β β β β β B B B B o (i) (ii) (iii)

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1 Output-Sensitive Algoithms fo Unifom Patitions of Points Pankaj K. Agawal y Binay K. Bhattachaya z Sandeep Sen x Octobe 19, 1999 Abstact We conside the following one and two-dimensional bucketing poblems: Given a set S of n points in R 1 o R 2 and a positive intege b, distibute the points of S into b equal-size buckets so that the maximum numbe of points in a bucket is minimized. Suppose at most (n=b) + points lies in each bucket in an optimal solution. We pesent algoithms whose time complexities depend on b and. No pio knowledge of is necessay fo ou algoithms. Fo the one-dimensional poblem, we give a deteministic algoithm that achieves a unning time of O(b 4 ( 2 + log n) + n). Fo the two-dimensional poblem, we pesent a Monte-Calo algoithm that uns in sub-quadatic time fo cetain values of b and. The pevious algoithms, by Asano and Tokuyama [1], seached the entie paameteized space and equied (n 2 ) time in the wost case even fo constant values of b and. We also pesent a subquadatic algoithm fo the special case of the two-dimensional poblem when b = 2. 1 Intoduction We conside geometic optimization poblems that do not seem to have any nice popeties like convexity and that have a lage numbe of distinct global optimal solutions. Consequently, it is had to develop a seach stategy that will avoid consideing all the optimum solutions (o moe likely nea-optimal solutions). Howeve, if the numbe of optimal solutions ae few, we may be able to pune the seach-space. This may lead to moe ecient algoithms that ae \output-sensitive" whee the notion of output is elated to the numbe Wok by the st autho was suppoted by Amy Reseach Oce MURI gant DAAH , by a Sloan fellowship, by NSF gants EIA{ , and CCR{ , and by a gant fom the U.S.-Isaeli Binational Science Foundation. Wok by the second autho was suppoted by an NSERC gant. Pat of this wok was done while the last two authos wee visiting Depatment of Compute Science, Univesity of Newcastle, Austalia. y Cente fo Geometic Computing, Depatment of Compute Science, Box 90129, Duke Univesity, Duham, NC , USA. pankaj@cs.duke.edu z School of Computing Science, Simon Fase Univesity, Bunaby, BC V5A 1S6, Canada. binay@cs.sfu.ca x Depatment of Compute Science and Engineeing, IIT Delhi, New Delhi , India. E- mail:ssen@cs.unc.edu 1

2 Intoduction 2 of optimal solutions. Since we do not know the optimum solution to begin with, we can ty to estimate the optima by some means, say, andom-sampling, and then use that to pune the seach space. The success of such an appoach depends on how eectively we estimate the optima. In this pape we conside the poblem of patitioning a set of points in R 1 o R 2 into equal-size buckets, so that the maximum numbe of points in a bucket is minimized. The st poblem that we conside is the following: Given a set S of n eal numbes and an intege 1 b n, patition S unifomly into b equal sized buckets, i.e., each bucket has the same width. The buckets ae dened by eal numbes i = L + i w, fo 0 i b whee L is the left endpoint of the left-most bucket and w is the width (size) of the buckets. The ith bucket B i is dened by the inteval [ i ; i+1 ) and S \ B i is the content of the ith bucket (fo a xed choice of L and w). We wish to minimize the maximum size of the contents in buckets. Two vesion of this poblem ae studied: (i) the tight case in which B 1 and B b ae equied to be nonempty, and (ii) the elaxed case in which they ae allowed to be empty. β β β β β o B B B B (i) (ii) (iii) Figue 1: (i) One-dimensional bucketing poblem; (ii) unifom-pojection poblem; (iii) twodimensional patitioning poblem. Next, we conside the two-dimensional poblem. Given a set S of n points in R 2 and an intege b n, we again wish to patition S into b equal-size buckets so that the maximum numbe of points in a bucket is minimized. We conside two types of buckets. Fist, we conside the case in which the buckets ae fomed by equally spaced b + 1 paallel lines, `0; : : : ; `b, with oientation, fo some 2 S 1. We equie S to lie between `1 and `b and each of `1; `b to contain at least one point of S. The buckets ae b stips dened by consecutive lines `i 1 and `i (1 i b); see Figue 1 (ii). This bucketing poblem is known as the unifom-pojection poblem. We next dene buckets to be the egions fomed by two families of equally-spaced p b + 1 lines. The extemal lines in both families ae equied to contain at least one point of S; see Figue 1 (iii). This poblem is called the two-dimensional patition poblem. Asano and Tokuyama [1] descibe O(n 2 ) and O(b 2 n 2 )-time algoithms fo the tight and elaxed cases of the one-dimensional poblem. We ae able to obtain an O(b 4 ( 2 +log n)+n)- time deteministic algoithm fo the tight case and O(b 5 ( 2 + log n) + bn)-time algoithm fo the elaxed case. The algoithm itself does not equie the value of ; the value is equied only fo the analysis. This poblem has applications to constuction of optimal

