Why it matters

Prim is alternative MST. Faster than Kruskal on dense graphs.

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The architecture

Start from any vertex.

Extract-min from priority queue.

Relax adjacent edges.

Prim MSTPQ startsource vertexExtract minclosest vertexRelax edgesupdate PQFibonacci heap gives O(E + V log V); rarely faster in practice
Prim.
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How it works end to end

Priority queue: min-heap by edge weight.

Extract-min V times.

Decrease-key on relax.

Fibonacci heap: better amortized decrease-key.