Related papers: On computing the total displacement number via wei…
Path integrals represent a powerful route to quantization: they calculate probabilities by summing over classical configurations of variables such as fields, assigning each configuration a phase equal to the action of that configuration.…
We give a simple combinatorial proof of a formula that extends a result by Grigorchuk (rediscovered by Cohen) relating cogrowth and spectral radius of random walks. Our main result is an explicit equation determining the number of `bumps'…
We address the problem of defining a network graph on a large collection of classes. Each class is comprised of a collection of data points, sampled in a non i.i.d. way, from some unknown underlying distribution. The application we consider…
We show that weighted path orders are special instances of a variant of semantic path orders. Exploiting this fact, we introduce a generalization of weighted path orders that goes beyond the realm of simple termination. Experimental data…
For $s$ $>$ 0, we consider an algorithm that computes all $s$-well separated pairs in certain point sets in $\mathbb{R}^{n}$, $n$ $>1$. For an integer $K$ $>1$, we also consider an algorithm that is a permutation of Dijkstra's algorithm,…
Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to…
Many problems in geometric optics or convex geometry can be recast as optimal transport problems: this includes the far-field reflector problem, Alexandrov's curvature prescription problem, etc. A popular way to solve these problems…
Modern computer networks support interesting new routing models in which traffic flows from a source s to a destination t can be flexibly steered through a sequence of waypoints, such as (hardware) middleboxes or (virtualized) network…
We present an algorithm for evaluating a linear ``intersection transform'' of a function defined on the lattice of subsets of an $n$-element set. In particular, the algorithm constructs an arithmetic circuit for evaluating the transform in…
We compute the number of ways a given permutation can be written as a product of exactly $k$ transpositions. We express this number as a linear combination of explicit geometric sequences, with coefficients which can be computed in many…
We introduce the notion of combinatorial encoding of continuous dynamical systems and suggest the first examples, which are the most interesting and important, namely, the combinatorial encoding of a Bernoulli process with continuous state…
We compare different analytical and numerical methods for studying the partitions of a finite system into fragments. We propose a new numerical method of exploring the partition space by generating the Markov chains of partitions based on…
This paper concentrates on the set $\mathcal{V}_n$ of weighted Dyck paths of length $2n$ with special restrictions on the level of valleys. We first give its explicit formula of the counting generating function in terms of certain weight…
We define a map from subspaces to Motzkin paths and show that the inverse image of every path is a disjoint union of symmetric Boolean subsets yielding an explicit symmetric Boolean decomposition of the subspace lattice.
A motion planning algorithm computes the motion of a robot by computing a path through its configuration space. To improve the runtime of motion planning algorithms, we propose to nest robots in each other, creating a nested quotient-space…
It is shown that the combinatorics of commutation relations is well suited for analyzing the convergence rate of certain Markov chains. Examples studied include random walk on irreducible representations, a local random walk on partitions…
Given two distributions $P$ and $S$ of equal total mass, the Earth Mover's Distance measures the cost of transforming one distribution into the other, where the cost of moving a unit of mass is equal to the distance over which it is moved.…
We develop a complete theory for the combinatorics of walk-counting on a directed graph in the case where each backtracking step is downweighted by a given factor. By deriving expressions for the associated generating functions, we also…
The Bin Packing Problem involves efficiently packing items into a limited number of bins without exceeding their capacity. In this paper, we try to answer a specific question in this field. Mathematically the combinatorial optimization…
This paper presents a minimum displacement motion planning problem wherein obstacles are displaced by a minimum amount to find a feasible path. We define a metric for robot-obstacle intersection that measures the extent of the intersection…