3 Optimal One-Dimensional Cuts 3 hash functions [1]. Come and O'Donnell [4] descibed an algoithm fo the unifom-pojection poblem that uns in O(bn 2 log n) time using O(n 2 + bn) spce. Asano and Tokuyama [1] gave an O(n 2 log n)-time algoithm, which uses O(n) space, by exploiting the dual tansfomation of the poblem. They also give altenative implementations that could be bette fo smalle b, but the wost-case unning time is (n 2 ) even fo constant values of b. Bhattachaya [2] also gave an altenate appoach fo this poblem, using the angle-sweep method. We st descibe a deteministic O(n 4=3 log 3=2 n)-time algoithm fo b = 2 fo the unifom-pojection poblem, thus impoving the quadatic uppe-bound. Fo lage values of b, we descibe a Monte Calo algoithm that computes an optimal solution in time O(minfbn 5=3 log n+b 2 n log n; n 2 g), with pobability at least 1 1=n. The dependence of unning time on is bone out by the fact that the numbe of possible optimal conguations (having the same value) depends on. The oveall appoaches fo both the poblems ae simila. Namely, we use a sample to \localize" the seach fo the global optimum. Although intuitively, this is a good heuistic, analyzing the bound on the numbe of \potential" candidates fo the global optimum, fom the optima of the sample, is athe technical. In the one-dimensional poblem, we can simply choose a a \deteministic" sample because the elements ae linealy odeed, but the two-dimensional algoithms ely on andom sampling. 2 Optimal One-Dimensional Cuts Fo a set S = fx 1 ; : : : ; x n g of eal numbes and an intege 1 b n, a pai c = (w; L) is called a cut if the set of b + 1 eal numbes j = L + j w, 0 j b, ae such that 0 x 1 x n < b. The inteval [ j 1 ; j ) is called the jth bucket and the set of x i 's lying (stictly) in this inteval is the contents of the jth bucket. We will denote the jth bucket by B j and the size of its contents, jb j \ Sj, by jbj c j fo a cut c. Let (c; S) = max jb c j j 1jb denote the cut-value of c. Let C be the set of all cuts. The optimal cut value, (S), is dened as (S) = min (c; S): c2c Any cut that achieves this cut value is an optimal cut. If we estict the cuts to satisfy the condition that jb 1 j; jb b j 1, i.e., the st and the last buckets cannot be empty, then it is called a tight cut. An optimal tight cut is dened analogously as above, esticted to the set of tight cuts. We will st descibe an algoithm fo nding an optimal tight cut. Denition 2.1 Two cuts c 1 and c 2 ae combinatoially distinct if thee is an i, 1 i b, such that jb c 1 i j 6= jb c 2 i j.

4 Optimal One-Dimensional Cuts 4 Figue 2: An aangement of lines, with one cell shaded. Denition 2.2 The aangement of a set L of lines in the plane, denoted A(L), is the plana subdivision induced by the lines of L; that is, A(L) is a plana map whose vetices ae the intesection points of lines in L, whose edges ae maximal (elatively open) connected potions of the lines S that do not contain a vetex, and whose faces ae the connected components of R 2 L; see Figue 2. We paameteize the poblem as follows. We epesent each cut c = (w; L) as a point in the plane. Abusing the notation slightly, we will use the tem \cut" to denote a point in the (w; L)-plane as well as the set of buckets induced by that cut. Let L = fx i = L + jw j 1 i n; 0 j bg be the set of (b + 1)n lines in the (w; L)-plane, which we efe to as the event lines. L consists of b + 1 families of paallel lines (one fo each xed j), each family containing n lines; see Figue 3 (i). Hence, evey face in A(L) contains at most 2(b + 1) edges. Fo all cuts c = (w; L) lying in the same face f of A(L), the cut value emains the same; we will denote this value by (f; S). Let j (f; S) = jb c j (S)j fo any c 2 f. The non-empty condition of exteme buckets implies that we have to conside only those cuts (w; L) that lie in the quadilateal Q dened by the intesection of the following fou constaints; see Figue 3. Q : x 1 L > x 1 w and x n x 1 b The above constaint leads to the following lemma. < w < x n x 1 b 1 : (2.1) Lemma 2.3 Fo evey point x i 2 S, thee exists an intege 1 j b 1, so that x i lies in one of the two buckets B j o B j+1 fo any tight cut. Poof: A point x i 2 S lies in the bucket B j of a cut c = (w; L) if and only if L + w (j 1) x i < L + w j:

5 Optimal One-Dimensional Cuts 5 Suppose thee ae two cuts c 1 = (w 1 ; L 1 ) and c 2 = (w 2 ; L 2 ) and two integes 1 k 1 < k < k 2 b such that x i lies in the bucket B k1 of the cut c 1 and in the bucket B k2 of c 2. Then we have the following two inequalities: Theefoe, On the othe hand, by (2.1), x i x 1 < k 1 w 1 and x i x 1 > (k 2 1) w 2 (k 1 + 1)w 2 : w 2 w 1 < k 1 k = 1 1 k : (2.2) w 2 > x n x 1 b 1 = 1 1 w 1 b x n x 1 b : (2.3) Compaing (2.2) and (2.3), we obtain k 1 > b 1, which contadicts the assumption that k 1 < k 2 1 b 2. Hence, the lemma is tue. 2 This lemma immediately implies that at most n lines of L intesect Q, and that Q intesects O(n 2 ) faces of A(L). The lines of L that intesect Q can be detemined in O(bn) time. We can theefoe seach ove Q \ A(L) in O(n 2 ) time to nd all combinatoially distinct optimal cuts. Lemma 2.4 Fo a set of m points, all the combinatoially distinct optimal cuts can be computed in O(m 2 ) time. L x 4 L x4 x3 x 2 x1 x3 x 2 x1 Q ( x 4- x 1)/b ( x 4- x 1)/(b-1) w w (i) (ii) Figue 3: (i) Set L and the feasible egion Q; (ii) Shaded egions denote C 22 ; C 23, and C 24, and the dak egion denotes C(2; 4; 2), the set of cuts fo which fx 2 ; x 3 ; x 4 g lie in the second bucket B 2. Fo an intege 1, let R S be the subset of points obtained by choosing evey n=th point of S. Fom ou pevious obsevation about diectly solving the poblem, we can compute the optimal solution fo R in O( 2 ) time.

