Sublinear Time Width-Bounded Separators and Their Application to the Protein Side-Chain Packing Problem

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1 Sublinear Time With-Boune Separators an Their Application to the rotein Sie-Chain acking roblem Bin Fu 1,2 an Zhixiang Chen 3 1 Dept. of Computer Science, University of New Orleans, LA 70148, USA. 2 Research Institute for Chilren, 200 Henry Clay Avenue New Orleans, LA fu@cs.uno.eu 3 Dept. of Computer Science, University of Texas - an American TX 78539, USA. chen@cs.panam.eu Abstract. Given > 2 an a set of n gri points Q in R, we esign a ranomize algorithm that fins a w-wie separator, which is etermine by a hyper-plane, in O(n 2 log n) sublinear time such that Q has at most ( +1 + o(1))n points one either sie of the hyper-plane, an at most c wn 1 points within w istance to the hyper-plane, where c 2 is a constant for fixe. In particular, c 3 = To our best knowlege, this is the first sublinear time algorithm for fining geometric separators. Our 3D separator is applie to erive an algorithm for the protein sie-chain packing problem, which improves an simplifies the previous algorithm of Xu [26]. 1 Introuction The work in this paper aims for efficient ientification of with-boune separators for a given set of points in the -imensional Eucliean space an their applications to intractable practical problems. Intuitively, a with-boune separator utilizes a simple structure hyper-plane to ivie the set into two balance subsets, while at the same time maintaining a low ensity of the set within a given istance to the hyper-plane. This new notion of separators was initially introuce by Fu in [11], an it was shown that these separators are very suitable in solving a number of istance-boune geometric problems such as the protein foling problem in the H moel in [10] an some other intractable problems in [11, 6]. The main contributions of this paper are summarize as follows: In section 5, we present an O(n 2 log n) sublinear time ranomize algorithm for fining a with-boune separator in the imensioinal Eucliean space R for > 2. To our best knowlege, this is the first sublinear time algorithm for fining geometric separators. For many other geometric problems, a higher This research is supporte by Louisiana Boar of Regents fun uner contract number LEQSF( )-RD-A-35, an in part by NSF Grant CNS

2 2 B. Fu an Z. Chen imension brings higher computational complexity. However, it is interesting to notice that the exponent of our algorithm s computational complexity is reversely propotional to the imension of the space. In section 6, we exhibit an application of our sublinear time separator to the protein sie-chain packing problem. One of the most funamental problems in the molecular biology is to preict a protein s 3D structure when given its 1D amino-aci sequence. Although much effort has been mae for ecaes, this problem remains unsolve. An important component of the general protein structure preiction problem is the protein sie-chain packing problem. It etermines the sie-chain positions onto the fixe backbone [23]. This problem has been prove to be N-complete [1]. Recently, a r O(n 2 3 log n) ave time algorithm was shown by Xu [26], where r ave is the average number of sie-chain rotamers in a protein. We apply with-boune separators to the protein sie-chain packing problem. The length of sie-chain of each amino aci is small compare to the size of one protein. Two sie-chains in a protein molecular o not interact with each other if their istance is slightly larger than the sum of their lengths accoring to moels use in (e.g. [4, 5, 26]). Using our with-boune separators, we obtain an algorithm with computational time r O(n 2 3 ) max, where r max is the maximal number of sie-chain rotamers among a protein. Since the number of rotamers is usually small, we assume both r ave an r max are constants, hence our new algorithm has a better complexity boun. 2 The Relate Work There have been extentive efforts on fining separators ue to their critical roles in many issues of algorithm esign an analysis. Because of space limit we cannot give a comprehensive review of the relate work but list some representative