How to Play a Coloring Game Against a Color-Blind Adversary

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1 How to Play a Coloring Game Against a Color-Blind Adversary Ke Chen March 5, 006 Abstract We study the problem of conflict-free (CF) coloring of a set of points in the plane, in an online fashion, with respect to halfplanes, nearly-eual axis-parallel rectangles, and congruent disks. As a warm-up exercise, the online CF coloring of points on the line with respect to intervals is also considered. We present randomized algorithms in the oblivious adversary model, where the adversary does not see the colors used. For the problems considered, the algorithms always produce valid CF colorings, and use O(log n) colors with high probability (these bounds are optimal in the worst case). Our randomized online algorithms are considerably simpler than previous algorithms for this problem and use fewer colors. We also present a deterministic algorithm for the CF coloring of points in the plane with respect to nearly-eual axis-parallel rectangles, using O(polylog(n)) colors. This is the first efficient deterministic online CF coloring algorithm for this problem. Introduction A range space (X, R) is defined by a ground set X and a family R of subsets of X, which are called ranges (for example, X = IR and R is the set of all disks in the plane). A coloring of a set P X is conflict-free (CF for short) with respect to R if for any range r R with P r, there is at least one point in P r that has a uniue color among the points of P r. Namely, there is a color that appears exactly once in the set P r for any range r R. The problem of CF coloring is motivated by freuency assignment in wireless networks. Specifically, wireless networks consist of two different types of nodes: base stations and clients. Base stations are interconnected by a backbone network and clients are served by base stations via radio links. Fixed freuencies are assigned to base stations to enable links to clients. Clients, usually mobile, may be within the reception range of several base stations. To minimize interference, the system must ensure that a client can always find a base station that has a uniue freuency among all the base stations within the client s reach. Due to the scarcity of freuency resources, the goal is to minimize the number of assigned freuencies. Thus, in our example, P is the set of base stations. Since a mobile client can communicate with all stations within a certain radius, the feasible region for a client is a disk, and R is a set of disks in the plane. To use the network, the client needs a base station with a uniue freuency within its reach. The problem was introduced by Even et al. [ELRS0]. They showed that one can find an assignment of O(log n) freuencies to the base stations which is conflict-free with respect to disks in the plane, and this is tight in the worst case. Har-Peled and Smorodinsky [HS05] extended those results by considering other cases, and provided some sufficient conditions for the CF chromatic Department of Computer Science, University of Illinois at Urbana-Champaign; 0 N. Goodwin Ave.; Urbana, IL 680; USA; kechen@uiuc.edu. Work on this paper was partially supported by a NSF award CCR-090.

2 number to be small for more general ranges. The dual version of the CF coloring problem was studied in [ELRS0, HS05], where the colors are assigned to the given ranges, so that for any point, the set of ranges that contain the point is conflict-free. Smorodinsky [Smo06] improved several results studied in [HS05] by providing deterministic coloring algorithms for those problems. Elbassioni and Mustafa [EM06] asked the following uestion: Given a set of points in the plane, if one is allowed to insert extra points into the plane, can the CF chromatic number (with respect to axis-parallel rectangles) be decreased? They showed that this is possible under certain conditions. Recently, Alon and Smorodinsky [AS06] showed that the CF chromatic number (with respect to disks) is O ( log k ) when each disk intersects at most k others. There is also interest in CF coloring in dynamic settings [FLM + 05, KS0], in which the points of P are inserted one by one starting with an empty set. When a point is inserted, a color is assigned to it and the color cannot be changed later. The coloring should remain conflict-free at all times. Fiat et al. [FLM + 05] considered the case where P is a set of n points on the line, and R is the set of all intervals on the line. They presented several deterministic and randomized algorithms for the problem. The best deterministic algorithm uses O ( log n ) colors, and the best randomized algorithm uses O(log n log log n) colors with high probability. (Throughout the paper, with high probability means that the success probability is at least /n c for an arbitrary constant c.) They also showed that online CF coloring of n points in the plane, with respect to arbitrary disks, reuires n colors in the worst case. Kaplan and Sharir [KS0] considered online CF coloring of n points in the plane with respect to halfplanes, congruent disks, and nearly-eual axis-parallel rectangles. They presented a fairly involved randomized algorithm that uses O ( log n ) colors with high probability for all three cases. In comparison, the known lower bound to the above problems, which also holds