Prims Algorithm

Prims Algorithm
Runtime	Heap: O(E log V) Array: O(V^2)
Class	Minimum_Spanning_Tree Graph Algorithm Greedy Algorithm

Approach: Greedy

See Minimum_Spanning_Tree

Implementation

for each u in V:
    key[u] = infinity   // cost array
    pi[u] = infinity    // from array
Q = new PriorityQueue(V)
key[root] = 0
while Q is not empty:
    u = extractMin(Q)
    # Reduce nodes
    for v in adj[u]:
        if v in Q and w[u,v] < key[v]:
            key[v] = w[u,v]

Analysis

Priority queue is slower than array when the graph is dense, so sometimes it's better to use Dijsktra's algorithm.

Proof: Greedy

Suboptimality

Given MST tree for the graph G

$OPT={\text{set of edges}}$

where the number of edges is 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 n - 1} , the weight of edges is minimum, and the tree is spanning.

Let 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 (x,y)} be a leaf edge in the tree.