results in this area. Lipton an Tarjan [16] prove that every n vertex planar graph has at most 8n vertices whose removal separates the graph into two isconnecte parts of size at most 2 3 n. Their 2 3-separator has been improve by a series of papers [7, 12, 2, 8] with the best recor 1.97 n by Djijev an Venkatesan [8]. Spielman an Teng [25] showe a 3 4 -separator with size 1.82 n for planar graphs. Separators for more general graphs were erive in [13, 3, 22]. A planar graph can be inuce by a set of non-overlapping iscs on the plane such that every vertex correspons to a isc center an each ege correspons to a tangent relationship between two iscs. The separator evelope by Miller, Teng an Vavasis [17] is a generalization of planar graph separators to the - imensional Eucliean space. Some O( k n) size separators for k-thick systems an the relate algorithms were erive in [18, 19, 17, 24]. The stuy of with-boune separators were initiate by Fu in [11] an has yiele successful applications in [10, 6]. Our with-boune geometric separator has some interesting avantages over previous geometric separators such as the popular geometric separator by Miller, Teng an Vavasis [17]. First, the withboune separator has a simple linear structure as the separator is etermine

3 Sublinear Time With-Boune Separators 3 by a hyper-plane an a with parameter w, but Miller et al. s separator is a sphere, which can be also foun in linear time [9]. The linear structure is very crucial for us in eriving sublinear time algorithm in this paper. Secon. the with-boune separator has a smaller constant in its size upper boun factor than other separators. The constant factor was not clearly given in Miller et al. s separator. Furthermore, their separator only has a balance conition boune n ue to their transformation to a higher imension, while The balance conition of the with-boune separator is boune by +1n. Thir, the withboune separator can be use to eal with an arbitrary set of points via using a set of gri points an weights to characterize the istribution of points from the input set. by Notations, Definitions, an With-Boune Separators For any finite set A, A enotes the number of elements in A. Let R be the set of all real numbers. For two points p 1, p 2 in the -imensional Eucliean space R, ist(p 1, p 2 ) is the Eucliean istance between p 1 an p 2. For a set A R, ist(p 1, A) = min q A ist(p 1, q). The iameter of any R is max p1,p 2 ist(p 1, p 2 ). For a > 0 an a set A of points in R, if the istance between every two points in A is at least a, then A is calle a-separate. For ɛ > 0 an a set Q of points in R, an ɛ-net of Q is another set of points in R such that each point in Q has a istance ɛ to some point in. We say is a net of Q if is an ɛ-net of Q for some constant ɛ > 0 (that oes not necessarily epen on the size of Q). A net set is usually a 1-separate set such as a gri point set. A weight function w : [0, ) is often use to measure the ensity of Q near each point in. Let f : R R be a smooth function. Its surface is the set L(f) = {v R f(v) = 0}. A hyper-plane in R through a fixe point p 0 R is efine by the equation (p p 0 ) v = 0, where v is a normal vector of the plane an. is the usual vector inner prouct. A hyper-plane in R is etermine by L(f) for some linear function f : R R. Definition 1. Given any Q R with a net R, a constant a > 0, an a weight function w : [0, ), an a-wie-separator is etermine by the surface L(f) for some linear function f : R R. The separator has two measurements for its quality of separation: (1) balance(l(f), Q) = max( Q 1, Q 2 ) Q, where Q 1 = {q Q f(q) < 0} an Q 2 = {q Q f(q) > 0}; an (2) ensity(l(f),, a 2, w), where in general ensity(a,, x, w) = p,ist(p,a) x w(p) for any A R an x > 0. When f is fixe or no confusion arises, we use balance(l, Q) an ensity(l,, a 2, w) to stan for balance(l(f), Q) an ensity(l(f),, a 2, w), respectively. Definition 2. A (b, c)-partition of R ivies the space into a isjoint union of regions 1, 2,..., such that each i, calle a regular region, has a volume of b an a iameter c. A (b, c)-regular point set A is a set of points in R with a (b, c)-partition 1, 2,..., such that each i contains at most one point from A.