in the static cases, is Ω(log n) colors [ELRS0, PT0]. Our results. We revisit the recent results of [FLM + 05, KS0], and show better and considerably simpler CF coloring algorithms for those problems. We present randomized algorithms for the aforementioned problems. The algorithms use O(log n) colors with high probability, and they always produce a valid coloring. Note that our results are worst-case optimal for all of the problems considered. Our randomized algorithms are against an oblivious adversary. Namely, the adversary cannot see the colors used by the coloring algorithm so far, when deciding where to insert the next point (that is, the adversary is color-blind). One way to think of an oblivious adversary is that the adversary knows the algorithm and decides where to place the points before the game begins. We also present a deterministic online algorithm for CF coloring with respect to nearly-eual axis-parallel rectangles in the plane. The algorithm uses O ( log n ) colors. This is the first efficient deterministic online CF coloring algorithm for this problem. Computational model. When analyzing the randomized online algorithms, there is a distinction between the oblivious adversary model and the adaptive adversary model. The oblivious adversary must construct the input seuence in advance, while the adaptive adversary chooses the input seuence based on the online algorithm s actions thus far. We refer the interested reader to [BE98]. As mentioned above, our randomized algorithms are considerably simpler than the previous algorithms of [FLM + 05, KS0]. The skeptical reader might assume that this holds because the adversary in our computational model is weaker (or maybe even unreasonably weaker). However, a careful inspection of the analysis of the randomized algorithms of [FLM + 05, KS0] reveals that they also only hold in this restricted model. In particular, if the adversary is not oblivious then the bounds on the CF chromatic number due to randomized algorithms of [FLM + 05, KS0] do not

3 hold [Sha05]. As such, one of the contributions of this paper is pointing out the importance of obliviousness when considering randomized online CF coloring. Paper organization. The paper is organized as follows. In section, we give a simple generic randomized algorithm for the problems studied in this paper. In section, we analyze the algorithm s performance for intervals. In section, we analyze the algorithm s performance for halfplanes. In section 5, we analyze the algorithm s performance for nearly-eual axis-parallel rectangles. In section 6, we analyze the algorithm s performance for congruent disks. In section 7, we present a deterministic algorithm for the problem of CF coloring points on the plane with respect to nearly-eual axis-parallel rectangles. We conclude in section 8. A generic randomized coloring algorithm In the online CF coloring problem, an oblivious adversary inserts a seuence of points P = p,... p n in the plane, one at a time. We need to color the points in an online fashion as the points arrive, so that the coloring is conflict-free with respect to a range space ( IR, R ) at all times, where future point locations are not known in advance. The goal is to minimize the number of colors used. We use positive integer numbers to designate colors, and say a color i is higher than a color i if i > i. Definition. Let P (t) = p,..., p t denote the set of points inserted so far at time t. Let C(i, t) denote the points of P (t) that are assigned color i, and let C( i, t) denote the points of P (t) that are assigned color i or greater. The points that are assigned color i are (i)-level points, and the points that are assigned color i or greater are ( i)-level points. If there exist a range r R and a point C(i, t) such that r contains p t+ and r C( i, t) = {}, then is an (i)-conflicting point for p t+. If there exist a range r R and a point C( i, t) such that r contains p t+ and r C( i, t) = {}, then is a ( i)-conflicting point for p t+. Note that an (i)-conflicting point for p t+ is a ( i)-conflicting point for p t+ with color i. The point p t+ is bad for color i if P (t) contains at least ω points that are ( i)-conflicting with p t+, where ω > is a prescribed constant that depends on the problem considered. The point p t+ is ineligible for color i if (i) the point p t+ is bad for color i, or (ii) there exists a point P (t) such that is an (i)-conflicting point for p t+. Note that (i) does not imply (ii), since all the ( i)-conflicting points for p t+ might be assigned colors strictly greater than i. Next, we provide a generic randomized algorithm that works for all the problems considered, and prove that the coloring it produces is always valid. For the incoming point p t+, the algorithm calls the procedure ColorPoint(p t+, ). In the procedure ColorPoint(, i), depicted in Figure, the point is assigned the color i when it passes a fair coin tossing and an eligibility test for color i. The following lemma shows that the algorithm always produces a valid CF coloring, for any range space. Lemma. At any time t, and for any range r R such that P (t) r, there is one and only one point that realizes the highest color in P (t) r. Proof: The proof proceeds by induction on t. The base case clearly holds. Assume that the claim is true at time t. We next show the claim holds at time t +.

