This paper studies the average complexity on the number of comparisons for sorting algorithms. Its information-theoretic lower bound is n lg n − 1.4427n + O(log n). For many efficient algorithms, the first n lg n term is easy to achieve and our focus is on the (negative) constant factor of the linear term. The current best value is −1.3999 for the MergeInsertion sort. Our new value is −1.4106, narrowing the gap by some 25%. An important building block of our algorithm is "two-element insertion," which inserts two numbers A and B, A < B, into a sorted sequence T . This insertion algorithm is still sufficiently simple for rigorous mathematical analysis and works well for a certain range of the length of T for which the simple binary insertion does not, thus allowing us to take a complementary approach with the binary insertion.

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Abstract. The counterfeit coin problem requires us to find all false coins from a given bunch of coins using a balance scale. We assume that the balance scale gives us only "balanced" or "tilted" information and that we know the number k of false coins in advance. The balance scale can be modeled by a certain type of oracle and its query complexity is a measure for the cost of weighing algorithms (the number of weighings). In this paper, we study the quantum query complexity for this problem. Let Q(k, N ) be the quantum query complexity of finding all k false coins from the N given coins. We show that for any k and N such that k < N/2, Q(k, N ) = O(k 1/4 ), contrasting with the classical query complexity, Ω(k log(N/k)), that depends on N . So our quantum algorithm achieves a quartic speed-up for this problem. We do not have a matching lower bound, but we show some evidence that the upper bound is tight: any algorithm, including our algorithm, that satisfies certain properties needs Ω(k 1/4 ) queries.

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We investigate the following sorting problem: We are given n bins with two balls in each bin. Balls in the ith bin are numbered n + 1 − i. We can swap two balls from adjacent bins. How many number of swaps are needed in order to sort balls, i.e., move every ball to the bin with the same number. For this problem the best known solution requires almost 4 5 n 2 swaps. In this paper, we show an algorithm which solves this problem using less than 2n 2 3 swaps. Since it is known that the lower bound of the number of swaps is 2n 2 /3 = 2n 2 3 − n 3 , our result is almost tight. Furthermore, we show that for n = 2 m + 1 (m ≥ 0) the algorithm is optimal.