4 4 B. Fu an Z. Chen For two regions A an B, if A B (A B ), we say B contains (intersects resp.) A. Let B (r, o) be the -imensional ball of raius r at center o. Its volume is V (r) = 2(+1)/2 π ( 1)/2 1 3 ( 2) r 2 if is o, or /2 π /2 2 4 ( 2) r otherwise. Let V (r) = v r, where v is a constant for the fixe imension. In particular, v 1 = 2, v 2 = π an v 3 = 4π 3. We will use the following well-known fact that can be easily erive from Helly Theorem (see [21]). Lemma 1. For an n-element set in the -imensional space R, there is a point q with the property that any half-space that oes not contain q, covers at most n elements of. (Such a point q is calle a centerpoint of.) +1 Definition 3. Let a > 0, p an o be two points in R. Define r (a, p 0, p) to be the probability that the point p has a perpenicular istance to a ranom hyper-plane L through the point p 0. Define function f a,p,o (L) = 1 if p has a istance a to the hyper-plane L through o, or 0 otherwise. The expectation of function f a,p,o (L) is E(f a,p,o (L)) = r (a, o, p). Assume = {p 1, p 2,..., p n } is a set of n points in R an each p i has weight w(p i ) 0. Define function F a,,o (L) = p w(p)f a,p,o(l). We give an upper boun for the expectation E(F a,,o (L)) for F a,,o (L) in the lemma below. Lemma 2. [11] Let 2. Let o be a point in R, a, b, c > 0 be constants an ɛ, δ > 0 be small constants. Assume that 1, 2,..., form a (b, c)-partition for R, an the weights w 1 > > w k > 0 satisfy k max k i=1 {w i} = O(n ɛ ). Let be a set of n weighte (b, c)-regular points in a -imensional plane with w(p) {w 1,..., w k } for each p. Let n j be the number of points p with w(p) = w j for j = 1,..., k. We have E(F a,,o (L)) (k ( 1 b ) 1 + δ) a k j=1 w j n 1 j + O(n 2 +ɛ ), where k = h k 2 = 4 π an k 3 = 3 2 ( 4π 3 ) v 1 with h = 2( 1)v 1 v. In particular, Definition 4. Let a 1,..., a > 0 be positive constants. A (a 1,..., a )-gri regular partition ivies R into a isjoint union of a 1 a rectangular regions. A (a 1,..., a )-gri (regular) point is a corner point of a rectangular region. Uner certain translation an rotation, each (a 1,..., a )-gri regular point is represente as (a 1 t 1,..., a t ) for some integers t 1,..., t. For a point p = (x 1,..., x ) R, if x 1,..., x are all integers, then p is simply calle a gri point (it is a (1,..., 1)-gri regular point). For each point q an a hyperplane L in R, efine s(q, L) to be the signe istance from q to L, which is s(q, L) = (q q 0 ) v L, where q 0 is a point on L, an v L is the normal vector of the plane L with the first nonzero coorinate to be positive. For a hyper-plane L in R, if L is through a point q 0 an has the normal vector v, then it has linear equation (u q 0 ) v = 0. If q R an = s(q, L), then the

5 Sublinear Time With-Boune Separators 5 hyper-plane L through q an parallel to L has equation (u (q 0 + v)) v = 0. We use L() to represent such a hyper-plane L. For an interval I R, I is the length of I. For example, [a, b) = b a. We often use r(e) to represent the probability of an event E. For a real number x, x is the largest integer y x, an x are the least integer z x. For an interval [a, b] R, efine center([a, b]) to be a+b 2. Lemma 3. Let be a finite set of points in R an q 0 be a fixe point in R. Then for a ranom hyper-plane L through q 0, r(s(p 1, L) = s(p 2, L) for p 1, p 2 with p 1 p 2 ) = 0. roof. A ranom hyper-plane L through a fixe point q 0 can be characterize by the equation (q q 0 ) v L = 0, where v L is the normal vector of L. Each unit vector can be consiere as a point of the surface of the unit ball B (1, o), where o = (0,..., 0) is the origin point. The surface area size of B (r, o) is equal to V (r) r = v r 1. The surface area of B (r, o) is of imension 1. For two fixe points p 1 an p 2, if s(p 1, L) = s(p 2, L), then (p 1 q 0 ) v L = (p 2 q 0 ) v L. It implies that (p 1 p 4 ) v L = 0. Consier