4 ColorPoint(, i) if coin-flip = TRUE ColorPoint(, i + ) elseif is ineligible for color i ColorPoint(, i + ) else Assign color i to. Figure : The Procedure ColorPoint. Suppose that at time t +, the point p t+ is assigned color i. The ranges that do not contain p t+ are not affected by the insertion of p t+. Thus, consider a range r R that contains p t+. The case that P (t) r = is trivial. Assume that a point is the uniue point that has the highest color in P (t) r, with color i (the uniueness of is due to the induction assumption). Depending on the relation between i and i, there are three cases. (i) If i > i, then p t+ is the uniue point of the highest color in P (t + ) r. (ii) If i < i, then is the uniue point of the highest color in P (t + ) r. (iii) Otherwise we have i = i. We show that this case is impossible. Indeed, since is the uniue point of the highest color i in P (t) r, then is a (i )-conflicting point for p t+. Therefore, p t+ is ineligible for color i, and it cannot be assigned the color i by the algorithm. Randomized online CF coloring for intervals In this section, we consider the problem of CF coloring for intervals. In this problem, points are inserted on the line, and the coloring has to be conflict-free with respect to all intervals on the line. As already noted, the algorithm and its correctness are presented in Section. We next analyze its performance. Claim. Pr[ C(i, n) C( i, n)] 8. Proof: Suppose that C( i, n). In ColorPoint(, i), the point passes the fair coin tossing with probability /. Thus it suffices to prove that the probability of being eligible for color i is / (when it is inserted). Note that there are at most two ( i)-conflicting points for when it is inserted: (i) the ( i)- level point to the left of, and (ii) the ( i)-level point to the right of. (Therefore, is not bad for color i, see Definition..) Each of them is of color > i with probability /, due to the fair coin tossing. As such, the probability that both of them are assigned a color > i is (/) = /. Recall that the (i)-conflicting points for p t+ are exactly the ( i)-conflicting points for p t+ with color i. This implies that the probability of being eligible for color i is /. Lemma. The coloring algorithm uses O(log n) colors, with high probability, for a seuence of n insertions of points on the line. Proof: Since C( i +, n) = C( i, n) \ C(i, n), we have Pr[ C( i +, n) C( i, n)] 8 = 7 8,

5 p t+ (a) p t+ (b) Figure : Examples of ( i)-conflicting points for p t+. In (a), p t+ is in the interior of CH(C( i, t)); the points and are ( i)-conflicting points for p t+. In (b), p t+ is in the exterior of CH(C( i, t)) and p t+ covers,, and ; the points,, and are ( i)-conflicting points for p t+. by Claim.. This implies E[ C( i +, n) ] 7 8 E [ C( i, n) ]. On the other hand, we have C(, n) = n. It follows that E[ C( j, n) ] any positive constant c and j (6c + 6) log n, we have E[ C( j, n) ] = O(/n c ). ( ) 7 j n. As such, for 8 Since C( j, n) takes only non-negative integer values, we have C( j, n) = 0 with high probability, by Markov s ineuality. Lemma. and Lemma. together establish the following theorem. Theorem. One can online CF color a seuence of n points on the line, against an oblivious adversary, such that the coloring is conflict-free with respect to intervals. The algorithm uses O(log n) colors with high probability. We remark that in this problem, the concept of an incoming point being bad for a color is actually not used. Thus, the algorithm is even simpler in this case. Randomized online CF coloring for halfplanes In this section, we consider the problem of CF coloring for halfplanes. In this problem, points are inserted in the plane, and the coloring has to be conflict-free with respect to all halfplanes. Definition. A point C( i, t) is maximal in C( i, t) if it is a vertex of the boundary of CH(C( i, t)), where CH(C( i, t)) denotes the convex hull of C( i, t). The maximal set of C( i, t) is the set of maximal points in C( i, t), and is denoted by V(C( i, t)). 5

