Dans les deux prochains chapitres, nous montrons deux visions diérentes de l'étude de la perçabilité. L'une d'elle consiste à construire des heuristiques qui évitent de calculer la matrice d'incidence A tout en garantissant un fac-teur raisonnable de l'optimal. Nous obtenons un algorithme adaptatif dont la précision dépend elle-même de la valeur de l'optimal. Les problèmes soulevés lors de l'étude d'heuristiques sont les suivants:
la relation exacte entre c et b
, o ù b
est la taille maximale d'un ensemble de boîtes mutuellement disjointes (nous obtenons à partir des nombres N(p;q;d) l'inégalité c b 1+d d ).
la di culté d'approximation des boîtes contraintes comparée aux boîtes non contraintes dans la hiérarchie polynomiale des problèmes.
Pouvons nous trouver un algorithme garantissant un certain facteur entre b, le nombre d'un ensemble maximal de boîtes mutuellement dis-jointes, et b
dans le cas de boîtes non contraintes?
Dans le papier de Imai et Asano [IA83], un algorithme est donné pour calculer une cellule maximale d'un arrangement denboîtes isothétiques en temps O(n
d 1
logn). Pouvons nous améliorer cet algorithme? Ceci permettrait d'augmenter l'e cacité de l'algorithme glouton.
Une autre approche de l'étude de la perçabilité est basée sur l'étude de propriétés combinatoires de classes d'objets convexes pour des valeurs don-nées dek. On peut voir les algorithmes donnés dans le chapitre 3 comme des algorithmes adaptatifs même si dans la complexité de ces algorithmes n'ap-parait qu'un seul paramétre. En eet, pour savoir si un ensemble est perçable par k points, on testera successivement s'il est perçable par 1, ..., k 1et k
points par des algorithmes de complexités respectives T 1 (n), ..., T k 1 (n) et T k (n). Soit k
0 le nombre minimal de points requis pour percer S, alors notre algorithme aura une complexité enP
k0 i=1
T i
(n). Nos algorithmes sont donc en ce sens adaptatif puisqu'ils dépendent de k
0. Parmi les problèmes connectés à cette étude, il serait intéressant d'obtenir des résultats similaires pour les nombres de Galai et les nombres C(p;q) [GW93a].
Chapitre 2
Fast Stabbing of Boxes in High
Dimensions
(Cette partie est parue en rapport de recherche INRIA Numéro 2854 [Nie96b] et fût présentée sous forme résumée au 12eatelier de travail européen de géométrie algorithmique [Nie96c] ainsi qu'à la 8econférence Canadienne de géométrie algorithmique [Nie96d]. Une version longue est actuellement soumise au journalTheoretical
Computer Science. Nous présentons ci-dessous une version incluant
certaines preuves et extensions omises dans [Nie96b].)
Abstract
We present in this technical report a simple yet e cient algorithm for stabbing a setS of naxis-parallel boxes in d-dimensional space with c points in output-sensitive time O(dn+nlogc) and linear space. Let c be the minimum number of points required to stab
S, then we prove that
cminf cd
d! + cd 1
(d 1)! 1;c(logn+ 1)d 1
(d 1)! g;
where xm is the rising factorial power: xm = Qm 1
i=0 (x + i) =
m! x+m 1
m
. Since nding a minimal set ofc points is NP-complete as soon as d > 1, we obtain a fast precision-sensitive heuristic for stabbingS in output-sensitive time and linear space. In the case of congruent or `constrained' isothetic boxes, our algorithm reports respectively c 2d 1c and c= O(c) stabbing points. Moreover, we show that the bounds we get on c are tight and corroborate
our results with some experiments. We also describe an optimal output-sensitive algorithm for nding a minimal-size optimal stab-bing point-set of intervals. Finally, we conclude with insights for further research.
Keywords:Computational geometry, Output-sensitive algorithms.
2.1 Setting the problem
LetSbe a set ofnd-dimensional geometric objects of constant descriptive-size. We say thatS is stabbed by kpoints if there exist kpoints so that each object of S contains at least one of these points. Thus, S can be immobi-lized under translation with k points. Given a set S as above, nding the minimum k so that S can be stabbed by k points has been shown to be NP-complete [FPT81] as soon as d 2
1. Therefore this problem is un-tractable for small values of n (say n ' 20 and d = 2 See section 2.3.4). This problem is also referenced in the literature as the set covering problem (or dually as the hitting set problem) where it is transformed into an opti-mization problem by means of matrix formulations. Let V = fS
i
ji 2 Ig be a collection of v = jVj = jIj subsets of 2
S for a set S of n elements. We want to nd a minimal covering collection, i.e. a sub-set I
0 I of indices so that S = S i2I 0 S i with jI 0
j as small as possible. In other words, we want to minimize e T x=jI 0 jsubject to Axe forx af0;1g n-vector, e=(1;:::;1)
and A a(vn)-binary matrix, each column of which is the incidence vector of one of the sets I
i, 1iv.