6 Optimal One-Dimensional Cuts 6 Lemma 2.5 Let n o ; o be the maximum size of a bucket in an optimal solution fo S and R, espectively. Then n o n o < 1 : Poof: Let c be an optimal cut fo R. Each bucket of c contains at most o points. Since R is chosen by selecting evey (n=)th point of S, each bucket of c contains at most ( o + 1)n= 1 points of S. Theefoe n o < ( o + 1)n=, o n o n o < 1 : Convesely, let c 0 be an optimal cut fo S. Then each bucket of c contains at most n o points of S, which implies that each bucket contains at most (n o + (n=) 1)=n points of R. Hence, n o + n n o o o < n o n < 1 : This completes the poof of the lemma. 2 We now descibe the algoithm fo computing an optimal solution fo S, assuming that we have aleady computed the value of o. Let C ij denote the set of points c = (w; L) in the (w; L)-plane so that the point x j 2 S lies in the bucket B i of the cut c. Then C ij = f(w; L) j L + (i 1)w x j < L + iwg is the cone with apex at (0; x j ); see Figue 3 (ii). Given thee integes 1 l n and 1 i b, the set of points in the (w; T L)-plane fo which the subset fx l ; x l+1 ; : : : ; x g of S lies in the ith bucket B i is C(l; ; i) = C j=l ij. C(l; ; i) is a cone fomed by the intesection of the halfplanes L + (i 1)w x l and L + iw > x. By Lemma 2.5, ( o 1) n < n o < ( o + 1) n : (2.4) Set m = ( o + 1)n= > n o. We will use this inequality to compute n o eciently. Dene n o = (n=b) + and m = (n=b) +. Using (2.4), we obtain that < + 2n=. Si-1 S i S i l i-1 β i-1 i l i βi i l i+1 β i+1 i+1 Figue 4: The bounday i can lie in the shaded inteval [l i ; i ). If b 2 n, then we use the O(n 2 )-time algoithm descibed ealie to compute an optimal cut, so assume that b 2 < n. If each bucket B i in a cut c contains at most m points of S, then, fo any 1 i b, the st i buckets in c contain at most i = mi points, theefoe i < x i. Similaly, the last b i buckets in c contain at most (b i)m points, theefoe

7 Optimal One-Dimensional Cuts 7 i x li, whee l i = n m(b i). Hence, i 2 [x li ; x i ). Set 0 = 1; see Figue 4. Note that i l i = b fo 1 i b. This implies that the subset S i = fx j j i 1 j < l i g always lies in the ith bucket B i (see Figue 4), fo all 1 i b. Hence, if thee is a cut = (w; L) T so that all buckets in contain at most m points, then lies in the egion b P (m) = C( i=1 i 1; l i 1; i), which is the intesection of b cones and is thus a convex polygon with at most 2b edges. Fo all cuts 62 P (m), (; S) > m. It thus suces to seach fo an optimal cut within P (m). Let H i L be a set of l i i = b lines dened as H i = fl + iw = x j j l i j < i g: Set H = S b i=1 H i; jhj = b 2. The same agument as in Lemma 2.3 shows that no line of H n L intesects the inteio of the polygon P (m). We constuct the aangement A(H) within the polygon P (m) in O(b 4 2 ) time. (Actually, we can clip A(H) inside P (m) \ Q, whee Q is the quadilateal dened in (2.1).) Let A P (H) denote this clipped aangement. By the above discussion, A P (H) is the same as A(L) clipped within P (m). Theefoe, fo any two points and 0 in a face f 2 A P (H), the contents of all buckets in the cuts and 0 ae the same. Let '(f) = h 1 (f; S); : : : ; b (f; S)i: If f and f 0 ae two adjacent faces of A P (H) sepaated by a line L + iw = x j, then the only dieence in the two cuts 2 f and 0 2 f 0 is that x j lies in B i 1 in one of them and it lies in B i in the othe. Theefoe '(f 0 ) and (f 0 ; S) can be computed fom '(f) and (f; S) in O(1) time. We compute an Euleian tou = hf 0 ; f 1 ; : : : ; f u i, u = O(b 4 2 ), of the dual gaph of A P (H) in time O(b 4 2 ). We compute '(f 0 ) and (f 0 ; S) in O(n) time. We then visit the faces of A P (H) along, and fo each i 1, compute '(f i ) and (f i ; S) fom '(f i 1 ) and (f i 1 ; S) in O(1) time. We can thus compute n o = (S) = min f2ap (H) (f; S) in O(b n) time. The total time spent in computing an optimal cut is O 2 + b 4 + n Choosing = db p n e, we obtain the following. 2 + n : Lemma 2.6 An optimal tight cut fo n points into b buckets can be found in O(b 4 2 +b 2 n) time. Instead of using the quadatic algoithm fo computing o, we can compute o ecusively. Let T () denote the maximum unning time of the algoithm fo computing an optimal cut fo the subset of S size chosen by selecting evey (n=)th point of S, then we have the following ecuence: T (n; ) = ( T (; 0 ) + O b 4 + n 2 + n O(n 2 ) if b 2 ( + 2n ) n; othewise:

8 The Unifom-Pojection Poblem 8 Choosing = dn=2e and using the fact that o n o =n + 1, we obtain that Hence, we can show that o b ; i.e., 0 =2 + 1 T (n; ) = O(b 4 ( 2 + log n) + n): Theoem 2.7 Given a set S of n points in R and an intege 1 b n, an optimal tight cut fo S with b buckets can be computed in O(b 4 ( 2 + log n) + n) time. We can use a simila analysis fo nding optimal cuts, including elaxed cuts. We simply eplace n by bn as thee ae bn event lines. Anothe way to view this is that the optimal cut can be detemined by tying out all non-edundant cuts fo buckets fo 2 b and selecting the best one. Coollay 2.8 An optimal elaxed cut fo a set of n points in R with b buckets can be found in O(b 5 ( 2 + log n) + bn) time. 3 The Unifom-Pojection Poblem In this section we descibe the algoithms fo the unifom pojection poblem. Let S = fp 1 ; : : : ; p n g be a set of n points in R 2 and 1 b n an intege. We want to nd b + 1 equally spaced paallel lines so that all points of S lie between the extemal lines, the exteme lines contain at least one points of S, and the maximum numbe of points in a bucket is minimized; see Figue 1 (ii). If the lines have slope, we efe to these buckets as the -cut of S. Fo each, thee is unique -cut of S. We st descibe a subquadatic algoithm fo b = 2. Next, we show how the unning time of the algoithm by Asano and Tokuyama can be impoved, and then we descibe a Monte Calo algoithm that computes (S), the optimum value, with high pobability, in subquadatic time fo cetain values of b and. a σ b h σ* h * a* b * pimal dual Figue 5: The duality tansfom in two dimensions. Vetical segment is the dual of the stip.

9 The Unifom-Pojection Poblem 9 It will be convenient to wok in the dual plane. The duality tansfom maps a point p = (a; b) to the line p : y = ax + b and a line ` : y = x + to the point ` = (; ); see Figue 5. Let `i denote the line dual to the point p i 2 S, and let L = f`i j 1 i ng. The dual of a stip bounded by two paallel lines `1 and `2 is the vetical segment = `1`2 ; a point p lies in if and only if the line intesects the segment. Figue 6: The 2-level in a line aangement. Let A(L) be the aangement of L as dened in Section 2. We dene the level of a point p 2 R 2 with espect to L, denoted by (p; L), to be the numbe of lines in L that lie below p. The level of all points within an edge o a face of A(L) is the same. Fo an intege 0 k < n, we dene the k-level of A(L), denoted by A k (L), to be the closue of the set of edges of A(L) whose levels ae k see Figue 6. A k (L) is an x-monotone polygonal chain with at most O(n(k + 1) 1=3 ) edges [5]. The lowe and uppe envelopes of A(L) ae the levels A 0 (L) and A n 1 (L), espectively. The total numbe of vetices on the uppe and lowe envelopes of A(L) is n because evey such vetex is the dual of the line suppoting an edge of the convex hull of S. Since we equie the exteme bucket boundaies to contain a point of S, the points dual to the exteme lines lie on the uppe and lowe envelopes of L. Fo a xed x-coodinate, let s() denote the vetical segment connecting the points on the lowe and uppe envelopes of L with the x-coodinate. We can patition s() into b equal-length subsegments s 1 (); : : : ; s b (). Let 0 (); : : : ; b () be the endpoints of these segments. These endpoints ae dual of the bucket boundaies of the -cut, and s i () is the dual of the jth bucket in the -cut. The line `j intesects s i (), i b, if and only if the point p j lies in the bucket B i coesponding to the -cut. Let i denote the path taced by the endpoint i () as we vay fom 1 to +1. If we vay, i (), fo 0 i b, taces along a line segment, as long as the endpoints of s() do not pass though a vetex of uppe o lowe envelopes. Theefoe each i is an x-monotone polygonal chain with at most n vetices; see Figue 7 fo an illustation. Since we will be looking at the poblem in the dual plane fom now, we will call i 's bucket lines. Let B = f 0 ; : : : ; b g. The intesection of a bucket line i with a line `j is an event at which the point p j switches fom B j 1 to B j o vice-vesa. Fo an x-coodinate and a subset A L, let i (A; ) denote the numbe of lines of A that intesect the vetical segment s i (); i (A; ) denotes the set of points dual to A that lie in the ith bucket of the -cut. Let (A; ) = max 1ib i (A; ): Set n o = (S) = (L) =

10 The Unifom-Pojection Poblem 10 Uppe envelope β 4 β 3 β 2 Lowe envelope β 1 β 0 Figue 7: The unifom-pojection poblem and the bucket lines in the dual setting. min (L; ). 3.1 Patitioning into two buckets We will st descibe a deteministic scheme that takes subquadatic time to nd an optimal solution fo patitioning S into two buckets. By ou convention, 0 ; 2 denote the uppe and lowe envelopes of L, espectively. To detemine n o, we will seach fo an x-coodinate o, whee 1 ( 0 ) is closest to the dn=2e-level of A(L). Fist, we compute = A dn=2e (L) in O(n 4=3 log 1+" n) time [3], fo any " > 0, and check whethe 1 intesects. If a point 1 ( o ) lies on, then we etun the o -cut. If 1 lies below, we compute the highest level in the inteval [1; dn=2e 1] of A(L) that 1 intesects, and set o to this level. This can be accomplished in O(n 4=3 log 2+" n) by pefoming a binay seach on the levels. Similaly, if 1 lies above, we nd in O(n 4=3 log 2+" n) time the smallest level in the inteval [dn=2e + 1; n 1] that 1 intesects and set o to this level. If 1 ( o ) is an intesection point of 1 and A o (L), then we etun the o -cut. Chan's algoithm computes the edges of a level incementally fom left to ight, so we can actually detect whethe 1 intesects the level while computing the level itself in O(n 4=3 log 1+" n) time using O(n) space. Hence, we obtain the following. Lemma 3.1 The optimal unifom pojection of n points in R 2 into two buckets can be computed in O(n 4=3 log 2+" n) steps, fo any " > 0, using O(n) space. 3.2 A deteministic algoithm In this section we pesent a deteministic algoithm fo the unifom-pojection poblem that has O(bn log n+k log n) unning time and uses O(n) stoage, whee K denotes the numbe of event points, i.e., the numbe of intesection points between L and B. This impoves the unning times of O(n 2 + bn + K log n) fo geneal b and O(b 0:610 n 1:695 + K log n) fo b < p n in [1]. As in Asano-Tokuyama's algoithm, we will sweep a vetical line though A(L), but unlike thei appoach we will not stop at evey intesection point of L and B. We st