6 Suppose that the algorithm is trying to assign color i to the incoming point p t+ ; that is, ColorPoint(p t+, i) is performing the eligibility test for color i for p t+. If a point P (t) is a ( i)- conflicting point for p t+, then there is a halfplane r such that r contains p t+ and r C( i, t) = {}. This implies that the point must be in the set V(C( i, t)). See Figure. Claim. If the point p t+ is in the interior of CH(C( i, t)), then P (t) contains at most three ( i)-conflicting points for p t+. Proof: If the boundary of CH(C( i, t)) is a triangle, then clearly the claim holds; otherwise, assume, without loss of generality, that,..., h are vertices on the boundary of CH(C( i, t)) in clockwise order, and p t+ has as one of its ( i)-conflicting points (see Figure ). Then p t+ must be within triangle, because otherwise any halfplane that contains and p t+ must contain or (or both), a contradiction with the assumption that is a ( i)-conflicting point for p t+. Now that p t+ is within, any halfplane that contains p t+ must contain at least one point of, and. This implies that no point other than, and can be ( i)-conflicting with p t+. Lemma. If the point p t+ is bad for a color i and it is assigned a color greater than i, then we have V(C( i, t + )) V(C( i, t)) (ω ). Proof: Since p t+ is bad for color i, there are at least ω points that are ( i)-conflicting with p t+ (see Definition.). As ω >, Claim. implies that p t+ must be in the exterior of CH(C( i, t)). Because at least ω points on the boundary of CH(C( i, t)) disappear from the boundary of CH(C( i, t + )) as p t+ is inserted, we have V(C( i, t + )) V(C( i, t)) (ω ) + = V(C( i, t)) (ω ). Claim. Among all the points of C( i, n), there are at most C( i, n) /(ω ) points that are bad for color i when they are inserted. Proof: Consider the scenario at time t: (i) If p t / C( i, t), namely, p t is assigned a color smaller than i, then we have as C( i, t) = C( i, t ). V(C( i, t)) = V(C( i, t )), (ii) If p t C( i, t) and p t is not bad for color i, then (iii) Otherwise, by Lemma., we have V(C( i, t)) V(C( i, t )) +. V(C( i, t + )) V(C( i, t)) (ω ). 6

7 s s s p t+ s s (a) s s (b) s Figure : The figure (a) depicts the points of C( i, t). The maximal points are black. A maximal point may be of multiple types. For example, is -maximal and -maximal, and as such appears in the multi-set V(C( i, t)) twice. The figure (b) illustrates the situation after p t+ is inserted and is assigned a color i, where and are not maximal points of ( i)-level any longer. Suppose that from time to n, the case (ii) occurs j times and the case (iii) occurs k times. We have j + k = C( i, n). Note that 0 V(C( i, n)) n = ( V(C( i, t)) V(C( i, t )) ) t= j (ω )k. It follows that k C( i, n) /(ω ). Now we are able to prove the main result of this section. Theorem.5 One can online color a seuence of n points in the plane, against an oblivious adversary, such that the coloring is conflict-free with respect to halfplanes. The algorithm uses O(log n) colors with high probability. Proof: As shown in the proof of Lemma., it suffices to prove that (in expectation) at least a constant fraction of points of C( i, n) are assigned color i, for any i. Consider the set C( i, n). By Claim., at least /(ω ) fraction of the points of C( i, n) had less than ω ( i)-conflicting points with them (when they were inserted). Thus, set ω =. Arguing as before, it follows that (in expectation) at least ( ) ( ) ω ω = fraction of the points of C( i, n) are assigned color i. 7

