Conflict-Free Colorings of Rectangles Ranges Khaled Elbassioni Nabil H. Mustafa Max-Planck-Institut für Informatik, Saarbrücken, Germany felbassio, nmustafag@mpi-sb.mpg.de Abstract. Given the range space (P; R), where P is a set of n points in IR 2 and R is the family of subsets of P induced by all axis-parallel rectangles, the conflict-free coloring problem asks for a coloring of P with the minimum number of colors such that (P; R) is conflict-free. We study the following question: Given P, is it possible to add a small set of points Q such that (P [ Q; R) can be colored with fewer colors than (P; R)? Our main result is the following: given P, and any ffl 0, one can always add a set Q of O(n 1 ffl ) points such that P [ Q can be conflict-free colored using ~ O(n 3 8 (1+ffl) ) 1 colors. Moreover, the set Q and the conflict-free coloring can be computed in polynomial time, with high probability. Our result is obtained by introducing a general probabilistic recoloring technique, which we call quasi-conflict-free coloring, and which may be of independent interest. A further application of this technique is also given. 1 Introduction A set of points P in IR 2, together with a set R of ranges (say, the set of all discs or rectangles in the plane) is called a range space (P; R). For a given range space (P; R), the goal is to assign a color to each point p 2 P such that for any range R 2Rwith R P 6= 0, the set R P contains a point of unique color. Call such a coloring of the points P a conflict-free coloring of (P; R). The problem then is to assign a conflict-free coloring to the range space (P; R) with the minimum number of colors. The study of the above problem was initiated in [ELRS03,Smo03], motivated by the problem of frequency-assignment in cellular networks. The work in [ELRS03] presented a general framework for computing a conflict-free coloring for shrinkable range spaces. In particular, for the case where the ranges are discs in the plane, they present a polynomial-time coloring algorithm that uses O(log n) colors for a conflict-free coloring. They also present an algorithm that conflict-free colors the set of points P when the ranges are scaled translates of a compact convex region in the plane. This result was then extended in [HS05] by considering the case where the ranges are rectangles in the plane. This seems harder than the disc case, and the work in [HS05] presented a simple algorithm that uses O( p n) 2 colors for a conflict-free coloring. They also show that for the case of random points in a unit square, O(log 4 n) colors suffice for rectangle ranges. Finally, they show that if the points lie on an exact uniform p n p n grid, then also O(log n) colors are sufficient. This result very strongly uses the degeneracy of the 1 Ignoring poly-logarithmic factors. 2 Ignoring poly-logarithmic improvements [PT03].
grid (i.e., p n points lie on each grid line), and it is not known what is the minimum number of colors needed if each point is perturbed within a small distance of its original location. In this paper, we study the following question: Given P, is it possible to add a small set of points Q such that (P [ Q; R) can be colored with fewer colors than (P; R)? It is instructive to look at the one-dimensional case as an example. Here we are given n points in IR, and would like to color them such that any interval contains a point of unique color. It is not hard to see that the geometric location of the points is irrelevant any set of n such points would require the same number of colors. Hence, adding points will only increase the total number of colors required. Does a similar situation hold in higher dimensions? For example, given n points in IR 2, and where the ranges are discs, is it possible to add f (n) points such that the total number of colors for a conflict-free coloring is at most o(log(n + f (n))) (as mentioned earlier, for disc ranges, it is always possible to conflict-free color using O(log n) colors). The result of Pach and Toth [PT03] answers the question negatively: any set of n points need Ω(log n) colors for a conflict-free coloring. A more interesting case is that of axis-parallel rectangle ranges, where the current best bounds are much worse than for the disc case: ~ O(n 1=2 ) colors for n points. We prove here that by adding a small (sub-linear, in fact) number of points, the total number of colors can be reduced substantially. We now state our results more precisely. Our results. Let P be a set of n points in the plane, and (P; R) the axis-parallel rectangle range space over P. Our main theorem is the following, stating that one can always decrease the number of colors needed by adding a small number of