The bottleneck

Binary heap decrease-key = O(log V). E decrease-keys → O(E log V). Fib heap collapses this to O(E).

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Fibonacci heap essentials

Lazy structure: consolidate only on extract-min. Trees of arbitrary rank. Amortized bounds via potential function.

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Complexity

Extract-min V times: O(V log V). Decrease-key E times: O(E). Total: O(E + V log V).

Practical reality

Constants are HUGE. Fibonacci heaps rarely beat binary heaps in practice for typical graph sizes. Binary heap wins to N ≈ 10⁷.