11 The Unifom-Pojection Poblem 11 compute the lowe and uppe envelopes of L, which ae the bucket lines 0 and b, espectively. We can then compute est of the bucket lines 1 ; : : : ; b 1 in anothe O(bn) time. We pepocess each i fo answeing ay-shooting queies in O(n log n) time so that a quey can be answeed in O(log n) time [8]. The total spaced used is O(bn). We sweep a vetical line fom x = 1 to x = +1, stopping at the intesection points of L and the bucket lines. At each x-coodinate, fo 1 i b, we maintain i (), and fo 1 j n, the index of the bucket j that contains the line `j in the -cut. These quantities emain the same fo all x-coodinates between two consecutive event points. We also maintain an event queue Q that stoes some of the event points that lie to the ight of the sweep line, but it is guaanteed to contain the next event point. Suppose we ae at an event point i () = i \ `j and `j lies above i to the ight of i (). Then `j moves fom B i to B i+1 at. We theefoe decease i () by 1, incease i+1 () by 1, and set j to i. The next intesection point of ` and B, if it exists, lies on eithe i o i+1. We compute in O(log n) time the intesection points of ` with i and i+1 that lie immediately afte i (), using the ay-shooting data stuctue and add them to Q. On the othe hand, if `j lies below i to the ight of i (), `j moves fom B i+1 to B i at. We decease i+1 () by 1, incease i () by 1, compute the next intesection points of `j with i and i 1, and add the two intesection points (if they exist) to Q. We spend O(log n) at each event point. Theefoe the total unning time of the algoithm is O((bn + K) log n). Q uses O(K) space and the ay-shooting data stuctues use O(bn) space. The size of Q can be educed to O(n) using the standad technique, namely, fo each line `j, stoe only one intesection point of `j with the bucket lines [6]. In paticula, suppose we want to inset a point 2 `j to Q. We check whethe Q aleady contains a point 0 on `j. If x() x( 0 ), we do not inset into Q. Othewise, we inset into Q and delete 0 fom it. The total time spent at each event point is still O(log n), but the size of Q is now O(n). Howeve, the ay-shooting data stuctue still equies O(bn) space. In ode to educe the oveall stoage to O(n), we patition the plane into u 2b vetical stips W 1 ; : : : ; W u so that each W i contains at most n vetices of the bucket lines. Note that each j contains at most n= vetices inside W i. We now un the above sweep-line algoithm in each W i sepaately. While sweeping a vetical line though W i, we have to pepocess only i \ W fo ay shooting, fo each 0 i b. Since each i has at most n=b vetices inside W i, the total space used by the ay-shooting data stuctues is O(n). The asymptotic unning time is still O((bn + K) log n). Hence, we obtain the following. Theoem 3.2 An optimum patitioning in the tight case can be detemined in O((bn + K) log n) time using O(n) stoage, whee K is the numbe of event points. 3.3 A Monte-Calo algoithm We now pesent a Monte-Calo algoithm that uns in sub-quadatic time, with high pobability, fo small values of b and, whee n o = (n=b) +. The oveall idea is quite staightfowad and simila to Section 2. Fom the given set L of n lines, we choose a andom subset R of size > 20 log n (a value that we will specify duing the analysis).

12 The Unifom-Pojection Poblem 12 Let R be the x-coodinates of all the intesection points of R and B, the set of bucket lines with espect to L. We compute o = min 2R (R; ). Note that we ae not computing (R) since we ae consideing buckets lines with espect to L. B can be computed in O(n log n + bn) time and o can be computed in additional O((b + n)) = O(n) time. We use o to estimate the oveall optimum n o with high likelihood. In the next phase, we use this estimate and the ideas used in the one-dimensional algoithm to sweep only those egions of B that \potentially" contain the optimal solution. In ou analysis, we will show that the numbe of such event points is o(n 2 ) if b and ae small. The eade can also view this appoach as being simila to the andomized selection algoithm of Floyd and Rivest. We choose two paametes and Va = Va() whose values will be specied in the analysis below. An event point with espect to L (esp. R) is a vetex of B o an intesection point of a line of L (esp. R) with a chain in B. The event points with espect to R patition the chains of B into disjoint segments, which we efe to as canonical intevals. Befoe descibing the algoithm we state a few lemmas, which will be cucial fo ou algoithm. In the following, we will assume that R is a andom subset of L of size > 20 log n. Ou st lemma establishes a elation between the event points of A(L) and those of A(R). Lemma 3.3 Let > 0 be a constant and let 1 i b be an intege. With pobability at least 1 1=n, at most O((n=) log n) event points of A(L) lie on any canonical inteval of i. Poof: The poof follows along the lines of a standad andom-sampling agument. Conside any event point of A(L). The pobability that moe than c(n=) log n lines of L ae not chosen befoe the st line is chosen to its ight is no moe than (1 n )cn log n= n c. The pobability that this holds fo any event point of A(L) (and hence fo A(R)) is less than K n c. Since K = O(n 2 ), by choosing c = + 2, the lemma follows. 2 Using a classical esult by Vapnik and Chevonenkis (see e.g. [11, Chapte 16]), which can also be poved using Cheno's bound, we can establish a elationship between the numbe of lines of L and of R intesecting a vetical segment. Lemma 3.4 Let e be a vetical segment and let L e L be the subset of n e lines that intesect e. Thee is a constant c such that with pobability exceeding 1 1=n 2, n e n jl e \ Rj log n c An immediate coollay of the above lemma is the following. Coollay 3.5 Thee is a constant c so that, with pobability exceeding 1 1=n, n o n o log n c : :

