Given a set @math of @math points in the plane, we study the computation of the probability distribution function of both the area and perimeter of the convex hull of a random subset @math of @math . The random subset @math is formed by drawing each point @math of @math independently with a given rational probability @math . For both measures of the convex hull, we show that it is #P-hard to compute the probability that the measure is at least a given bound @math . For @math , we provide an algorithm that runs in @math time and returns a value that is between the probability that the area is at least @math , and the probability that the area is at least @math . For the perimeter, we show a similar algorithm running in @math time. Finally, given @math and for any measure, we show an @math -time Monte Carlo algorithm that returns a value that, with probability of success at least @math , differs at most @math from the probability that the measure is at least @math .
More recently, in 2014, @cite_8 considered a set of @math points in the plane colored with @math colors, and studied, among other computation problems, the computation of the expected area or perimeter of the convex hull of a random sample of the points. Such random samples are obtained by picking for each color a point of that color uniformly at random. They proved that both expectations can be computed in @math time. We note that their arguments can be used to compute both @math and @math , each one in @math time. In the case of the expected perimeter, similar arguments were discussed by @cite_6 .
@cite_6 @cite_8
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Given a set of @math n points, each is painted by one of the @math k given colors, we want to choose @math k points with distinct colors to form a color spanning set. For each color spanning set, we can construct the convex hull and the smallest axis-aligned enclosing rectangle, etc. Assuming that each point is chosen independently and identically from the subset of points of the same color, we propose an @math O ( n 2 ) time algorithm to compute the expected area of convex hulls of the color spanning sets and an @math O ( n 2 ) time algorithm to compute the expected perimeter of convex hulls of the color spanning sets. For the expected perimeter (resp. area) of the smallest perimeter (resp. area) axis-aligned enclosing rectangles of the color spanning sets, we present an @math O ( n log n ) (resp. @math O ( n 2 ) ) time algorithm. We also propose a simple approximation algorithm to compute the expected diameter of the color spanning sets. For the expected distance of the closest pair, we show that it is @math # P-complete to compute and there exists no polynomial time @math 2 n 1 - ? approximation algorithm to compute the probability that the closest pair distance of all color spanning sets equals to a given value @math d unless @math P = N P , even in one dimension and when each color paints two points. In this paper we present several results on the expected complexity of a convex hull of @math points chosen uniformly and independently from a convex shape. (i) We show that the expected number of vertices of the convex hull of @math points, chosen uniformly and independently from a disk is @math , and @math for the case a convex polygon with @math sides. Those results are well known (see rs-udkhv-63,r-slcdn-70,ps-cgi-85 ), but we believe that the elementary proof given here are simpler and more intuitive. (ii) Let @math be a set of directions in the plane, we define a generalized notion of convexity induced by @math , which extends both rectilinear convexity and standard convexity. We prove that the expected complexity of the @math -convex hull of a set of @math points, chosen uniformly and independently from a disk, is @math , where @math is the largest angle between two consecutive vectors in @math . This result extends the known bounds for the cases of rectilinear and standard convexity. (iii) Let @math be an axis parallel hypercube in @math . We prove that the expected number of points on the boundary of the quadrant hull of a set @math of @math points, chosen uniformly and independently from @math is @math . Quadrant hull of a set of points is an extension of rectilinear convexity to higher dimensions. In particular, this number is larger than the number of maxima in @math , and is also larger than the number of points of @math that are vertices of the convex hull of @math . Those bounds are known bkst-anmsv-78 , but we believe the new proof is simpler. Motivated by a frequency assignment problem in cellular networks, we introduce and study a new coloring problem that we call minimum conflict-free coloring (min-CF-coloring). In its general form, the input of the min-CF-coloring problem is a set system @math , where each @math is a subset of X. The output is a coloring @math of the sets in @math that satisfies the following constraint: for every @math there exists a color @math and a unique set @math such that @math and @math . The goal is to minimize the number of colors used by the coloring @math . Min-CF-coloring of general set systems is not easier than the classic graph coloring problem. However, in view of our motivation, we consider set systems induced by simple geometric regions in the plane. In particular, we study disks (both congruent and noncongruent), axis-parallel rectangles (with a constant ratio between the smallest and largest rectangle), regular hexagons (with a constant ratio between the smallest and largest hexagon), and general congruent centrally symmetric convex regions in the plane. In all cases we have coloring algorithms that use O(log n) colors (where n is the number of regions). Tightness is demonstrated by showing that even in the case of unit disks, @math colors may be necessary. For rectangles and hexagons we also obtain a constant-ratio approximation algorithm when the ratio between the largest and smallest rectangle (hexagon) is a constant. We also consider a dual problem of CF-coloring points with respect to sets. Given a set system @math , the goal in the dual problem is to color the elements in X with a minimum number of colors so that every set @math contains a point whose color appears only once in S. We show that O(log |X|) colors suffice for set systems in which X is a set of points in the plane and the sets are intersections of X with scaled translations of a convex region. This result is used in proving that O(log n) colors suffice in the primal version. Given n points in the plane, we present algorithms for finding maximum perimeter or area convex k-gons with vertices k of the given n points. Our algorithms work in linear space and time @math . For the special case @math we give @math algorithms for these problems. Several related issues are discussed.