the sub-area on the surface of B(1, o): {v (p 1 p 2 ) v = 0 an v v = 1}, which is the intersection between a plane (p 1 p 2 ) v = 0 an B (1, o), an is of imension 2. It is easy to see that it has area size 0 in the -imensional space. The lemma follows since the union of a finite number of areas of area size 0 still has 0 area size. 4 An Overview of Our Techniques Given any set Q of points in R with a net, the iea of our techniques for fining an a-with-boun separator is to transform the problem from the - imensional space to the 1-imensional space. By Lemma 1 an Lemma 2, we can see the existence of a hyper-plane that satisfies both the balance an the ensity conitions. Lemma 2 gives an upper boun on the expectation of F a,,o (L). By Markov s inequality, r(f a,,o (L) > (1 + α)e(f a,,o (L))) 1 1+α. Thus, a ranom hyper-plane L has probability α = α 1+α that F a,,o(l) (1 + α)e(f a,,o (L)). The chance is amplifie if we repeat the ranom selection of the hyper-plane L multiple times. Let n = an n Q = Q. After a hyper-plane L is fixe, we try to fin another hyper-plane L that is parallel to L. We want L to guarantee the esire balance an ensity conitions. To o so, we compute signe istances for all the points in Q an to the hyper-plane L. Those signe istances are all ifferent for the points in Q an, respectively, for the points in (by Lemma 3). These signe istances are all in the 1-imensional real axis, an fining L can be one via fining a right position among these istances, hence this transforms the problem from the -imensional space into to the 1-imensional space as follows: Fin the interval [D 1,+1, D,+1 ] such that both the left sie (, D 1,+1 ) an the right sie (D,+1, + ) have roughly n Q +1 signe istances from Q to L. So, every hyper-plane L (parallel to L) with a signe istance in [D 1,+1, D,+1 ]

6 6 B. Fu an Z. Chen to L guarantees the balance conition. For an interval I, we compute its weight as the sum of the weights of the points of with their signe istances in I. We then look for an interval [x a, x + a] that has x [D 1,+1, D,+1 ] an the smallest weight. Finally, we let L be a hyper-plane with a signe istance x to L. The balance bounaries D 1,+1 an D,+1 can be etecte by sampling a small number of points from Q. Using the Chernoff boun, we have a high probability that there is a small fraction ifference from the exact bounaries. Similarly, the esire interval can be also etecte by sampling a small number of points from. 5 The Sublinear Time Ranomize Algorithm We use the following well-known Chernoff boun (see [20] for a proof) an simplie version in Lemma 4. Theorem 1. [20] Let X 1,, X n be n inepenent ranom 0, 1 variables, where X i takes 1 with probability p i. Let X = n i=1 X i, an µ = E[X]. [ Then for any ] µ. δ > 0, (1) r(x < (1 δ)µ) < e 1 2 µδ2 e, an (2) r(x > (1+δ)µ) < δ (1+δ) (1+δ) Lemma 4. Let X 1,, X n be n inepenent ranom 0, 1 variables, where X i takes 1 with probability p. Let X = n i=1 X i. Then for any 1 3 > ɛ > 0, (1) r(x < pn ɛn) < e 1 2 nɛ2, an (2) r(x > pn + ɛn) < e 1 3 nɛ2. Theorem 2. Let 2 be the fixe imension number an v be a positive parameter. Let a, b, c > 0 be constants an δ, s 1, s 2 > 0 be small constants. Let Q be another set of n Q points in R, an be a set of n (b, c)-regular points, which form a net for Q. Let w 1 > w 2 > w k > 0 be positive weights with k w 1 = O(n s 1 ), w 1 w k = o(n 1 k ), w k = O(n s 2 ), an w be a mapping from to {w 1,, w k }. There exists an O(v 2 (n 2 +2(s 1+s 2 ) log n + log n Q )) time ranomize algorithm to fin a hyper plane M with probability such that (1) each half v space has ( +1 + δ)n Q points from Q, an (2) p an ist(p,m) a ( ) w(p) k b 1 + δ a k j=1 w jn 1 j + O(n 2 +s 1 ) for all large n, where n j 1 is the number of points p with w(p) = w j (j = 1,, k). roof. We use two phases to fin the separator hyper-plane. The first phase etermines the orientation of the hyper-plane by selecting a ranom hyper-plane, an fins the region of the separator hyper-plane for a balance partition. The secon phase fins the position of the separator plane with a small sum of weights for the points of the set close to it. Without loss of generality, we assume that 0 < δ < 1. Since n j 1(j = 1,, k), we have k n. Let b = i=1 a i. Select constant c 0 > 0 an let δ 1 = c 0 δ so that (k b 1 +3δ 1 )(1+δ 1 ) 2 (k b 1 + δ 2 ). Let a 1 = a(1 + δ 1 ) an α = δ 1. Let c 1 be a constant such that k w 1 c 1 n s1 an k w k c 1 n s 2. (1)

