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Minimum spanning tree

Find a minimum spanning tree of a graph.

Minimum spanning tree

In a weighted graph, the tree connecting all nodes with the smallest total weight is called the minimum spanning tree (MST). A tree with n nodes has n-1 edges.

Kruskal's algorithm sorts edges by weight and adds those that do not form a cycle, building the MST.

Edge list

Kruskal works with an edge list. Each edge is a (u, v, w) triple: from u to v with weight w. The graph is undirected.

We will find the MST in an example graph with five edges. Edges are defined in raw form before sorting.

Practice init_edges function

Write init_edges(). Return the edge list. Print it with print(init_edges()).

Starter code

solution.py
def init_edges():
    # Return edge list

Sample solution

solution.py
def init_edges():
    # Return edge list
    return [(0, 1, 10), (0, 2, 6), (0, 3, 5), (1, 3, 15), (2, 3, 4)]
print(init_edges())

Expected output

[(0, 1, 10), (0, 2, 6), (0, 3, 5), (1, 3, 15), (2, 3, 4)]

Questions and answers

Review the lesson questions together with their answers. To test yourself, solve the lesson interactively.

1 How many edges does an MST with 4 nodes have?

  • 3 Correct answer
  • 2
  • 4
  • 5

Explanation: n-1 = 4-1 = 3 edges.

2 Smallest edge

Question 
min_edge = (____, 3, 4)
Answer 
min_edge = (2, 3, 4)

Explanation: 2 is right.

3 Minimum spanning tree: match the terms

Term Meaning
(2,3,4) Smallest
(0,3,5) Second
(0,1,10) Fourth
(1,3,15) Largest

All lessons in Advanced

1 DP - Memoization

  1. What is memoization?
  2. Fibonacci memoization
  3. Dict-based cache
  4. Summary - Memoization

2 DP - Tabulation

  1. What is tabulation?
  2. Building the table
  3. Fibonacci tabulation
  4. Summary - Tabulation

3 LCS

  1. LCS definition
  2. DP table
  3. Backtracking the table
  4. Summary - LCS

4 Knapsack

  1. Knapsack problem
  2. 0/1 knapsack DP
  3. Decision table
  4. Summary - Knapsack

5 DP project

  1. Project: Overview
  2. Cost table
  3. DP solution
  4. Print the result

6 Matrix chain

  1. Multiplication order
  2. DP table
  3. Split point
  4. Summary - Matrix chain

7 Floyd-Warshall

  1. All pairs
  2. Triple loop
  3. Table update
  4. Summary - Floyd-Warshall

8 Dijkstra

  1. Single source
  2. Priority queue
  3. Relaxation
  4. Summary - Dijkstra

9 Bellman-Ford

  1. Negative weights
  2. Edge relaxation loop
  3. Negative cycle
  4. Summary - Bellman-Ford

10 Kruskal

  1. Minimum spanning tree
  2. Edge sorting
  3. Cycle check
  4. Summary - Kruskal

11 Prim

  1. Prim idea
  2. Priority queue
  3. Visited set
  4. Summary - Prim

12 Topological sort

  1. Dependency order
  2. DFS-based sort
  3. Kahn algorithm
  4. Summary - Topological sort

13 Union-find, trie, segment tree

  1. Union-find
  2. Trie
  3. Segment tree
  4. Summary - Advanced structures

14 Advanced final

  1. Challenge 1
  2. Challenge 2
  3. Challenge 3
  4. Challenge 4