Top-down Mergesort with a sorted check before each merge has merge cost M ≤ (H+3)n for any input, where H is the run-length entropy.
Efficient Algorithms for Envy-Free Stick Division With Fewest Cuts
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abstract
Given a set of n sticks of various (not necessarily different) lengths, what is the largest length so that we can cut k equally long pieces of this length from the given set of sticks? We analyze the structure of this problem and show that it essentially reduces to a single call of a selection algorithm; we thus obtain an optimal linear-time algorithm. This algorithm also solves the related envy-free stick-division problem, which Segal-Halevi, Hassidim, and Aumann (AAMAS, 2015) recently used as their central primitive operation for the first discrete and bounded envy-free cake cutting protocol with a proportionality guarantee when pieces can be put to waste.
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Top-Down Mergesort with Sorted Check Has Mergecost $\le(\mathcal H+3)n$
Top-down Mergesort with a sorted check before each merge has merge cost M ≤ (H+3)n for any input, where H is the run-length entropy.