new points. Theorem 1. Given a set P of n points in IR 2, and any ffl 0, it is always possible to add a set Q of O(n 1 ffl ) points such that P [ Q can be conflict-free colored using ~ O(n 3 8 (1+ffl) ) colors. Furthermore, the set Q and the conflict-free coloring can be computed in polynomial time with high probability. We prove the above theorem by using a probabilistic re-coloring technique that can be used to get a coloring with weaker properties, which we call a quasi-conflict-free coloring. As another application of quasi-conflict-free colorings, we show that points that are regularly placed can be colored using fewer colors than for the general case. More precisely, if one can partition P by a set of vertical and horizontal lines such that each cell contains at least one point (i.e., a non-uniform grid), then one can color P using few colors: Theorem 2. Given any p n p n grid G such that each cell contains one point of P, there exists a conflict-free coloring of P using O(n 3=8+ffl +2 8=3ffl ) colors, for any constant ffl>0. Furthermore, this coloring can be computed in expected time O(n 5=4 ). Outline. We start by definitions and briefly describing the framework for conflict-free coloring proposed in [ELRS03] in Section 2. Section 3 describes our quasi-conflictfree coloring technique. We then prove, using quasi-conflict-free colorings, our main Theorem 1 in Section 4, and Theorem 2 in Section 5.
2 Preliminaries Definitions. Let P = fp 1 ;::: ;p n g be a set of n points in the plane, and (P; R) any shrinkable range space over P. Define the conflict-free graph (with vertex set V and edge set E) of the range space (P; R) as follows: each vertex v i 2 V corresponds to the point p i 2 P, and (v i ;v j ) 2 E iff there exists a range T 2Rsuch that T contains both p i and p j and no other point of P. Call a range T 2Rconflict-free if it contains a point of unique color. Let G be a p r p r (non-uniform) grid consisting of r cells. Let P (G) denote the set of r points when each grid cell contains exactly one such point, and where the point p ij 2 P lies in the i-th row and j-th column of G. Let C i P (respectively, R i P ) denote the sequence of p r points lying in the i-th column (respectively, i-th row) of the grid. Namely, C i = hp 1i ;p 2i ;::: ;p p ri i, and R i = hp i1 ;p i1 ;::: ;p i p r i. Note that C i and R i are ordered sequences. Observe that for axis-parallel rectangle ranges, if we perturb the points such that the sorted order of the points in both the x and y-coordinates is unchanged, then the conflictfree graph would also be unchanged. Hence, for the general case, one can assume that the points lie at the vertices of an n n integer grid and no two points have the same x or y coordinates. A general framework. In [ELRS03], a general framework for computing conflict-free colorings of (P; R) is presented, as follows. Iteratively, compute a large independent set, say I 1, in the conflict-free graph of (P; R) and color all the points in I 1 with the same color, say c 1. Now repeat the above procedure for the range (P n I 1 ; R), coloring the next independent set with color c 2, and so forth. It is shown in [ELRS03] that firstly, for certain range spaces, discs for example, one can guarantee an independent set of size linear in the number of points. Therefore, for disc ranges, the above procedure finishes after at most a logarithmic number of steps, and hence the number of colors used are logarithmic. Second, the above coloring procedure yields a conflict-free coloring. The above framework therefore reduces the conflict-free coloring problem to one of showing large independent sets in certain graphs. This can then be applied to computing conflict-free coloring for other, more general range spaces. In [HS05], a conflict-free coloring of rectangle ranges using ~ O(n 1=2 ) colors is presented. The underlying lemma used to derive this result is the following. Lemma 1 (Erdős-Szkeres Theorem). Given any sequence S of n numbers, there exists a monotone subsequence of S of size at least p n. Furthermore, one can partition S into O( p n) monotone subsequences in time O(n 3=2 log n). Now take the set of points P, and sort them by their increasing x-coordinates. Take the corresponding y-coordinates of points in this sequence as the sequence S, and compute the largest monotone subsequence. It is easy to see that the points corresponding to this subsequence form a monotone sequence in both x (because of the initial sorting) and y (monotone subsequence). Now, picking every other point in this sequence forms a set of p n=2 points which are independent in the conflict-free graph. Iteratively repeating this gives the above-mentioned result.
