ﻻ يوجد ملخص باللغة العربية
The classic Ham-Sandwich theorem states that for any $d$ measurable sets in $mathbb{R}^d$, there is a hyperplane that bisects them simultaneously. An extension by Barany, Hubard, and Jeronimo [DCG 2008] states that if the sets are convex and emph{well-separated}, then for any given $alpha_1, dots, alpha_d in [0, 1]$, there is a unique oriented hyperplane that cuts off a respective fraction $alpha_1, dots, alpha_d$ from each set. Steiger and Zhao [DCG 2010] proved a discrete analogue of this theorem, which we call the emph{$alpha$-Ham-Sandwich theorem}. They gave an algorithm to find the hyperplane in time $O(n (log n)^{d-3})$, where $n$ is the total number of input points. The computational complexity of this search problem in high dimensions is open, quite unlike the complexity of the Ham-Sandwich problem, which is now known to be PPA-complete (Filos-Ratsikas and Goldberg [STOC 2019]). Recently, Fearley, Gordon, Mehta, and Savani [ICALP 2019] introduced a new sub-class of CLS (Continuous Local Search) called emph{Unique End-of-Potential Line} (UEOPL). This class captures problems in CLS that have unique solutions. We show that for the $alpha$-Ham-Sandwich theorem, the search problem of finding the dividing hyperplane lies in UEOPL. This gives the first non-trivial containment of the problem in a complexity class and places it in the company of classic search problems such as finding the fixed point of a contraction map, the unique sink orientation problem and the $P$-matrix linear complementarity problem.
Inspired by connections to two dimensional quantum theory, we define several models of computation based on permuting distinguishable particles (which we call balls), and characterize their computational complexity. In the quantum setting, we find th
This short survey of recent work in tile self-assembly discusses the use of simulation to classify and separate the computational and expressive power of self-assembly models. The journey begins with the result that there is a single universal tile s
Consider a graph with a rotation system, namely, for every vertex, a circular ordering of the incident edges. Given such a graph, an angle cover maps every vertex to a pair of consecutive edges in the ordering -- an angle -- such that each edge parti
Diffusion-Limited Aggregation (DLA) is a cluster-growth model that consists in a set of particles that are sequentially aggregated over a two-dimensional grid. In this paper, we introduce a biased version of the DLA model, in which particles are limi
A emph{2-interval} is the union of two disjoint intervals on the real line. Two 2-intervals $D_1$ and $D_2$ are emph{disjoint} if their intersection is empty (i.e., no interval of $D_1$ intersects any interval of $D_2$). There can be three different