7 Sublinear Time With-Boune Separators 7 Let o be the center point from Lemma 1 (our algorithm oes not nee to fin such a center point o, but will use its existence). By Lemma 2, E(F a1,,o) (k b 1 +δ 1 ) a 1 k j=1 w jn 1 j +O(n 2 +s 1 ). By Markov inequality, r(f a1,,o(l) (1 + α)e(f a1,,o)) 1 1+α. This tells us that a ranom hyper-plane L has the probability at least α such that there exists a separator hyper-plane L (it may be through o) that satisfies the conitions of the theorem an is parallel to L. We assign the values to some parameters: r = c 4 v, where c 4 is a constant to be fixe later (2) δ 2 = δ 1 a c 1 (3) ɛ = δ 2 3c 1 n 1 +s1+s2 ɛ 0 = δ 6 (5) ɛ 1 = 5ɛ 0 (6) m 1 = 3(ln r + log n Q) ɛ 2 0 (7) m 2 = (ln log n + r) ɛ 2 (8) hase 1 of the algorithm: The input of our algorithm is, Q, n Q = Q, an n =. Each input point p has the format < (x 1,, x ), w(p) >, where p = (x 1,, x ) an w(p) is the weight of p. The algorithm starts with the following steps: Select a fixe point o R an a ranom plane L through (ranom hyper-plane can be selecte via selecting a ranom normal vector). Select m 1 ranom points q 1,, q m1 from Q an let Q =< q 1,, q m1 > represent the list of these points (one point may appear multiple times). For each q j Q, compute its signe istance qi = s(q i, L) to L. Fin the ( 1 +1 ɛ 1)m 1 -th least point D1,+1 = s(q 1, L) for q1,, qm1. Fin the o ( +1 + ɛ 1)m 1 -th least point D,+1 = s(q 2, L) for q1,, qm1. Select m 2 ranom points p 1,, p m2 from an let =< p 1,, p m2 > represent the list of these points. For each p i, compute pi = s(p i, L). It is wellknown that fining the i-th element from a list takes linear steps. The computation above takes (m 1 + m 2 ) steps. In the rest of the algorithm, we locate the position of the separator hyper-plane by fining its signe istance to L. Its position will be at the center of an interval of size 2a. In the rest of the proof, we treat both an Q as lists of points from R. Each point appears only at most once on both an Q. Let t = k b 1 + δ. For q R an A R, efine r(a, L, q) = {q q A an s(q,l) s(q,l)} A. For a list of points B =< x 1,, x m > from R an a point q R, efine X B,L,q (i) = 1 if s(q i, L) s(q, L), or 0 otherwise. We also efine Y (B, L, q) = m i=1 X B,L,q(i). (4)

8 8 B. Fu an Z. Chen Lemma 5. It has probability 1 e r 50 such that r(q, L, q1) 1 [ +1 1 δ, +1 δ 6 ] an r(q, L, q 2) [ +1 + δ 6, +1 + δ]. hase 2 of the algorithm: In this phase, we will fin a position of L (parallel to L) with the signe istance to L in the range [D1,+1, D,+1 ]. Lemma 5 guarantees (with high probability) that each position in the interval [D1,+1, D,+1 ] gives a balance partition. We look for the position that has the small sum of weights for the points of close to L. For a list A =< x 1,, x m >, A = m is enote to be the length of A an x A means that x is one of the elements in A (x = x i for some 1 i m). For a real subset J R an a list A of finite points in R, efine r (A, L, J, w j ) = { p p A an w(p) = w j an s(p, L) J}, A an Z(A, L, J, w j ) = p A X L,p,J,w j, where XL,p,J,w j = 1 if s(p, L) J an w(p) = w j, or 0 otherwise. We also efine W (A, L, J) = p A an s(p,l) J w(p). By the efinitions, It is easy to see that W (A, L, J) = k w j Z(A, L, J, w j ) = j=1 k w j r (A, L, J, w j ) A. (9) Since k j=1 n j = n, we have that