8 5 Randomized online CF coloring for nearly-eual axis-parallel rectangles A (possibly infinite) family of axis-parallel rectangles are said to be nearly-eual, if there exists some positive constant α, such that the ratio between the largest width and the smallest width of the rectangles in the family is less than α, and the ratio between the largest height and smallest height of the rectangles is less than α. Consider a family Q of nearly-eual axis-parallel rectangles. Without loss of generality, we assume that Q contains a unit suare. Let S be an axis-parallel suare grid with suare side length of /α. We tile the plane with S. This ensures that any rectangle Q Q is larger than a suare tile of S (that is, Q has both larger width and larger height than the suare tile). We assign each grid suare a color class, so that no rectangle of Q intersects two distinct grid suares with the same color class. It is easy to verify that a constant number (indeed, O ( α ) ) of color classes suffices. We assign colors to points within each grid suare independently, using the colors of the class assigned to the suare. Let S be an arbitrary suare tile. By the discussion above, we can assume (without loss of generality) that all the points of P are in the interior of. Let s, s, s, s be the corners of. Note that if a rectangle Q Q intersects, then Q contains at least one corner of. This is because the rectangle Q is larger than. Definition 5. A point C( i, t) is j-maximal in C( i, t) if there exists a rectangle Q Q such that Q C( i, t) = {} and Q contains the corner s j, for j =,,,. We say is maximal if it is j-maximal for some j =,,,. The maximal set of C( i, t) is the multi-set of maximal points in C( i, t), and is denoted by V(C( i, t)). Suppose that the algorithm is trying to assign color i to the incoming point p t+. Clearly, ( i)-conflicting points for p t+ can only be from the set V(C( i, t)). See Figure. In the following, we say b < x c if a point b has a smaller x coordinate than a point c. The relations < y, > x, and > y are defined similarly. Set ω = 0 for this problem. We have the following lemma. Lemma 5. If the point p t+ is bad for a color i and it is assigned a color greater than i, then we have ( ω ) V(C( i, t + )) V(C( i, t)). Proof: Since the point p t+ is bad for color i, there exist (at least) ω points of C( i, t), say,..., ω, and ω rectangles of Q, say Q,..., Q ω, such that Q j contains p t+ and Q j C( i, t) = { j }, for j =,..., ω. Suppose, without loss of generality, that ω/ of the rectangles, say Q, Q,..., Q ω/, contain the corner s (we remind the reader that any rectangle of Q contains at least one corner of if it intersects ). Suppose further that < x... < x ω/. This implies that > y... > y ω/. We claim that points,..., ω/ are not -maximal after the point p t+ is assigned a color larger than i. Suppose the contrary, say, is still -maximal. Then we must have > x p t+ or > y p t+. But neither is possible. If > x p t+, then the rectangle Q ω/ must contain both and ω/, a contradiction with the assumption Q ω/ C( i, t) = { ω/ }. If > y p t+, then the rectangle Q must contain both and, a contradiction with the assumption Q C( i, t) = { }. The proof of the following theorem is similar to that of Theorem.5, and we omit the easy details. 8

9 Theorem 5. One can online CF color a seuence of n points in the plane, against an oblivious adversary, such that the coloring is conflict-free with respect to a family of nearly-eual axis-parallel rectangles. The algorithm uses O(log n) colors with high probability. 6 Randomized online CF coloring for congruent disks In the problem of CF coloring for congruent disks, points are inserted in the plane, and the coloring has to be conflict-free with respect to a family of congruent disks. Using the machinery established in the previous sections and the geometric properties of congruent disks observed by Kaplan and Sharir [KS0], it is not difficult to design the algorithm and give the analysis for this case. In particular, Kaplan and Sharir pointed out that the boundaries of congruent disks behave as pseudo-lines within a small suare region. This property helps prove an analog of Lemma 5.. Thus, by using the algorithm presented in Section and the arguments of [KS0], one gets the following result. We omit the easy but tedious details. Theorem 6. One can online CF color a seuence of n points in the plane, against an oblivious adversary, such that the coloring is always conflict-free with respect to a family of congruent disks. The algorithm uses O(log n) colors with high probability. 