LAS POLÍTICAS DE LA PROCURACIÓN GENERAL Y DE LA SUPREMA CORTE DE JUSTICIA

Zhongping Qin1,2 _{and Binhai Zhu}2

1 _{Department of Mathematics, Huazhong University of Science and Technology,}

Wuhan, [email protected].

2 _{Department of Computer Science, Montana State University, Bozeman, MT}

59717-3880, [email protected].

Abstract. In this paper we present a simple approximation algorithm for the following NP-hard map labeling problem: Given a set S of n

distinct sites in the plane, one needs to place at each site an axis-parallel sliding square of maximum possible size (i.e., a site can be anywhere on the boundary of its labeling square) such that no two squares overlap and all the squares have the same size.By exploiting the geometric properties of the problem, we reduce a potential 4SAT problem to a 2SAT problem. We obtain a factor-2 approximation which runs inO(n2_{logn) time using}

discrete labels.This greatly improves the previous factor of 4.

1 Introduction

Map labeling is an old art in cartography and finds new applications in recent years in GIS, graphics and graph drawing [1,4,5,12,14,15,19,17,18,21,25,26,27]. About a decade ago, an interesting relation between 2SAT and map labeling was found by Formann and Wagner [12]. This finding leads to a series of exact and approximate solutions for different map labeling problems; among them, the first factor-2approximation for labeling points with maximum discrete squares [12], the first polynomial time solution for labeling a set of disjoint rectilinear line segments [21], a factor-3.6 approximation for labeling points with maximum circles [7], and a factor-2approximation for labeling points with maximum square pairs [22]. (Recently, we obtain a very simple factor-3 approximation for labeling points with maximum circles [23].)

The idea of using 2SAT in map labeling is very simple. Suppose that we have two points p, q and somehow we want to label p, q using two candidate labels each, one upper and one lower (Figure 1), we need to pick one label for each point. Assume that we use a binary variableX (Y) to encode the labeling ofp

(q) in Figure 1, i.e., if we pick the upper label forp(q) thenX = 1 (Y = 1) else

X = 0 (Y = 0), then in the example of Figure 1 we need to satisfy the following formula ¬(X ∧Y)∧ ¬(¬X∧Y)∧ ¬(¬X∧ ¬Y), where ¬(u∧v) means ‘we do not want u, v to be true at the same time’. Simplifying the above formula, we

_{This research is partially supported by Hong Kong RGC CERG grant Ci-}

tyU1103/99E, NSF CARGO grant DMS-0138065 and a MONTS grant.

M. Penttonen and E. Meineche Schmidt (Eds.): SWAT 2002, LNCS 2368, pp. 100–109, 2002. c

have (¬X ∨ ¬Y)∧(X ∨ ¬Y)∧(X ∨Y). It is easy to find a truth assignment

X = 1, Y = 0 for this 2SAT formula, which implies that we should choose the upper label forpand lower label for q.

Y

p

q

X

Fig. 1.An example of using 2SAT in map labeling.

At this point let us say a few words about the discrete/sliding models used in map labeling. In [12], for each site one has 4 candidate labels (axis-parallel squares each of which has a vertex anchored at the site) and the problem is to select one out of the 4 candidates for each site so as to maximize the labels’ size yet make sure that no two labels intersect. (We simply call each of the 4 candidate labels a discrete label.) In the past several years more generalized models have been proposed. The basic idea is to allow each site to have an infinite number of possible candidate labels (see [8,13,16,24,26]). This model is more natural than the previous discrete model (like the one in [12]) and has been coined as the

sliding modelin [16]. Certainly, designing efficient algorithms for map labeling under the sliding model becomes a new challenge.

In this paper, we investigate the following map labeling problem under the sliding model.Given a setSofnsites in the plane, label sites inSwith maximum uniform axis-parallel squares, namely how to placenaxis-parallel squares of the same size such that no two squares intersect except at their boundary; each square is associated with a pointpi inS such thatpilies on the boundary of that square;

and the size of these uniform squares is maximum.