3 Quasi-Conflict Free Colorings Given a set P of n points in IR 2 and a parameter k, denote by G k a n k n k grid such that every row and column of G k has exactly k points of P. Note that each cell of G k need not contain any point; see Figure 1(a). Given P and k, we call a coloring of P quasi-conflict-free with respect to k, ifev- ery rectangle which contains points from the same row or the same column of G k is conflict-free. Note that every conflict-free coloring of P is quasi-conflict-free, though the converse might not be true. We now prove that P can be quasi-conflict-free colored with fewer colors. The coloring procedure is probabilistic [MR95]: we show that with high probability, a certain coloring is quasi-conflict-free. The existence of such a coloring then follows. Theorem 3. Given any point-set P and a parameter k n c for c > 0, there exists a quasi-conflict-free coloring of G k using ~ O(k 3=4 ) colors. This coloring can be computed, with high probability, in polynomial time. PROOF. Set r = n=k to be the number of rows and columns. We use the following coloring procedure: Step 1. By Lemma 1, partition the points in column j, for j = 1;::: ;r, into h = (p k) independent sets, each of size at most p k. Note that by simply iteratively extracting the largest monotone subsequence, one can get O( p k) independent sets, where, however, some sets can be of size much larger than p k. This considerably complicates the analysis later on, by forcing the consideration of large and small independent sets separately. To avoid that (without affecting the worst-case analysis), at each iteration we only extract an independent set of size (p k), forcing a decomposition into ( p k) independent sets, each of size (p k). Step 2. Pick, independently for each column j, j =1;::: ;r, ß j 2 S h to be a random permutation. Equivalently, we can think of ß j as an assignment of h distinct colors to the independent sets of column j (all columns share the same set of h colors). From here on, it is assumed that assigning a color to a given set of points means assigning the same color to all points in the set. Thus ß j (S) is the color assigned to set S and ß j (p) is the color assigned to the independent set containing point p.forl =1:::h, let S l j be the set of points in column j with the color l (i.e., belonging to the same independent set). Let X 1 ;::: ;X h be a family of h pairwise-distinct sets of colors. We shall recolor the points of G k using these sets of colors, such that the final coloring is quasi-conflict free: we assign colors from X l to points which were colored l above. Step 3. Fixarowi 2 [r], and a color ` 2 [h]. Let A l i be the set of points with color `, taken from all the cells of row i, i.e., A l i = r [ j=1 S l j R i
We recolor the set of points A l i using colors from X`, such that any rectangle containing points from this set is conflict-free. The number of colors needed for this step is, from Lemma 1, def = hx `=1 Now, the theorem follows from the following two lemmas. q maxf jaì j : i =1;::: ;rg: (1) Lemma 2. The coloring procedure described above is quasi-conflict-free. PROOF. Take any rectangle T that lies completely inside a row or a column of G k : The column case. IfT contains only points belonging to a single column C j of G k, then the fact that the coloring of Step 1 was conflict-free within each column implies that T contains a point p 2 C j such that ß j (p) 6= ß j (p 0 ) for all p 0 6= p inside T. But then p maintains a unique color inside T also after recoloring in Step 2, since points with different ß j values are colored using different sets of colors. The row case. Now assume that T contains only points belonging to a single row i of G k. If there is an ` 2 [h] such that T Aì 6= ;, then by the conflict-free coloring of Aì in Step 3 above, and the fact that X` is distinct from all other colors used for row i, we know that there exists a point of A l i inside T having a unique color. Lemma 3. With probability 1 o(1), the procedure uses ~ O(k 3=4 ) colors. PROOF. Fix i 2 [r] and ` 2 [h]. Define t = k=h to be the size of the largest independent set in any column (as defined in Step 1.). We now upper-bound by estimating the maximum size of the union of sets of a fixed color ` in row i. The sizes of these sets vary from 1;::: ;t, so we first partition Aì into approximately same-sized sets, and estimate each separately as follows. For m =1; 2;::: ;log t, let A m i;j = [h`=1 fs : S = C j Aì ; 2m 1»jSj»2 m g be the family of sets in cell ij with size in the interval [2 m 1 ; 2 m ]. Note that rx j=1 ja m i;j j» k ; (2) 2m 1 since the total number of points in row i is at most k, and each set in A m i;j has at least 2 m 1 points.