7 Deterministic online CF coloring for nearly-eual axis-parallel rectangles In this section, we present an efficient deterministic online algorithm for CF coloring a seuence of points P = p,..., p n in the plane with respect to a family Q of nearly-eual axis-parallel rectangles. As discussed in Section 5, we assume that the points of P are within a suare, which is smaller than any rectangle of Q. Definition 7. A uadrant is an unbounded region of the plane whose boundary is formed by two rays, one parallel to the x-axis and the other parallel to the y-axis. The common source point of these two rays is the corner of the uadrant. A uadrant Q is a right-top uadrant if Q = {c(q) + (x, y) x, y 0}, where c(q) denotes the corner of Q. Left-top, right-bottom, and left-bottom uadrants are defined similarly. We need the following simple decomposition lemma. Lemma 7. Let (X, R) be a range space that can be decomposed into β range spaces (X, R j ), for j =,..., β, where R = R j. Assuming that algorithm Alg j can CF color a seuence of n points j with respect to (X, R j ) using f j (n) colors, then one can CF color a seuence of n points with respect to (X, R) using f j (n) colors. j Proof: We assign an incoming point p the color A(p) = A (p),..., A β (p), where A j (p) denotes the color assigned to p by the jth algorithm Alg j. The correctness is immediate. Indeed, consider a range r and the point set P (t) after t insertions. Suppose r R k ; then there exists a point r P (t) such that has the uniue color A k () among the colors assigned (by Alg k ) to the points of r P (t), by the correctness of Alg k. Clearly, A() A(p) for all p r P (t) and p. 9

10 We remind the reader that every rectangle of Q is larger than ; as such every rectangle of Q intersects like a uadrant of the following four types of uadrants: right-top, left-top, leftbottom, and right-bottom. Therefore, Lemma 7. implies that it is sufficient to show how to CF color the points with respect to, say, right-top uadrants. Indeed, by using the same algorithm independently four times (with appropriate reflections of the plane) to color the points with respect to the four different types of uadrants, we get a new algorithm that can CF color for rectangles of Q. 7. Preliminaries We need an algorithm for the following problem DynCFProb of coloring points on the line: At each time, we either (i) insert a new point onto the line, or (ii) replace all the points in an interval with a new point; and we wish to color the points online using positive integers, so that there exists a uniue highest colored point in any interval at all times. Fiat et al. [FLM + 05] presented a deterministic algorithm for the following related problem, using O ( log n ) colors: At each time, we insert a new point onto the line, and we wish to color the points online using positive integers, so that there exists a uniue highest colored point in any interval at all times. Note that the algorithm of [FLM + 05] can be immediately adapted to solving DynCFProb: For the insertion operation, just apply the algorithm of [FLM + 05]; for the replacement operation, supposing that p t+ is replacing all the points of interval I, then we assign the highest color in I to p t+. (Note that the replacement operation is only performed conceptually, by collapsing all the points of I into p t+.) We refer to this adapted algorithm as DynCFAlg. We also need an algorithm for the following problem SuffixCFProb of coloring points on the line: At each time, we insert a new point onto the line to the right of last point inserted; and we wish to color the points using positive integers, so that there exists a uniue highest colored point for each suffix at all times. There is a simple algorithm for this problem: For i, let b(i) be the position of the rightmost bit in the binary representation of i. (For example, b(),..., b(0) is,,,,,,,,,, respectively.) If p is the ith point inserted onto the line then we assign b(i) to p. We refer to this algorithm as SuffixCFAlg. It is easy to verify the following claim. Claim 7. SuffixCFAlg uses at most log n + colors for n points, such that for each suffix of the list of the points on the line, there exists a uniue point of the highest color. 