Sharif Digital Repository

In this thesis, we consider the problem of online conflict-free coloring for regions. First, we analyze online variant of the conflict-free coloring of intervals on the line, where each newly inserted interval must be assigned a color upon insertion such that the coloring remains conflict-free, i.e. for every point p in the union of the current intervals, there must be an interval I with a unique color among all intervals covering p. we present deterministic algorithm for achieving this goal which uses O(log^3⁡n) color. Next, we consider the online conflict-free coloring of nearly-equal axis-parallel rectangles in the plane and present a deterministic algorithms which uses O(log^12⁡n)... 

A coloring of regions is conflict-free if for any covered point in the plane, there exist a region that cover it, with a unique color. In this thesis, we examine the problem of conflict-free coloring of regions in dynamic model, with minimum colors, and with the possibility of recoloring the optimal numbers of colored regions. That means, regions are inserted one by one on the plane, and by inserting any region, inserted regions shall be conflict-free. Dynamic model are divided into three categories: offline model, online with absolute positions model and online with relative positions model. The difference between these three categories is in the information contained in insertion of... 

The visibility is one of the most important problems in computational geometry. Two points are said to be visible to each other, if the line segment that joins them does not intersect any obstacles. The art gallery problem is a well-studied visibility problem in computational geometry. It originates from a real-world problem of guarding an art gallery with the minimum number of guards who together can observe the whole gallery. In many of applications, the “guards” are “landmarks” deployed in an environment to help provide navigation and localization service to mobile robots. The mobile device communicates with these landmarks through wireless, or other “line-of-sight” signaling... 

Let S be a set of n regions of some fixed type (such as discs, axis-parallel rectangles, etc). A conflict-free coloring (CF-coloring) of S is an assignment of colors to each region such that for every point p covered by a non-empty subset of S, there is a region with a unique color among the regions containing p. We try to maintain a CF-coloring as the objects move. Regions are moving, so after an intersection, the coloring might not be conflict-free anymore and we need to recolor some regions. The goal is to recognize the intersections and to minimize both the total number of colors used and the number of recolorings needed. In this thesis we study kinetic CF-coloring for moving... 

Let S be a set of n regions of some fixed type (such as discs, axis-parallel rectangles, etc). A conflict-free coloring (CF-coloring) of S is an assignment of colors to each region such that for every point p covered by a non-empty subset of S, there is a region with a unique color among the regions containing p. The goal is to use minimum number of colors. In this thesis we study online CF-coloring for axis-parallel rectangles in the plane which rectangles are on x-axis. We present a randomized algorithm with O(〖log〗^6 n) colors. We also prove that this algorithm works with high probability  

The conflict-free coloring is one of the computational geometry problems that has received attention in the last two decades. The root of this problem comes from the frequency allocation problem to telecommunications antennas, where we have several telecommunication antennas in two-dimensional space aiming to assign to each of them a frequency so that every point on the plane, which is at least inside one of the antenna ranges, stays in a range with a frequency different from the other antennas containing point.One of the generalizations of this problem is the k-strong case, where we have n regions (equivalent to the circular region of the antennas) that we want to color so that for each... 

Conflict-free coloring problem is one of the computational geometry problems that has been studied in the last two decades. The root of this problem comes from the frequency assignment to cellular networks problem. In that research, we have a cellular network containing several regions in two-dimensional plane and we want to assign a frequency to each of the regions so that if we look at any intersection of them, there is a region with a unique color compared to other regions of that intersection. In other words, a person’s cellular phone in the range of some regions (in the intersection of them) would be able to connect to one antenna with a unique frequency and function properly. This...