Minimum Spanning Tree

A minimum spanning tree is

• a tree, meaning it has no cycle
• minimum, meaning it has minimum weight
• spanning, meaning it connects all nodes

MST Problem

The MST problem takes a connected graph Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} and outputs an MST for that graph.

Approach: Greedy

The approach is to try to add the smallest edges as long as they do not create a cycle; add an edge to the tree that is minimum across the cut of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} vs. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V - T}

Given the MST of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{n - 1} = v_1, v_2, \ldots, v_{n-1} } , the MST of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} should be that of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{n-1}} plus the edge that connects to $v_{n}$ that is the shortest.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle OPT(n) = OPT(n-1) + min( (v_i, v_n) \in E ) }

Implementation

MST(G):
    mst_nodes = start_node
    mst = adjacency list with no edges
    # candidate_edges is a min heap
    candidate_edges = (start_node, G[start_node])
    while mst_nodes is not spanning:
        new_edge = candidate_edges.pop()