Lecture 2-3-17

Spanning Tree Algorithms

• m > n - 1
  • otherwise graph is not connected
• Algorithms:
  • Find a cycle and delete the longest edge
    • DFS: O(n + m)
    • Delete O(mn)
    • Total time: O(m²n)
  • Find arbitrary vertex
    • Find adjacent edge out of T and
    • Until all vertices in T

Prim’s Algorithm

• O(nm)
  • natural algorithm
• Psuedo-proof
  • Suppose it makes 1st mistake at step i … …
  • 1st mistake: T was part of a MST, at step i, some edge from T to x was added …
  • Not a mistake … … …
  • Therefore never makes 1st mistake
• Proof:
  • T needs to go to x from some vertex U
  • No mistake was made before so T_min but once added X, mistake was made
  • Add edge from T_min + u to x
    • Now T has some cycle
    • At some point, the cycle has to go back into the tree T
  • Claim there’s another spanning tree:
    • T_min minus v-w is a ST
    • Only one way to get between pair of points in a tree
    • Cost(v-w) < Cost(u-x)
      • C(u-x) <= C(vw) because algorithm took the smallest edge out of T
        • Since u-x was chosen over v-w
      • T_min + u-x \ v-w <= C(T_min)
      • Now, T_min + u-x \ v-w is also another MST
        • Contradiction
    • Therefore: adding u-x is not a mistake

Proof of Greedy Algorithms

• Show that they never make a 1^st mistake
• Induction
  • Proof BWOC

Kruskal’s Algorithm

• Sort edges in increasing order
• Add next u-v iff does not create a cycle
  • Personal: I like u-v does not already have a path better than “cycle”
• Not obvious that Prim’s/Kruskal’s is better
• Both are subproofs of:
  • Any current tree and add the smallest edge out of it, then the greedy algorithm based on any of the trees will always give you a MST
• Analysis:
  • Runtime: O(mlogm) = O(mlogn) since m can be at most n² and log(n²) = 2log(n)
  • Repeated m times and have to check cycle
    • Adding to current solution and can’t have more than n-1 edges = O(n)
  • O(mn + mlogn) also O(mn)
• Worth learning because
  • Cover’s union find algorithm

Sollin’s Algorithm

• Would be faster than O(mn)
• Natural implementation: O(mlogn)
• Boruvka: Czech railroad engineer in the 1920’s