13 The Unifom-Pojection Poblem 13 Poof: Suppose the -cut is an optimal cut fo R. Apply Lemma 3.4 to the segments s 1 (); : : : ; s b (). Since b n and each segment s i () intesects less than n lines of L, the claim follows. 2 Coollay 3.6 Let be a -cut so that evey bucket of contains at most m points of S. Fo 1 i b 1, let l i = (b i)m n cp log n and i = im n + cp log n; whee c is an appopiate constant. Then with pobability exceeding 1 1=n, l i ( i (); R) i : (3.1) Poof: If each bucket of at most m points, then the st i-buckets of contain at most mi points of S and the last (b i) buckets of contain at most (b i)m points of S. The lemma now follows by an application of Lemma 3.4 to the segments 0 () i () and i () b (). 2 We also need the following esult by Matousek on simplex ange seaching. Lemma 3.7 (Matousek [9]) Given a set P of points and a paamete m, n m n 2, one can pepocess P fo tiangle ange seaching in time O(m log n), to build a datastuctue of O(m) space and then epot queies in O((n log 2 n)= p m + K) time, fo output size K, whee K is numbe of points in the quey tiangle. Remak. If m = ( 2 log 2 n) and K (n=) log n, then the output size dominates the quey time, so the quey time becomes O(K) in this case. We now descibe the algoithm in detail. We st compute in O(n log n + bn) time the uppe and lowe envelopes of L and the bucket lines 0 ; : : : ; b. Next, we choose a andom sample R of size, whee > 20 log n is a paamete to be xed late, and compute o = min (R; ), whee vaies ove the x-coodinates of all the event points of B with espect to R. As mentioned ealie, we ae not computing an optimal solution fo R, since the bucket lines ae dened by L. We can compute o in O(n) time as descibed in [1]. This completes the st phase of the algoithm. The total time equied by this phase is O(n log n + bn) + O(n) = O(( + b)n): (3.2) In the following we assume that the set R satises Lemmas 3.3 and 3.4 and Coollaies 3.5 and 3.6. This holds with pobability exceeding 1 1=n. By Coollay 3.5, Set m L = max ( n log n o cn n log n o cn ; n b ). n log n n o o + cn :

14 The Unifom-Pojection Poblem 14 We st nd the smallest 0 i dlog ne, by testing fo i = 0; 1; : : : in inceasing ode, such that m L + 2 i < n o m L + 2 i+1. We then pefom a binay seach in the inteval [m L +2 i ; m L +2 i+1 ] to compute the optimal value n o. We thus need a pocedue that, given an intege m 2 [m L + 2 i ; m L + 2 i+1 ], can detemine whethe n o m o n o > m. Suppose n o = (n=b) + and m = (n=b) +. Since m L n=b and m L + 2 i < n o, we have > 2 i. Theefoe m m L + 2 i+1 n o + 2 i < n b + 2: (3.3) We un the decision algoithm O(log n) times. β2 β 1 X Figue 8: 1 ; 2, and X. Solid lines belong to R and dashed lines belong to L n R. Shaded egions denote the segments s 1 () fo 2 X. Lage (small) bullets ae the intesection points of L with the bucket lines that lie (esp. do not lie) inside X R. Aowed segments epesent the canonical intevals in I 1. We now descibe the decision algoithm. If each bucket of a -cut contains at most m points of S, then by Coollay 3.6, l i ( i (); R) i. Fo each 1 i < b, let X i = f j l i ( i (); R) i g: Let X = T b 1 i=1 X i, and let jxj be the numbe of connected components in X. Fo any 62 X, at least one of the i does not satisfy (3.1), so (L; ) > m fo any such -cut. We theefoe estict ou seach to the -cuts fo which 2 X and compute m o = min 2X (L; ). If m o m, then n o m. Othewise, we conclude that n o > m. Hence, it suces to descibe an algoithm fo computing m o. Fo each 0 i b, let I i be the set of canonical intevals of i whose x-pojections intesect X (see Figue 8), and let i = f 2 X j i () is an event point with espect to Lg: Set I = S b i=0 I i, = jij, and = S b i=0 i. Since evey event point whose x-coodinate is in i lies on a canonical inteval in I i, by Lemma 3.3, jj = O((n=) log n). Since the contents of buckets change only at the event points, m o = min 2X (L; ) = min (L; ): 2

