MSTs

• Graph cycles

Spanning Trees

• Given undirected graph w/ weighted edges, spanning tree T is a subgraph of G where T:
  • Is connected & acyclic → tree
    • Not connected → multiple separate trees → forest
  • Includes all of the vertices → spanning
• Minimum spanning tree = spanning tree w/ total edge weight minimized

MST vs SPT

• MST = fundamentally non-sourced object → not every graph has node for which SPT is also MST

Cut Property

• Cut = assign every node either color gray or white → assignment of graph's nodes to 2 non-empty sets
• Crossing edge = edge connecting node from one set to node from other set
• Cut property: Given any cut, minimum weight crossing edge guaranteed to be in MST
  • If multiple edges have same minimum weight → have to deal w/ tie-breaking, otherwise if unique edge weights, MST will contain minimum weight crossing edge
• Color/set assignment doesn't have to be contiguous → can be any assignment → flip coin $\forall$ vertices to determine what color/set

Cut Property Proof

• Suppose minimum weight crossing edge $e$ not in MST
  • Adding $e$ to MST creates cycle → can eliminate one edge in order to make it an MST again
  • Some other edge $f$ must also be crossing edge
  • If spanning tree doesn't contain minimum weight crossing edge → can replace currently used crossing edge w/ minimum crossing edge → will still be spanning tree w/ lower weight
  • Replacing $f$ w/ $e$ results in lower weight spanning tree → contradiction
• If minimum weight crossing edge $e$ not in MST, can use $e$ instead of some other crossing edge

Generic MST Finding Algorithm

• Need efficient way of finding cut w/ no crossing edges in MST
  • If cut randomly, may end up finding same minimum crossing edge each time

Prim's Algorithm

• Fringe contains all vertices in other set (other side of cut)
• Difference between Dijkstra's = considers edge weight instead of total path weight

Prim's vs Dijkstra's

• Dijkstra's algorithm explores based on distance from source, SPT expands outwards
• Prim's algorithm explores based on distance from MST under construction, jumps around

Prim's Runtime

• Same as Dijkstra's

Kruskal's Algorithm

• Compares distance/weight of edges (edge based) instead of distance/weight from vertex to spanning tree (Prim's, vertex based)
• Use weighted quick union to determine whether edge creates cycle

Prim's vs Kruskal's

• If unique weight edges → should get same answer for both (b/c only 1 MST)
• Both ultimately rely on cut property

Kruskal's Runtime

• $V$ connect operations b/c MST has total $V - 1$ edges/connections

Shortest Paths & MST Algorithms Summary