Data structures & algorithmsLesson 2 of 2 in the course Graph algorithms

Minimum Spanning Trees: Properties, Algorithms, Complexity, and Applications

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

This lesson introduces spanning trees, minimum spanning trees, and the minimum spanning forest. These ideas help model economical ways to connect locations or systems.

An edge-weighted undirected graph has vertices, undirected edges, and a weight assigned to each edge. A weight can represent distance, time, construction cost, or another quantity. It need not be a geometric distance and does not have to satisfy the triangle inequality.

All flashcards

What is the comparison model for edge weights?
A model in which an algorithm learns about weights only by comparing pairs of edges.
What is a soft heap?
An approximate priority queue that may corrupt a small number of keys.
What does it mean for a comparison-based MST algorithm to be optimal?
It uses the fewest comparisons possible in the worst case under its decision-tree formulation.
What is a Steiner tree problem?
Finding a minimum-weight tree connecting a specified subset of vertices.
How can an MST support single-linkage clustering?
It provides a compact structure for representing connections among data points.

Quiz questions

  1. In a cycle, one edge is strictly heavier than every other edge. What can be concluded about that edge?Answer: It belongs to no minimum spanning tree.
  2. An edge is the unique lightest edge crossing a cut. What does the cut property imply?Answer: The edge belongs to every minimum spanning tree.
  3. How does Kruskal's algorithm decide whether to add an edge?Answer: It adds the edge when its endpoints are in different components.
  4. What is the main repeated step in Borůvka's algorithm?Answer: Choose a cheapest outgoing edge for each current component, then contract connected components.
  5. Which statement correctly relates minimum spanning trees and minimum bottleneck spanning trees?Answer: Every minimum spanning tree is a minimum bottleneck spanning tree, but not conversely.

Based on the Wikipedia article Minimum spanning tree, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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