7. Deterministic online CF coloring for right-top uadrants For simplicity of exposition, we assume that the points of P have distinct x and y coordinates. We would like to color P with respect to the range space ( IR, Q RT), where Q RT is the set of all right-top uadrants. We use the RGB color metaphor to describe colors of points. As such, a point is assigned a R-value and a G-value, which together as a pair form the color of the point. Definition 7. A point p dominates a point if p > x and p > y ; namely, any right-top uadrant that contains must also contain p. A point p P (t) is maximal in P (t) if no point in P (t) dominates p. The hull of P (t) is the sorted list of the maximal points of P (t) in increasing x-coordinate order, and is denoted by H(P (t)). Observation 7.5 Let H(P (t)) =,..., k, where < x... < x k. The following holds: 0

11 p t+ 5 p t+ 5 p t+ 5 (a) (b) (c) Figure : Illustrating the hull H(P (t + )) after p t+ is inserted. The points of H(P (t + )) are black. In (a), p t+ is dominated by,,. In (b), p t+ dominates,. In (c), p t+ neither dominates nor is dominated by,,,, or 5. (i) If a point p dominates some points of H(P (t)), then it must dominate a consecutive block of points of H(P (t)). That is, if b and c are dominated by p, then b,..., c are dominated by p. (ii) If a uadrant Q Q RT contains some points of H(P (t)), then Q H(P (t)) consists of a consecutive block of points of H(P (t)). Suppose that H(P (t)) =,..., k. When the point p t+ is inserted, there are three possibilities. (i) If p t+ is dominated by a point of H(P (t)), then the hull H(P (t + )) remains the same as H(P (t)). (ii) If p t+ dominates a consecutive block of points of H(P (t)), say b,..., c, then H(P (t + )) =,..., b, p t+, c+,..., k. (iii) Otherwise, there must exist an index b such that b < x p t+ < x b+, and we have H(P (t + )) =,..., b, p t+, b+,..., k. See Figure. Observe that one can view the hull as an instance of DynCFProb: When inserting p t+, case (i) above does not change the hull; case (ii) corresponds to a replacement operation in DynCFProb; and case (iii) corresponds to an insertion operation in DynCFProb. Therefore, when inserting p t+, in case (i), we assign 0 to be the R-value of p t+ ; and in cases (ii) and (iii), we use DynCFAlg to assign a positive integer (with respect to the hull) to be the R-value of p t+. The points with R-values eual to zero are trivial points. This following claim is implied directly from the correctness of DynCFAlg. Claim 7.6 At any time t, for a consecutive block of points of the hull H(P (t)), there exists a uniue highest R-value in the block. In particular, in case (ii) above, in the block of points on H(P (t)) dominated by p t+, there is an uniue point, say i, that realizes the highest R-value (also note that p t+ inherits the R-value from i ). We add a directed edge from i to p t+. Therefore, we create a number of directed paths of the points of P. For a point p, the target of the directed path to which p belongs is the leader of p, and is denoted by leader(p). (If a directed path has only a single point p then the leader of p is itself.) It is easy to verify that the R-values of the points on a directed path are all the same. See Figure 5. Lemma 7.7 Consider a uadrant Q Q RT and let i be the highest R-value assigned to a point of Q P (t). Then all the points with R-value i in Q P (t) form a suffix of a directed path. Furthermore, the leader of the points of this directed path is on the hull H(P (t)).