In [16] van Kreveld et al. proved the NP-hardness of this problem, i.e., it is NP-hard to decide whether a set of points can all be labeled with axis-parallel unit squares under the sliding model. A careful look at their proof results in a better hardness-result, i.e., it is NP-hard to decide whether a set of points can all be labeled with axis-parallel squares of size greater than 0.75. (Notice that van Kreveld et al. tried to maximize the number of sites labeled instead of the size of the labels in [16].) Clearly, it is meaningful to study approximation algorithms for this problem. In [27], two approximation algorithms, with factor-5√2and factor-4 respectively, were proposed.

Our idea is as follows. We first compute the minimum diameterD5,∞(S), un- der theL∞metric, of a 5-subset ofSand use it to bound the optimal solutionl∗. We then design a decision procedure, which, forL≤D5,∞(S), decides whether a labeling ofS using discrete labels of sizeL/2exists or not and ifL≤l∗ _then the answer is always yes. This can be done as follows. We identify all the feasible

regions (to be defined formally) to place a label of size L forpi. (There are at

most 4 such feasible regions forpi.) We prove that if we shrink all of the optimal

sliding labels by a half then we can label points inSwith discrete labels at least half of the optimal size at 2stages: some of them can be labeled at a unique discrete position and will not interact with other labels, others can be labeled at one of the 2discrete positions such that either one of the candidates of such a pointpi is in the optimal label forpi or one of the candidates ofpi intersects

at most one candidate of another pointpj.

Our detailed algorithm again uses 2SAT. For each pointpi, letCi(L) be the

square that has pi as its center and is of edge length L ≤D5,∞(S). We thus have anO(n)-size intersection graphG(L) for allCi(L),1≤i≤n. (G(L) can be

constructed inO(nlogn) time.) Consequently, for fixedLwhether we can label

S using sliding labels of size at leastL/2can be determined inO(n) time and moreover; if L ≤ l∗ _{then the answer is always positive and the corresponding} approximate labeling can be determined also in O(n) time. With this decision procedure, we can either have anO(n2_{logn) time approximation.}

2 Preliminaries

In this section we make some necessary definitions related to our algorithm. Thedecision versionof the MLUS-AP problem is defined as follows:

Instance: Given a set S of points (sites) p1, p2, ..., pn in the plane, a real

number l >0.

Problem: Does there exist a set of n uniform axis-parallel squares of edge length l, each of which is placed at each input site pi ∈ S such that no two

squares intersect, a site can be anywhere on the boundary of its labeling square and no site is contained in any square.

This problem is NP-hard [16]. From now on we will focus on themaximiza- tion versionof this problem (i.e., to compute/approximate the optimal solution with sizel∗_{). We say that an approximation algorithm for a (maximization) op-} timization problemΠprovides aperformance guaranteeofρif for every instance

I of Π, the solution value returned by the approximation algorithm is at least 1/ρof the optimal value forI. (For the simplicity of description, we simply say that this is a factor-ρapproximation algorithm forΠ.)

If we allow a small number of sites to be unlabeled then it is possible to obtain a bicriteria polynomial time approximation scheme (PTAS) for this problem [8]. The best known approximation factor is 4 and the running time of the corresponding algorithm is O(nlogn) [27]. In this paper we present an

O(n2_{logn) time, factor-2approximation algorithm for this problem.}

Letk≥2be an integer. Given a set S of k points (sites) in the plane, the

k-diameter of S under theL∞-metric is defined as the maximum L∞-distance between any two points in S. Given a setS of at leastk sites in the plane, the

min-k-diameter ofSunder theL∞metric, denoted asDk,∞(S), is the minimum

k-diameter over all possible subsets ofS of size k.

In the following section we present an approximation solution for MLUS- AP. We use D5,∞(S), the min-5-diameter of the set S under L∞, to bound the optimal solution l∗_{. Given a set of n sites S,D}

5,∞(S) can be computed in

O(nlogn) time [6,10].