Let Y m;` i;j be the indicator random variable that P takes value 1 if and only if there exists a set S 2A m i;j with ß j (S) =`. Let Y m;` r i = i;j. Then, j=1 Y m;` E [Y m;` i;j ]=Pr[Y m;` ja m j h 1 i;j ja m j 1 (ja m i;j i;j j 1)! i;j =1]= h ja m j ja = jam i;j j m i;j i;j j! h E[Y m;` i ]= rx j=1 ja m i;j j h» k h2 m 1 = t 2 ; m 1 where the last inequality follows from (2). Note that the variable Y m;` i is the sum of independent Bernoulli trials, and thus applying the Chernoff bound 3, we get Pr[Y m;` i > t log k t log k ln t log k 4 2m 1 E[Y ]» e m;` ] 2m 1 i : (3) 2m 1 Using E[Ym i;` ]» t=2m 1 and 2 m» t, we deduce from (3) that Pr[Y m;` i > t log k 2 m 1 ]» (log k) (log k)=2 : Thus, the probability that there exist i, m, and ` such that Y m;` i >tlog k=2 m 1 is at most rh(log t)(log k) (log k)=2 = o(1). Therefore with probability 1 o(1), Y m;` i» t log k=2 m 1 for all i, `, and m. In particular, with high probability, log Xt jaì j» m=1 Y m;` i 2 m» 2t log k log t: Combining this with (1), we get» h p 2t log k log t = p 2hk log k log t = ~ O(k 3=4 ), as desired. 4 Conflict-Free Coloring of General Point Sets In this section, we prove Theorem 1 which states that by adding a small number of points, the total number of colors for a conflict-free coloring can be significantly reduced. First note that it is quite easy to add a super-linear number of points to get better colorings. Indeed, by adding n 2 points, one can get a conflict-free coloring using O(log n) colors: simply partition the point set into a n n grid, where each row and column 3 In particular, the following version [MR95]: Pr[X (1 + ffi)μ]» e (1+ffi)ln(1+ffi)μ=4, for ffi>1 and μ = E[X].
