest Path Algorithm

• “Single source”
  • From a starting point to all other points in the graph
• “Single source single destination”
  • No advantage to above
  • Same order of complexity as finding cheapest path to all points
• Dijkstra
  • P & V in OS / synchronizations
• Dijkstra’s Algorithm
  • Assumptions:
    • Non-negative weight edges
      • For rewards in conjunction with costs
    • Digraph
      • If knew how to solve for undirected, no way to solve digraph
      • Conversely, knowing digraph knows how to solve for undirected
  • Greedy algorithm
  • Steps:
    1. Takes the one with the least cost
    2. Look at all edges out of s & x
       • Only place you can go wrong
  • Prim’s algorithm
    • Simply weight of edge
  • In this:
    • Weight of the edge + the cost of getting to the head of the edge
  • Runtime: O(n²)
  • Finalizes distances by order of cheapest to most costly

“I am going to lie to you”

• Don’t write this down
• Show you something kind of important to know
• Single source single destination shortest path
  • Simple variant of dijkstra’s: bidirectional shortest path
• Find vertices in order of distance from the start vertex
• In the worst case, O(n²)
• Average case:
  • Expected: O(sqrt(n))
  • Order: O(n * sqrt(n))
• All this would be true if weights of edges are random numbers

Lie

• Why do I teach you this?
• Expected time analysis can be very dangerous
• You can often be convinced that things are equally likely to happen
• Problem with model not math