Spanning Tree Running Time

Prim’s Algorithm

• Some problems require intuitive sense for algorithms / how the approach would be if done manually
• Algorithm:
  • Keep track (also stores where the edge originates from) of best edge to each vertex initialized with infinity
  • Initialize a starting vertex to 0
  • Keep adding best edge by scanning and finding smallest and then updating best edges
• Using an array, the running time is from O(nm) => O(n²)
• Updating:
  • Update costs = O(n)
    • In fact, actually degree(x)
    • Sum of all degrees: O(m)
  • Scan matrix O(n)
    • Always taking constant * n
    • Bottleneck is in finding the smallest value
• Current implementation: running time of O(n²)
• For complete graphs:
  • Prim’s algorithm is as good as can hoped to be
  • Cannot do better than O(n²)

Kruskal’s Algorithm

• To get the algorithm to run faster than it does, it is not something immediately obvious
  • I think he’s hinting at union find
• Algorithm:
  1. Sort edges O(mlogm)
  2. From smallest to largest x-y, add x-y iff does not create a cycle O(mn)
     • Saying creates cycle is the same thing as already in a tree
• Since Prim’s also runs in O(mn), this can also be improved
• Not showing proof of correctness because it’s the same proof as proving Prim’s
• Improving the bottleneck:
  • Bottleneck is determining whether a new edge creates a cycle
  • sets(trees)