We present two new versions of the chromatic art gallery problem that can improve upper bound of the required colors pretty well. In our version, we employ restricted angle guards so that these modern guards can visit α-degree of their surroundings. If α is between 0 and 180 degree, we demonstrate that the strong chromatic guarding number is constant. Then we use orthogonal 90-degree guards for...

The paper presents a parallel algorithm for polygon approximation targeted at reconfigurable multi-ring hardware. The proposed algorithm grows the edges of polygon approximation that is based on principle of merging. The edge/s are grown simultaneously at point/s where the minimum merging error is produced. The merging process is made faster by carrying it out in two stages: i) During first sta...

Let Fq be a finite field of cardinality q and characteristic p. Let P (x) be any one-variable Laurent polynomial over Fq of degree (d1, d2) respectively and p d1d2. For any fixed s ≥ 1 coprime to p, we prove that the q-adic Newton polygon of the L-functions of exponential sums of P (xs) has a tight lower bound which we call HodgeStickelberger polygon, depending only on the d1, d2, s and the res...

Given a graph G with n vertices and a set S of n points in the plane, a point-set embedding of G on S is a planar drawing such that each vertex of G is mapped to a distinct point of S. A straight-line point-set embedding is a point-set embedding with no edge bends or curves. The point-set embeddability problem is NP-complete, even when G is 2-connected and 2-outerplanar. It has been solved poly...

In Chapter 1 it was shown that Fisk's proof of the art gallery theorem can be converted into an algorithm that covers a polygon of n vertices with [n/3\ guards in O{n logn) time. Although [n/3\ is necessary in some cases, often this is far more guards than are needed to cover a particular polygon. For example, convex polygons only require one guard, but Avis and Toussaint's algorithm would stil...

Let Fq be a finite field of cardinality q and characteristic p. Let P (x) be any one-variable Laurent polynomial over Fq of degree (d1, d2) respectively and p d1d2. For any fixed s ≥ 1 coprime to p, we prove that the q-adic Newton polygon of the L-functions of exponential sums of P (xs) has a tight lower bound which we call Hodge-Stickelberger polygon, depending only on the d1, d2, s and the re...

How many different classes of partially distinguishable landmarks are needed to ensure that a robot can always see a landmark without simultaneously seeing two of the same class? To study this, we introduce the chromatic art gallery problem. A guard set S ⊂ P is a set of points in a polygon P such that for all p ∈ P , there exists an s ∈ S such that s and p are mutually visible. Suppose that tw...

The art gallery problem asks for the smallest number of guards required to see every point of the interior of a polygon P . We introduce and study a similar problem called the chromatic art gallery problem. Suppose that two members of a finite point guard set S ⊂ P must be given different colors if their visible regions overlap. What is the minimum number of colors required to color any guard s...

In 1973, Victor Klee posed the problem of determining the minimum number of guards sufficient to cover the interior of an n-wall art gallery room (Honsberger 1976). He posed this question extemporaneously in response to a request from Vasek Chvatal (at a conference at Stanford in August) for an interesting geometric problem, and Chvatal soon established what has become known as "Chvatal's Art G...