Some heuristics that give approximation of the minimum stabbing num-ber c
have been given. V. Chvátal [Chv79] gave a polynomial time (cubic) greedy algorithm to nd a cover set of sizecsuch thatcc
(1 + logk)wherek
is the maximum column sum (kn). Very recently, this bound was improved tocc
(logk loglogk+(1))by P. Slavík [Sla96]. D.S. Hochbaum [Hoc82] proposed another cubic algorithm with a cover set of size at most c
f, where
f is the maximum row sum. Interestingly, Bellare et al. [BGLR93] showed that no polynomial time algorithm can approximate the optimal solution wi-thin a factor of (
1 8
)logjSj, unlessNP DTIME[n loglogn
], where >0. This result has been recently succesfully extended to the best possible logn
bound by U. Feige [Fei96].
One major drawback from the computational geometry point of view is that these methods do not consider geometrical objects nor their shapes.
1. More precisely, Fowler et al. [FPT81] showed that covering a set of points with xed-size squares (the so-called BOX-COVER problem) is NP-complete as soon asd>1.
(Although it has been shown that the intersection graph2 of d dimensional convex objects can be arbitrary as soon asd3 [Weg67]). This means that we have to supply matrixA. One way to proceed is to consider from the whole arrangement of the objects all the sets de ned by vertices. More precisely, to each vertex we associate the set of objects containing it (thus, the size of the matrix is O(n
d
)n and these algorithms require O(n d+2
)time and O(n d+1
)
space).
D.S. Hochbaum and W. Maass [HM84] considered the case of geome-trical objects and give polynomial approximation scheme (note that no fully approximation scheme exists unless P=NP we refer the reader to the com-prehensive text book [GJ79] for a complete explanation). Their method is in-novative since it is general and takes into account the nature of the objects. It should be noted that their method applies only to geometric objects. Unfor-tunately, the running time of these algorithms are at least cubic and thus can-not handle a huge amount of data. Moreover, their algorithm considers sets of identical convex objects T, or dually covering sets of points with convex translates T
. (T
is the centrally symmetric convex object of T). Many applications coming from VLSI design, image processing and point location have to deal with large inputs [TF80]. Recently, H. Brönninman and M.T. Goodrich [BG94] investigate these problems using the Vapnik-Chervonenkis dimension (VC-dimension). They obtain precision-sensitive set covers if the VC-dimension3 is bounded as it is generally the case when considering geo-metric objects. Their algorithm uses nontrivial concepts (and subroutines) such as set systems, -net, net nder, ... (see also [Mat91a]) and still relies on the fact that matrix A is computed beforehand.
In this paper, we are even more restrictive by considering the case of axis-parallel boxes in high dimensions (that are often considered in VLSI design, image processing and point location in d-dimensional euclidean space); for example, we are given a set of points inE
d and some hypercube H
d. We want to associate to each point a hypercube that contains it so that we minimize the number of hypercubes. In other words, we want to cover the point set with a minimum number of patches, i.e. translates of H
d. Throughout the paper, the boxes are considered to be closed, i.e. points on the boundary of boxB stab B. Our main algorithm, described in section 2.3, will not require to compute the arrangement of the isothetic boxes4. Note that, we do not
2. The intersection graph of a set of objects is de ned as follows: we associate to each object a node and there exists an edge between two nodes i the corresponding objects intersect.
3. The VC-dimension ofd-dimensional isothetic boxes is2.
4. Computing the arrangement of a set of n isothetic boxes cost O(n d
) time and space [PS85].
consider d as a constant in the sequel.
We give in this paper a simple algorithm and study its approximation factor5.
This fast algorithm may be useful in many applications. More precisely, we give a truly output-sensitive O(dn+nlogc)-time algorithm that computes a set of c points stabbing the set of n d-dimensional boxes. Interestingly, we show that cminfcd
d! + cd 1
(d 1)! 1;c
(logn+1) d 1 (d 1)!
g wherec is the optimal value and xm is the rising factorial power: xm = Qm 1
i=0 (x+i) = m! x+m 1
m
and x0 = 1. Moreover, we exhibit a generic example where this bound is matched. We can re ne the complexity analysis to show that c 2d 1c
and c=O(c) when dealing respectively with congruent isothetic boxes and `constrained' boxes.
The paper is organized as follows:
In section 2.2, we consider the case of a family of n intervals and give an optimal (n(logc+ 1))-time algorithm that gives an optimal stabbing set of c points. We use this basic case in order to devise another algorithm and analyze its behavior.
In section 2.3, we enhance the algorithm in higher dimensional space and study both its running time and its approximation factor. We show that the given bounds are tight. We re ne the analysis for sets of congruent isothetic and constrained isothetic boxes. We corroborate our theoritical results with experiments.
Finally, in section 2.4, we conclude and give several guidelines for future research.