Theorem
Any MST is also a minimum bottleneck spanning tree. Converse false — an MBST is not necessarily minimum-sum.
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Faster than MST
MBST can be found in O(E) via Camerini's algorithm (median-based).
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Camerini's sketch
Find median-weight edge. If lighter half + heavier half connect graph, recurse. Otherwise adjust.
Applications
Robust network design where worst-case link matters (max-min throughput).