We introduce new stable natural merge sort algorithms, called 2-merge sort and α-merge sort. We prove upper and lower bounds for several merge sort algorithms, including Timsort, Shivers' sort, α-stack sorts, and our new 2-merge and α-merge sorts. The upper and lower bounds have the forms c · n log m and c · n log n for inputs of length n comprising m monotone runs. For Timsort, we prove a lower bound of (1.5 − o(1))n log n. For 2-merge sort, we prove optimal upper and lower bounds of approximately (1.089 ± o(1))n log m. We prove similar asymptotically matching upper and lower bounds for α-merge sort, when ϕ < α < 2, where ϕ is the golden ratio.Our bounds are in terms of merge cost; this upper bounds the number of comparisons and accurately models runtime. The merge strategies can be used for any stable merge sort, not just natural merge sorts. The new 2-merge and α-merge sorts have better worst-case merge cost upper bounds and are slightly simpler to implement than the widely-used Timsort; they also perform better in experiments. We report also experimental comparisons with algorithms developed by Munro-Wild and Jugé subsequently to the results of the present paper.

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Let 𝑓 : {0, 1} 𝑛 → {0, 1} be a boolean function, and let 𝑓 ∧ (𝑥, 𝑦) = 𝑓 (𝑥 ∧𝑦) denote the AND-function of 𝑓 , where 𝑥 ∧𝑦 denotes bit-wise AND. We study the deterministic communication complexity of 𝑓 ∧ and show that, up to a log 𝑛 factor, it is bounded by a polynomial in the logarithm of the real rank of the communication matrix of 𝑓 ∧ . This comes within a log 𝑛 factor of establishing the log-rank conjecture for AND-functions with no assumptions on 𝑓 . Our result stands in contrast with previous results on special cases of the logrank conjecture, which needed significant restrictions on 𝑓 such as monotonicity or low F 2 -degree. Our techniques can also be used to prove (within a log 𝑛 factor) a lifting theorem for AND-functions, stating that the deterministic communication complexity of 𝑓 ∧ is polynomially related to the AND-decision tree complexity of 𝑓 .The results rely on a new structural result regarding boolean functions 𝑓 : {0, 1} 𝑛 → {0, 1} with a sparse polynomial representation, which may be of independent interest. We show that if the polynomial computing 𝑓 has few monomials then the set system of the monomials has a small hitting set, of size poly-logarithmic in its sparsity. We also establish extensions of this result to multi-linear polynomials 𝑓 : {0, 1} 𝑛 → R with a larger range. CCS CONCEPTS• Theory of computation → Communication complexity; Oracles and decision trees.

Eleven X-chromosomal STRs from two multiplex PCR approaches (DXS6807, DXS8378, DXS7132, DXS6800, DXS9898, DXS7424, DXS101, DXS7133, HPRTB, DXS8377, and DXS7423), located in four different X-chromosomal linkage groups, were typed in two population samples from Africa, Morocco, and Madagascar. Forensic efficiency parameters such as polymorphism information content and mean exclusion chance were calculated. A deviation from the Hardy–Weinberg equilibrium could not be found. The investigation of four father–daughter and five mother–son meioses (from Morocco) revealed no mutations in any STR analyzed. Our data were compared with European, African-American, and Asian populations from the literature.

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