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This paper studies the average complexity on the number of comparisons for sorting algorithms. Its information-theoretic lower bound is n lg ⁡ n − 1.4427 n + 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 elements 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 together with the binary insertion. [ABSTRACT FROM AUTHOR]

• Study the minsum k -sink problems on dynamic flow path networks for the confluent/non-confluent flow model. • Develop algorithms which run in almost linear time regardless of the number of sinks k. • Improve upon the previous results for the same problem with the confluent flow model. • Provide the first polynomial time algorithm for the problem with the non-confluent flow model. • The main theoretical contribution is to construct novel data structures for solving subproblems in polylogarithmic time. We address the facility location problems on dynamic flow path networks. A dynamic flow path network consists of an undirected path with positive edge lengths, positive edge capacities, and positive vertex weights. A path can be considered as a road, an edge length as the distance along the road and a vertex weight as the number of people at the site. An edge capacity limits the number of people that can enter the edge per unit time. In the dynamic flow network, given particular points on edges or vertices, called sinks , all the people evacuate from the vertices to the sinks as quickly as possible. The problem is to find the location of sinks on a dynamic flow path network in such a way that the aggregate evacuation time (i.e., the sum of evacuation times for all the people) to sinks is minimized. We consider two models of the problem: the confluent flow model and the non-confluent flow model. In the former model, the way of evacuation is restricted so that all the people at a vertex have to evacuate to the same sink, and in the latter model, there is no such restriction. In this paper, for both the models, we develop algorithms which run in almost linear time regardless of the number of sinks. It should be stressed that for the confluent flow model, our algorithm improves upon the previous result by Benkoczi et al. [Theoretical Computer Science, 2020], and one for the non-confluent flow model is the first polynomial time algorithm. [ABSTRACT FROM AUTHOR]

We present improved exponential time exact algorithms for Max SAT. Our algorithms run in time of the form O ( 2 ( 1 − μ ( c ) ) n ) for instances with n variables and m = c n clauses. In this setting, there are three incomparable currently best algorithms: a deterministic exponential space algorithm with μ ( c ) = 1 O ( c log ⁡ c ) due to Dantsin and Wolpert (2006) [9] , a randomized polynomial space algorithm with μ ( c ) = 1 O ( c log 3 ⁡ c ) and a deterministic polynomial space algorithm with μ ( c ) = 1 O ( c 2 log 2 ⁡ c ) due to Sakai, Seto and Tamaki (2015) [30] . Our first result is a deterministic polynomial space algorithm with μ ( c ) = 1 O ( c log ⁡ c ) that achieves the previous best time complexity without exponential space or randomization. Furthermore, this algorithm can handle instances with exponentially large weights and hard constraints. The previous algorithms and our deterministic polynomial space algorithm run super-polynomially faster than 2 n only if m = O ( n 2 ) . Our second results are deterministic exponential space algorithms for Max SAT with μ ( c ) = 1 O ( ( c log ⁡ c ) 2 / 3 ) and for Max 3-SAT with μ ( c ) = 1 O ( c 1 / 2 ) that run super-polynomially faster than 2 n when m = o ( n 5 / 2 / log 5 / 2 ⁡ n ) and m = o ( n 3 / log 2 ⁡ n ) respectively. Our third result is a randomized exponential space algorithm for Max SAT with μ ( c ) = 1 O ( c 1 / 2 log 3 / 2 ⁡ c ) that run super-polynomially faster than 2 n when m = o ( n 3 / log 5 ⁡ n ) . [ABSTRACT FROM AUTHOR]