Three Puzzles on Mathematics, Computation, and Games

In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure

Experimental investigation of an interior search method within a simple framework

A steepest gradient method for solving Linear Programming (LP) problems, followed by a procedure for purifying a non-basic solution to an improved extreme point solution have been embedded within an otherwise simplex based optimiser. The algorithm is designed to be hybrid in nature and exploits many aspects of sparse matrix and revised simplex technology. The interior search step terminates at a boundary point which is usually non-basic. This is then followed by a series of minor pivotal steps which lead to a basic feasible solution with a superior objective function value. It is concluded that the procedures discussed in this paper are likely to have three possible applications, which are (i) improving a non-basic feasible solution to a superior extreme point solution, (iii) an improved starting point for the revised simplex method, and (iii) an efficient implementation of the multiple price strategy of the revised simplex method

Analysis of pivot sampling in dual-pivot Quicksort: A holistic analysis of Yaroslavskiy's partitioning scheme

The final publication is available at Springer via http://dx.doi.org/10.1007/s00453-015-0041-7The new dual-pivot Quicksort by Vladimir Yaroslavskiy-used in Oracle's Java runtime library since version 7-features intriguing asymmetries. They make a basic variant of this algorithm use less comparisons than classic single-pivot Quicksort. In this paper, we extend the analysis to the case where the two pivots are chosen as fixed order statistics of a random sample. Surprisingly, dual-pivot Quicksort then needs more comparisons than a corresponding version of classic Quicksort, so it is clear that counting comparisons is not sufficient to explain the running time advantages observed for Yaroslavskiy's algorithm in practice. Consequently, we take a more holistic approach and give also the precise leading term of the average number of swaps, the number of executed Java Bytecode instructions and the number of scanned elements, a new simple cost measure that approximates I/O costs in the memory hierarchy. We determine optimal order statistics for each of the cost measures. It turns out that the asymmetries in Yaroslavskiy's algorithm render pivots with a systematic skew more efficient than the symmetric choice. Moreover, we finally have a convincing explanation for the success of Yaroslavskiy's algorithm in practice: compared with corresponding versions of classic single-pivot Quicksort, dual-pivot Quicksort needs significantly less I/Os, both with and without pivot sampling.Peer ReviewedPostprint (author's final draft

On the Asymptotic Average Number of Efficient Vertices in Multiple Objective Linear Programming

AbstractLeta1,…,am,c1,…,ckbe independent random points in Rnthat are identically distributed spherically symmetrical in Rnand letX≔{x∈Rn|aTix⩽1,i=1,…,m} be the associated random polyhedron form⩾n⩾2. We consider multiple objective linear programming problems maxx∈XcT1x, maxx∈XcT2x,…,maxx∈XcTkxwith 1⩽k⩽n. For distributions with algebraically decreasing tail in the unit ball, we investigate the asymptotic expected number of vertices in the efficient frontier ofXwith respect toc1,…,ckfor fixedn,kandm→∞. This expected number of efficient vertices is the most significant indicator for the average-case complexity of the multiple objective linear programming problem

Analysis of Quickselect under Yaroslavskiy's Dual-Pivoting Algorithm

There is excitement within the algorithms community about a new partitioning method introduced by Yaroslavskiy. This algorithm renders Quicksort slightly faster than the case when it runs under classic partitioning methods. We show that this improved performance in Quicksort is not sustained in Quickselect; a variant of Quicksort for finding order statistics. We investigate the number of comparisons made by Quickselect to find a key with a randomly selected rank under Yaroslavskiy's algorithm. This grand averaging is a smoothing operator over all individual distributions for specific fixed order statistics. We give the exact grand average. The grand distribution of the number of comparison (when suitably scaled) is given as the fixed-point solution of a distributional equation of a contraction in the Zolotarev metric space. Our investigation shows that Quickselect under older partitioning methods slightly outperforms Quickselect under Yaroslavskiy's algorithm, for an order statistic of a random rank. Similar results are obtained for extremal order statistics, where again we find the exact average, and the distribution for the number of comparisons (when suitably scaled). Both limiting distributions are of perpetuities (a sum of products of independent mixed continuous random variables).Comment: full version with appendices; otherwise identical to Algorithmica versio

Compositional data analysis applied to a study of movement behaviours of recent retirees

Studies of movement behaviours have numerous applications. A recent approach involves studying the time spent on different activity types using compositional data analysis. In compositional data analysis, several variables are constrained to an arbitrary sum and the primary interest is their proportions of the whole. This thesis explores the mathematical foundations of the study of compositional data and their practical applications. First, mathematical operations are defined for compositions using Aitchison geometry. Methods are presented for transforming compositions into real-valued coordinates and back. Various statistical methods are also defined for compositions and compositional data. Some of the techniques presented are demonstrated by applying them to a study of movement behaviours. REACT is a randomized controlled trial study focusing on whether commercial activity trackers affect movement behaviours among the recently retired. By using compositional data analysis, the proportions of time spent on different activity types can be studied. Based on the results, it would appear that those who used activity trackers spent a slightly higher portion of their day on physical activity than those who did not