15 The Unifom-Pojection Poblem 15 It thus suces to compute (L; ) fo all 2. We will descibe late how to compute X and I, but we st descibe how to compute and an optimal cut fom X and I. We pepocess S in O( 2 log 2 n) time into a data stuctue of size O( 2 log n) fo answeing tiangle ange queies using Lemma 3.7. Fo each canonical inteval I 2 I i, we compute the subset L I L of lines that intesect I in O((n=) log n) time using the ange-seaching data stuctue, because, in the pimal plane, I coesponds to a double-wedge and it contains a point of p i 2 S if and only if I intesects `i. We then compute the intesection points of I and L I these ae the event points with espect to L that lie on I. We epeat this step fo all intevals in I. The total time spent in computing these intesection points is O( 2 log 2 n + (n=) log n). We discad those event points whose x-pojections do not lie in X. is the set of the emaining event points. We sot in an inceasing ode in time O(jj log n). The total time spent in computing and soting is O( 2 log 2 n + (n=) log n) + O(jj log n) = O( 2 log 2 n + (n=) log 2 n): (3.4) We sweep a vetical line ove X fom left to ight, stopping at the x-values in. Fo a 2 X, we maintain () = h 1 (L; ); : : : ; b (L; )i: The vecto () emains the same fo all x-values in X lying between two consecutive values in. Suppose we ae at a point 2, which belongs to i. Let I be the connected component of X that contains. If is the leftmost event point in I, we compute the numbe of lines if L intesecting the vetical segment s i () (i.e., the points of S lying in the ith bucket of the -cut), fo 1 i b, using the ange-seaching data stuctue in time O((n=) log n), and set i (L; ) to this value. We can theefoe compute () fo such an event point in O(b(n=) log n) time. If is not the st event point in I, then we update () as follows. Suppose i () = i \ `j and `j lies above i afte i (). Then the point p j moves fom the bucket B i to B i+1 at, We decease i (L; ) by 1 and incease i+1 (L; ) by 1. Similaly, if `j lies below i to the ight of i (), we incease i (L; ) by 1 and decease i+1 (L; ) by 1. The total time spent by the sweep-line algoithm is O bn log n jxj + O(jj) = O Finally, we descibe how to compute X and I i. Set l i = (b i)m n cp log n and i = im n + cp log n: (bjxj + ) n log n : (3.5)

16 The Unifom-Pojection Poblem 16 Dene = i l i = bm n + 2cp log n n p b c log n b n (by equation (3.3)) 2b n + 2cp log n: β i i l i Figue 9: i and the plana subdivision M i ; the shaded egion denotes M i ; the thick shaded line is the clipped i. Recall that X i is the x-pojection of the potion of i that lies between A li (R) and A i (R). We compute A li (R) and A i (R) and clip the potion of i between these two levels; see Figue 9. X i consists T of O(n+ 4=3 ) connected components and can be computed within b 1 this bound. We set X = X i=1 i; jxj = O(b(n+ 4=3 )). Next, we compute the levels A j (R), l i j i. Let M i be the esulting plana subdivision induced by the edges and vetices of A li (R); : : : ; A i (R). By a esult of Dey [5], jm i j = O( 4=3 ( i l i ) 2=3 ) = O( 4=3 2=3 ): M i can be computed in time O( log + jm i j) = O( 4=3 2=3 ) [7]. Since i is an x-monotone polygonal chain and M i consists of edge-disjoint x-monotone polygonal chains, the numbe of intesection points between i and M i is O(n + jm i j) = O(n + 4=3 2=3 ), and they can be computed within that time bound. We can thus compute the set I 0 i of all canonical intevals of i whose x-pojections intesect X i in time O(n + 4=3 2=3 ). We discad those canonical intevals of I 0 i whose x-pojections do not intesect X. The emaining intevals of I 0 gives the set i I i. Theefoe X i ji 0 i j = O(b(n + 4=3 2=3 )): Repeating this pocedue fo all bucket lines, the total time in computing jxj and I is O(b(n + 4=3 2=3 )): (3.6)

17 Two-Dimensional Patitioning 17 Summing up (3.2), (3.4), (3.5), and (3.6); substituting the values of and X; and using the fact that we un the decision algoithm O(log n) times, the total time in computing n o is thus T (n) = O(( + b)n) + O( 2 log 3 n) + b(n + 4=3 2=3 ) n log3 n + O(b 2 (n + 4=3 ) n log2 n + b(n + 4=3 2=3 )) n log2 n + O(b(n + 4=3 2=3 )) log n = O(n) + O(b(n + 4=3 2=3 )) n log3 n + O(b 2 (n + 4=3 ) n log2 n): In the last equality, we use the fact that b n 1=3 and that we will be choosing b 2=3 n 2=3. Substituting the value of, we obtain bn 2 T (n) = O(n) + O log3 n b p n + log n + Setting = O(b 1=3 n log 3 n) b p 2=3 n + log n + O n 2 b 2 + n1=3 log 2 n = O((b 2 )n log 3 n) + O n + bn2 p log 7=2 n + b 2=3 n log 10=3 n n 2 O b 2 l b 2=3 n 2=3 log 7=3 n + n1=3 log 2 n m, we obtain the following. : Theoem 3.8 Thee is a Monte Calo algoithm to compute the optimal unifom pojection of a set of n points in R 2 onto b equal-sized buckets in time O(minfbn 5=3 log 7=3 n + (b 2 )n log 3 n; n 2 g); with pobability at least 1 1=n, whee the optimal value is (n=b) +. In paticula, ou algoithm can detect whethe thee is a unifom pojection (i.e., with = 0) in O(minfbn 5=3 log 7=3 n; n 2 g) time; if thee exsits a unifom pojection, the algoithm etuns one. 4 Two-Dimensional Patitioning In this section we conside the poblem of patitioning a set S of n points in R 2 into \ectangula" buckets. Moe pecisely, given S and an intege b 1, we want to compute two families of equally spaced p b + 1 lines L = f`0; : : : ; `pb g and L0 = f`01 ; : : : ; `0p b g, so that the following conditions hold (i) If the oientation of the lines in L is 2 [0; =2), then the oientation of the lines in L 0 is =2 +. +