(a) (b) Fig. 1. (a) The grid G k, k =6. The solid points are part of the input P, while the empty ones are the newly added ones, Q (that lie on a uniform grid). (b) Partitioning the grid into k 2 grids; here k =3. The points in these grids will be colored with distinct sets of colors (indicated by the cell colors). contains exactly one point. Add the n 2 points on an exact uniform grid such that each cell contains one point, and conflict-free color these points using O(log n) colors 4. We now prove our main theorem. Proof of Theorem 1. Let k be an integer, and set r = n k. We start by putting the set of points P into an r r grid G, in which each row and each column contains exactly k points. See Figure 1(a) for an illustration. We let Q be a set of r 2 points organized in a uniform grid G 0, such that every cell of G contains a point of G 0 in its interior. Next we partition G into 4 grids G 0;0, G 0;1, G 1;0 and G 1;1, where G i;j consists of cells lying in even rows (resp. columns) if i (resp. j) is0, and odd rows (resp. columns) if i (resp. j) is 1. Finally, we color these grids by 5 pairwise-disjoint sets of colors, such that Q(G 0 ) is conflict-free and G 0;0 ;::: ;G 1;1 are quasi-conflict-free. Clearly the resulting coloring of P [ Q is conflict-free: any rectangle spanning more than two rows and two columns of G must contain a point of Q. Otherwise it contains points from a single row or single column of one of G i;j, which are quasi-conflict-free colored. Since Q lies on the uniform grid G 0, the number of colors needed for G 0 is O(log r) [HS05]. Furthermore, by Theorem 3, the number of colors needed for the other four grids is 4 In fact, as pointed out by an anonymous referee, it follows from [HS05] that one can add a set Q of O(n log n) points such that P [ Q can be colored using O(log 2 n) colors: project all the points onto the vertical bisector in the x-ordering, color the projected points using O(log n) colors, and recurse on the two sides.
~O(k 3=4 ). Finally, setting k = n (1+ffl)=2, it follows that one can add the set Q of n 2 =k 2 = n 1 ffl points to P to get a conflict-free coloring of P [ Q using ~ O(k 3=4 )= ~ O(n 3 8 (1+ffl) ) colors. Remark: The set Q of points we add lies on an exact uniform grid. This can be relaxed by instead placing points anywhere inside the cells of G k, and using Theorem 2 to compute the conflict-free coloring of Q. This allows one greater freedom to place the new points, although the bounds become correspondingly worse. 5 Conflict-Free Colorings of Points in a Grid We are given a set P of n points and a p n p n grid G such that each cell of G contains exactly one point of P. We now first show that Theorem 3 can be used, together with shifted dissection, to obtain a conflict-free coloring of P (G) which uses fewer colors than the general case. We also observe that this case is not much easier than the general case, i.e., coloring a grid with fewer than O(n 1=4 ) colors would imply that the general case can be colored with fewer than O(n 1=2 ) colors. Given G and an integer k» p n, we can partition G into a set of k 2 grids in a natural way (see Figure 1(b)): For x =0; 1;::: ;k 1 and y =0; 1;::: ;k 1, grid G x;y consists of all the points p ij of G such that i mod k = x and j mod k = y, namely, P (G x;y )=fp x+ik;y+jk 2 P (G) j i; j 2 Z :0» x + ik; y + jk < p ng: Now we color the points of G as follows. Let r = n k. Each of the k 2 different grids 2 (of size p n=k p n=k) will be colored with a different set of colors. One grid, say G 0;0 is recursively conflict-free colored. For the other k 2 1 grids, we use Theorem 3 to get a quasi-conflict-free coloring of the points of each grid using O(r 3=8 ) colors. Lemma 4. The coloring of P (G) described above is conflict-free. PROOF. Let T be a rectangle in the plane containing at least one point of P (G). Consider the following two cases: First, if T contains only points from at most k 1 rows of G, then let R be any row of G such that R T is non-empty. Take any grid G x;y such that P (G x;y ) R T 6= ;. Then T is a rectangle containing points belonging to only one row of G x;y (since T spans less than k 1 rows), and consequently a point with a unique color exists inside T, since (i) the grid G x;y is quasi-conflict-free colored, and (ii) the points belonging to all other grids are colored with a different set of colors than the one used for G x;y. The case when T contains points from at most k 1 columns of G is similar. Second, if T is a large rectangle, i.e., contains points from at least k rows and at least k columns of G, then T must contain a point from G 0;0. By the conflict-free coloring of G 0;0 we know that T P (G 0;0 ) contains a point with a unique color. The same point has a unique color among all other points in T P (G) since, again, the different grids are colored with different sets of colors.