12 (a) (b) (c) Figure 5: Illustrating the directed paths of the points of P. The numbers beside the points show the order they are inserted. The points on the hull are black. In (a), the R-values of p, p, p, p, p 5 are,,, 0,, respectively. In (b), p 6 dominates p and is assigned the R-value of p, which is. In (c), p 7 dominates p and p 6, and is assigned the higher of the R-values of p and p 6, which is. Proof: Let U denote the set of points in Q P (t) with R-value eual to i. We first show that for any p U, leader(p) must be on the hull. Suppose for the contrary that is the leader of p and is not on the hull. Since is not on the hull, it must have been dominated by some point, say,, in Q P (t). Since is the leader of p, there is no directed edge from to, which implies that inherited a R-value higher than has. But this contradicts the assumption that has the highest R-value i in Q P (t). Note that for any p U, all the points after p in the directed path to which p belongs must also be in U. This implies that U is formed by a union of suffices of directed paths. Next, assume, for the sake of contradiction, that p, U, and p is not an ancestor of and is not an ancestor of p. Then leader(p) and leader() must be distinct points on the hull. But this is impossible since it would imply that the interval encompassing leader(p) and leader() on the hull has the maximal R-value (i.e., i) appearing twice (which contradicts the correctness of DynCFAlg). Thus, U must be formed by a suffix of a single directed path. Having assigned the R-values to the points, the algorithm still fails to provide us with a valid CF coloring of the points, since several points on a directed path (having the same R-values) are still conflicting with each other. However, when inserting a new (non-trivial) point to the hull, either we add this point to the end of a specific directed path, or alternatively, it is the first vertex in a new directed path. Thus, for a new (non-trivial) point, we assign it a G-value according to its position in its directed path. In particular, we assign a point a G-value by using SuffixCFAlg on the directed path the point is being added to. The color of a point p is the pair (R p, G p ), where R p and G p are the R-value and G-value assigned to p, respectively. Lemma 7.8 The generated coloring is conflict-free with respect to right-top uadrants. Proof: Consider any uadrant Q Q RT and let i be the maximal R-value in Q Q RT. By Lemma 7.7, all the points that have R-value i in Q form a suffix of a directed path π. We know that there is a uniue G-value assigned to one of those points on π by Claim 7.. Thus, the point on π that realizes this G-value has a uniue color in Q. Lemma 7.9 For a seuence of n points, the algorithm uses O ( log n ) colors.

13 Proof: The algorithm assigns O ( log n ) distinct R-values, because DynCFAlg uses this number of colors for a seuence of n points. To complete the proof, observe that the algorithm assigns at most log n + different G-values, by Claim The result Combining Lemma 7., Lemma 7.8 and Lemma 7.9 together implies the following result. Theorem 7.0 One can deterministically online color a seuence of n points in the plane, such that the coloring is always conflict-free with respect to a family of nearly-eual axis-parallel rectangles. The algorithm uses O ( log n ) colors. 8 Conclusions In this paper, we presented randomized online CF coloring algorithms (against oblivious adversaries) for several ranges in the plane, using O(log n) colors with high probability. The algorithms use fewer colors and are considerably simpler than the previous algorithms [KS0]. We also presented the first efficient deterministic algorithm for CF coloring points in the plane with respect to nearly-eual axis-parallel rectangles (which works against a non-oblivious adversary). Interestingly, we were unable to extend the deterministic algorithm to other ranges (in particular, halfplanes, and congruent disks) in the plane, and we leave this as an open problem for further research. Another open problem is to obtain randomized algorithms with comparable performances against non-oblivious adversaries. 9 Acknowledgments The author is grateful to Sariel Har-Peled for helpful discussions on the problems studied in the paper and useful comments on the manuscript. In particular, the simplified presentation of Section 7 follows his suggestions. The author thanks the anonymous reviewers for many valuable comments on improving the presentation of the paper. The author also thanks John Fischer for useful comments. References [AS06] [BE98] N. Alon and S. Smorodinsky. Conflict-free colorings of shallow discs. In Proc. nd Annu. ACM Sympos. Comput. Geom., 006. To appear. A. Borodin and R. El-Yaniv. Online computation and competitive analysis. Cambridge University Press, New York, NY, USA, 998. [ELRS0] G. Even, Z. Lotker, D. Ron, and S. Smorodinsky. Conflict-free colorings of simple geometric regions with applications to freuency assignment in cellular networks. SIAM J. Comput., ():9 6, 00. [EM06] K. Elbassioni and N. H. Mustafa. Conflict-free colorings of rectangle ranges. In B. Durand and W. Thomas, editors, Proc. rd International Sympos. Theoretical Aspects of Comp. Sci., volume 88 of LNCS, pages 5 6, 006.

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