This article extends recent results on log-Coulomb gases in a $$p$$ -field $$K$$ (i.e., nonarchimedean local field) to those its projective line $$\mathbb{P}^1(K)$$ , where the latter is endowed with $$PGL_2$$ -invariant Borel probability measure and spherical metric. Our first main result an explicit combinatorial formula for canonical partition function of arbitrary charge values. second call...

Recently, Pawlik et al. have shown that triangle-free intersection graphs of line segments in the plane can have arbitrarily large chromatic number. Specifically, they construct triangle-free segment intersection graphs with chromatic number Θ(log log n). Essentially the same construction produces Θ(log log n)-chromatic triangle-free intersection graphs of a variety of other geometric shapes—th...

A basic question in computational geometry is how to find the relationship between a set of points and a line in a real plane. In this paper, we present multidimensional data structures for N points that allow answering the following queries for any given input line: (1) estimate in O(log N) time the number of points below the line; (2) return in O(log N + k) time the k ≤ N points that are belo...

We give a tight analysis of an old and popular sweep-line heuristic for constructing a spanning tree of a set of n points in the plane. The algorithm sweeps a vertical line across the input points from left to right, and each point is connected by a straight line segment to the closest point left of (or on) the sweep-line. If W denotes the weight the Euclidean minimum spanning tree (EMST), the ...

In this paper, we consider the problems for covering multiple intervals on a line. Given a set B of m line segments (called “barriers”) on a horizontal line L and another set S of n horizontal line segments of the same length in the plane, we want to move all segments of S to L so that their union covers all barriers and the maximum movement of all segments of S is minimized. Previously, an O(n...

The paper deals with on-line routing in WDM (wavelength division multiplexing) optical networks. A sequence of requests arrives over time, each is a pair of nodes to be connected by a path. The problem is to assign a wavelength and a path to each pair, so that no two paths sharing a link are assigned the same wavelength. The goal is to minimize the number of wavelengths used to establish all co...

An edge-ordered graph is a with total ordering of its edges. A path $$P=v_1v_2\ldots v_k$$ in an called increasing if $$(v_iv_{i+1}) < (v_{i+1}v_{i+2})$$ for all $$i = 1,\ldots,k-2$$ ; and it decreasing > . We say that P monotone or decreasing. rooted tree T either every from the root to leaf Let G be graph. In straight-line drawing D G, vertices are drawn as different points plane edges straig...