CS 411 Fall 2025  >  Outline for October 31, 2025

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CS 411 Fall 2025
Outline
for October 31, 2025

Outline

Greedy Techniques [L Ch 9 intro]

• Idea: be short-sighted. At each step, make what seems to be the best choice right now.
• Greedy techniques, at each step, make a choice that is:
  • Feasible (legal, within constraints of problem).
  • Locally optimal.
  • Irrevocable.
• Being greedy usually gives incorrect results.
  • Example of maximum weight independent set in vertex-weighted graph.
• We will discuss algorithms in which working greedily always gives a correct result. Greedy techniques are also usually very fast.

Spanning Trees I: Prim’s Algorithm [L 9.1]

• Spanning Trees
  • Connected graph
  • Tree
  • Spanning tree
  • (Edge-)weighted graph
  • Minimum spanning tree
  • Minimum Spanning Tree Problem
• Two greedy algorithmic ideas
  • Idea #1 (Prim’s Algorithm—today). Choose a vertex, and count it as reachable. Then repeat: add to the spanning tree the least-weight edge from a reachable vertex to an unreachable vertex—which then becomes reachable.
  • Idea #2 (Kruskal’s Algorithm—next time). Repeat: Add to a collection of edges the least-weight edge between two vertices that we cannot currently travel between (that is, the least-weight edge that, when added to the collection, will not form a cycle).
• Prim’s Algorithm: Correctness
• Prim’s Algorithm: Implementation
  • For efficiency, use Priority Queue to select least-weight edge.
• Prim’s Algorithm: Efficiency