18 Two-Dimensional Patitioning 18 (ii) S lies between `0 and `pb as well as between `00 and `0p b. (iii) Each of the exteme lines `0; `00 ; `pb ; `0p b contains at least one point of S. (iv) The buckets ae ectangles B ij dened by `i 1 ; `i; `0j 1 ; `0j, fo any pai 1 i; j p b. The maximum numbe of points in a bucket is minimum. See Figue 1 (iii) fo an example. If the slope of lines in L is (and of lines in L 0 is 1=), we efe to the esulting buckets as the -cut. Let ij (L; ) be the numbe of points in the bucket B ij of the -cut. In the dual setting, the stip fomed by the lines `i 1 and `i of the -cut is the vetical segment s i () as dened in the pevious section. Similaly, the dual of the stip fomed by `0j 1 and `0j is the segment s j( 1=). Hence, a point p k belongs to the bucket B ij of the -cut if `k intesects both s i () and s j ( 1=). Let B = f 0 ; : : : ; p g be the set of bucket b lines as dened in Section 3.2 (a vetical segment s() whose endpoints lie on the lowe and vetical envelopes of A(L) is patitioned into p b equal segments s 1 (); : : : ; s p ()). b As noted by Asano and Tokuyama, we can still compute an optimal solution by a sweepline algoithm. We sweep two vetical lines L and L 0. L sweeps the plane fom x = 0 to x = +1. When L is at x =, L 0 is at x = 1=. We stop when eithe L o L 0 cosses an intesection point of L and B. At each, we maintain, fo evey 1 i; j p b, the numbe of points of S that lie in the bucket B ij of the -cut, and fo each line `k 2 L, the pai (i; j) if p k 2 B ij. If L passes though an event point lying on i, then a line moves fom a bucket B ij to B (i+1)j at, o vice-vesa. Similaly, if L 0 passes though an event point lying on j, then a lines moves fom a bucket B ij to the bucket B i(j+1) at, o vice-vesa. As in Section 3.2, we can update the invaiant and the event queue at each event point in O(log n) time. Hence, we conclude the following: Theoem 4.1 An optimum two-dimensional patitioning in the tight case can be detemined in O((bn+K) log n) time using O(n) stoage, whee K is the numbe of event points. We can also extend the Monte-Calo algoithm to this poblem. If (S; ) m, then the stips dened by two consecutive lines of L (o L 0 ) contain at most p bm points. If we choose a andom sample R as in Section 3.3, dene o = min max i;j ij (R; ), and compute it using the deteministic algoithm, then Lemmas 3.3 and 3.4 and Coollay 3.5 still hold. Coollay 3.6 can now be e-stated as follows. Coollay 4.2 Let be a -cut so that evey bucket of contains at most m points of S. Fo 1 i p b 1, let l i = ( p b i) p bm n cp log n and i = i p bm n + cp log n; whee c is an appopiate constant. Then with pobability exceeding 1 1=n, l i ( i (); R); ( i ( 1=); R) i : (4.1)

19 Refeences 19 We can now poceed along the same lines as in Section 3.3. whethe n o m fo a given intege m, we st dene the set In ode to detemine X = f j l i ( i (); R); ( i ( 1=); R) i ; 81 i p g: We sweep two vetical lines though X as in the deteministic algoithm, but using the ideas fom Section 3.3 to compute event points, to move diectly fom one connected component of X to anothe, and to compute X and I. Since thee ae p b + 1 bucket lines in this case, we have = Pp b ji i=0 ij = O( p b(n + 4=3 2=3 )), whee = i l i 2b(=n) + 2c p log n. Caying out the analysis of Section 3.3 with the new value of, we can conclude the following. Theoem 4.3 Given a set of n points in the plane and an intege b, thee exists a Monte- Calo algoithm to nd an optimal two-dimensional patition in time O(minfb 1=2 n 5=3 log 7=3 n+ (b 3=2 )n log 3 n; n 2 g), with pobability at least 1 1=n, whee the optimal value is (n=b)+. Refeences [1] T. Asano and T. Tokuyama, Algoithms fo pojecting points to give the most unifom distibution and applications to hashing, Algoithmica 9 (1993), 572{590. [2] B. Bhattachaya, Usefulness of angle sweep, Poc. of Foundations of Softwae Technology and Theoetical Compute Scienc, 1991, pp.???. [3] T. Chan, A emak on computing the level in line aangements, manuscipt, [4] D. Come and M.J. O'Donnell, Geometic poblems with applications to hashing, SIAM Jounal on Computing 11 (1982), 217{226. [5] T. K. Dey, Impoved bounds on plana k-sets and elated poblems, Discete Comput. Geom., 19 (1998), 373{382. [6] H. Edelsbunne, Algoithms in Combinatoial Geomety, Spinge-Velag, Heidelbeg, [7] S. Ha-Peled, Talking a walk in a plana aangement, Poc. 40th IEEE Annual Sympos. Foundations of Compute Science, 1999, to appea. [8] J. Heshbege and S. Sui, A pedestian appoach to ay shooting: Shoot a ay, take a walk, J. Algoithms, 18 (1995), 403{431. [9] J. Matousek, Ecient Patition tees, Discete and Computational Geomety 8 (1992), 315{334. [10] R. Motwani and P. Raghavan, Randomized Algoithms, Cambidge Univesity Pess, New Yok, [11] J. Pach and P. K. Agawal, Combinatoial Geomety, John Wiley and Sons, New Yok, 1995.