Proof of Theorem 2. Using the above coloring scheme, the number of colors needed is at most f (n) = We prove by induction that min 1»k» p n ρk 2 nk2 3=8 + f ( nk2 ) ff f (n)» n 3=8+ffl +2 8=3ffl The base case is when the number of points is small, more precisely, when n» 2 16=3ffl. Using Lemma 1, one can color them with at most p n» 2 8=3ffl colors. When n 2 16=3ffl, then by setting k 2 = n ffl=(1+ffl), and applying the inductive hypothesis, we have : f (n)» k 2 n k 2 3=8 +( n k 2 )3=8+ffl +2 8=3ffl =(n ffl=(1+ffl) )(n 1=(1+ffl) ) 3=8 +(n 1=(1+ffl) ) 3=8+ffl +2 8=3ffl =2(n 1=(1+ffl) ) 3=8+ffl +2 8=3ffl =(2 8=(3+8ffl) n 1=(1+ffl) ) 3=8+ffl +2 8=3ffl» (2 8=3 n 1=(1+ffl) ) 3=8+ffl +2 8=3ffl» (n ffl=(1+ffl) n 1=(1+ffl) ) 3=8+ffl +2 8=3ffl = n 3=8+ffl +2 8=3ffl where the last inequality follows from the assumption that n 2 16=3ffl. Remark 1: One can also consider computing an independent set of the points in a grid. In fact, using similar ideas, one can show the existence of an independent set of size Ω(n 5=8 ). However, this does not immediately yield a conflict-free coloring of P (G) using O(n 3=8 ) colors. This is because, once we extract an independent set of points P 0 from P (G) of size Ω(n 5=8 ), the remaining set P (G) n P 0 is no longer a grid. However, we have seen above, within an arbitrarily small additive constant in the exponent, such a coloring can indeed be achieved. Remark 2: Having partitioned the grid G into k 2 grids, we need not color all these grids with distinct sets of colors. In fact, we can assign only O(k) sets of colors to the different grids such that the whole grid G can be conflict-free colored. Finally, observe that this grid case is not much easier than the general case. Observation 1 If there always exists an independent set of size at least n c for a point set P (G) lying on a p n p n grid, then, for any general set of points, there exists an independent set of size at least n 2c 1. PROOF. Assume otherwise. Then there exists a general point set Q of size p n whose maximum independent set has size at most n c 1=2. We construct P (G) of size n by putting p n copies of Q in each column, i.e., set each C i to be a (translated copy)
of Q. By our earlier observation, this can be done since we can always move the y- coordinates of each point in Q to lie in a different row. The maximum independent set of this new point set can contain at most n c 1=2 points from each column, and therefore has maximum independent set of size less than n c, a contradiction. In particular, if one could show the existence of a linear sized independent set for the grid case, that would imply a linear sized independent set for the general case, for which the current best bound is roughly ~ O(n 1=2 ). Finally, we have used the algorithm for conflict-free coloring general pointsets as a black-box to get the decomposition into a small number of independent sets (in the conflict-free graph). Therefore, any improved bounds on the general problem imply an improvement in our bounds. Acknowledgements. We would like to thank Sathish Govindarajan for many helpful discussions, and two anonymous reviewers for useful suggestions that improved the content and presentation of this paper. References [ELRS03] G. Even, Z. Lotker, D. Ron, and S. Smorodinsky. Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks. SIAM J. Comput., 33:94 136, 2003. [HS05] S. Har-Peled and S. Smorodinsky. Conflict-free coloring of points and simple regions in the plane. Discrete & Comput. Geom., 34:47 70, 2005. [MR95] R. Motwani and P. Raghavan. Randomized Algorithms. Cambridge University Press, New York, NY, 1995. [PT03] J. Pach and G. Toth. Conflict free colorings. In Discrete & Comput. Geom., The Goodman-Pollack Festschrift. Springer Verlag, Heidelberg, 2003. [Smo03] S. Smorodinsky. Combinatorial problems in computational geometry. PhD thesis, Tel-Aviv University, 2003.