Long Common Subsequences and the Proximity of Two Random Strings

Let $( x_1 ,x_2 , \cdots x_n )$ and $( x\u27_1 ,x\u27_2 , \cdots x\u27_n , )$ be two strings from an alphabet $mathcal{A}$, and let $L_n$ denote their longest common subsequence. The probabilistic behavior of $L_n$ is studied under various probability models for the x’s and $x\u27$’s

Shortest Paths Through Pseudo-Random Points in the dd-Cube

A lower bound for the length of the shortest path through n points in [0, Ild is given in terms of the discrepancy function of the n points. This bound is applied to obtain an analogue for several pseudorandom sequences to the known limit behavior of the length of the shortest path through n independent uniformly distributed random observations from [0, l]d

Darrell Huff and Fifty Years of How to Lie with Statistics

Over the last fifty years, How to Lie with Statistics has sold more copies than any other statistical text. This note explores the factors that contributed to its success and provides biographical sketches of its creators: author Darrell Huff and illustrator Irving Geis

Probability and Problems in Euclidean Combinatorial Optimization

This article summarizes the current status of several streams of research that deal with the probability theory of problems of combinatorial optimization. There is a particular emphasis on functionals of finite point sets. The most famous example of such functionals is the length associated with the Euclidean traveling salesman problem (TSP), but closely related problems include the minimal spanning tree problem, minimal matching problems and others. Progress is also surveyed on (1) the approximation and determination of constants whose existence is known by subadditive methods, (2) the central limit problems for several functionals closely related to Euclidean functionals, and (3) analogies in the asymptotic behavior between worst-case and expected-case behavior of Euclidean problems. No attempt has been made in this survey to cover the many important applications of probability to linear programming, arrangement searching or other problems that focus on lines or planes

The Bohnenblust–Spitzer Algorithm and its Applications

The familiar bijections between the representations of permutations as words and as products of cycles have a natural class of “data driven” extensions that permit us to use purely combinatorial means to obtain precise probabilistic information about the geometry of random walks. In particular, we show that the algorithmic bijection of Bohnenblust and Spitzer can be used to obtain means, variances, and concentration inequalities for several random variables associated with a random walk including the number of vertices and length of the convex minorant, concave majorant, and convex hull

Complete Convergence of Short Paths and Karp\u27s Algorithm for the TSP

Let Xi, 1 ≤ i \u3c ∞, be uniformly distributed in [0, 1]2 and let Tn be the length of the shortest closed path connecting {X1, X2, …, Xn}. It is proved that there is a constant 0 \u3c β \u3c ∞ such that for all ϵ \u3e 0 ∑n=1∞p(|Tn/n−β‾‾‾‾‾√|\u3eϵ)\u3c∞.n1∞pTnnβϵ∞ This result is essential in justifying Karp\u27s algorithm for the traveling salesman problem under the independent model, and it settles a question posed by B. W. Weide

A Counterexample Related to a Criterion for a Function to be Continuous

A construction is given of a right-continuous function which satisfies an arithmetic continuity condition, but which is not continuous. The problem is motivated by a familiar criterion for the continuity of sample paths of a stochastic process

Subadditive Euclidean Functionals and Nonlinear Growth in Geometric Probability

A limit theorem is established for a class of random processes (called here subadditive Euclidean functionals) which arise in problems of geometric probability. Particular examples include the length of shortest path through a random sample, the length of a rectilinear Steiner tree spanned by a sample, and the length of a minimal matching. Also, a uniform convergence theorem is proved which is needed in Karp\u27s probabilistic algorithm for the traveling salesman problem

Sums of Squares of Edge Lengths and Spacefilling Curve Heuristics for the Traveling Salesman Problem

The sum of squares of the edge lengths of the tour provided by the spacefilling curve heuristic applied to a random sample of n points from the unit square is proved to be asymptotically equal to a periodic function of the logarithm of the sample size