n j n k for some 1 j k. By (1), we have j=1 that n s 2 k c 1 w k for some 1 j k. By (3), for some 1 j k, k 1 c 1 w j. This implies that n 1 δ 2 n 1 s2 s2 c 1 w j ( n k ) 1 c 1 w j n 1 j δ 1 a w j n 1 j. (10) Lemma 6. Let f n be an integer an H 1, H 2,, H f R be f real intervals. It has probability e r such that W (, L, H i ) [W (, L, H i ) n m 2 δ 2 n 1 s 2, W (, L, H i ) n m 2 + δ 2 n 1 s 2 ]) for i f an j k. Case 1. D1,+1 D,+1 3an 2. artition [D1,+1, D,+1 ] into isjoint intervals [l 1, l 2 ), [l 2, l 3 ),, [l u 1, l u ), [l u, l u+1 ] such that each l i+1 l i (i = 1,, u) is equal to D 1,+1 D,+1 g 1(n ) 3a, where g 1 (n ) = u = n 2. Let J i = [l i, l i+1 ) if i < u, an J u = [l u, l u+1 ]. Compute W (, L, J i ) for i = 1,, u, which takes O(m 2 + g 1 (n )) = O(m 2 ) steps. The algorithm selects J = J i0 that has the least W (, L, J i0 ) an let L = L(center(J i0 )), which takes O(g 1 (n )) = O(m 2 ) steps. Assume that J i1 is the interval with the least W (, L, J i1 ). Lemma 7. It has probability e r such that W (, L, J i0 ) a k j=1 w j n 1 j. ( ) k b 1 + δ

9 Sublinear Time With-Boune Separators 9 Case 2. D1,+1 D,+1 < 3an 2. Let J be interval such that center(j ) [D1,+1, D,+1 ] an J = 2a 1 = 2a(1 + δ 1 ) an W (, L, J ) is the least. Subcase 2.1. D1,+1 D,+1 δ 1a. Let J = [D1,+1 a, D 1,+1 + a] an let L = L(D1,+1 ) (In other wors, L = L(center(J))). Clearly, J J an W (, L, J) W (, L, J ). Subcase 2.2. δ 1 a < D1,+1 D,+1 < 3an. Let g 2 (n ) be the least integer v 2 such that D,+1 D 1,+1 +2a v δ 1a 3. Since v 2 an D,+1 D 1,+1 +2a v 1 > δ 1a 3, we have D,+1 D 1,+1 +2a v = v 1 D,+1 D 1,+1 +2a v v 1 > v 1 δ 1a v 3 δ1a 6. Therefore, v D,+1 D 1,+1 +2a δ 1 a D,+1 D 1,+1 +2a g 2 (n ) 6 3an 2 +2a δ 1 a 6 = 6(3n 2 +2) δ 1 = O(n 2 ). Let s = [ δ 1a 6, δ 1a 3 ]. artition [D 1,+1 a, D,+1 + a] into the union of g 2 (n ) isjoint intervals of size s: [r 1, r 2 ) [r 2, r 3 ) [r v 1, r v ) [r v, r v+1 ], where v = g 2 (n ) an r i+1 = r i + s for i = 1,, v. Let I i = [r i, r i+1 ) for i = 1,, v 1 an I v = [r v, r v+1 ]. Let Ji = I i I i+1 I i+h 1 for i = 1,..., v h + 1, where h is an integer with 2a < h s < 2a + 2s. The algorithm selects the interval J = Ji 2 that has the least W (, L, Ji 2 ). Finally, the algorithm outputs L = L(center(J)) for the separator hyper-plane. We analyze the algorithm for the case 2. Lemma 8. Assume that J is the interval output from the case 2 (either subcase 2.1 or subcase 2.2). It has probability e r such that W (, L, J) W (, L, J ) + 2δ 1 aw j n 1 j for some j k. For a list A of finite points in R an a hyper-plane M 1, efine F 1 (M 1, a, A) = p i A an ist(p i,m 1 ) a w(p i). If M 1 an M 2 are two hyper-planes with signe istance M1,M 2 = s(p, M 1 ) for some point p in the M 2, then F 1 (M 2, a, A) = W (A, M 1, J), where J is the interval [ M1,M 2, a, M1,M 2 + a]. The the hyperplane L(center(Ji 2 )) output by the algorithm has that F 1 (L(center(Ji 2 )), a, ) F 1 (L(center(J )), a 1, ) + 2δ 1 aw j n 1 j for some j k. See the section?? for the algorithm escription in the Appenix. Time an accuracy of the algorithm: After the hyper-plane L is selecte in phase one, by Lemma 5 we have the probability at least 1 e r that both r(q, L, q1) 1 [ +1 δ, 1 +1 δ 6 ] an r(q, L, q 2) [ +1 + δ 6, +1 + δ]. This means every L (parallel to L) with the signe istance in the interval [D1,+1, D,+1 ], it has at most ( +1 + δ)n Q points of Q in each of the half spaces. In phase 2, we have probability ( at ) least 1 e r to output the separator L such that F 1 (L, a, ) k b 1 + δ a k j=1 w j n 1 j (case 1 of phase 1, see Lemma 7) or F 1 (L, a, )) F 1 (L(J ), a 1, ) + 2δ 2 w j n 1 j (case 2 of phase 2, see Lemma 8), where J is the interval of length 2a 1 with the least F 1 (L(J ), a 1, ) an center between D1,+1 an D,+1. Assume that L is a fixe hyper-plane an L is a another hyper-plane that is parallel to L an F 1 (L, a 1, ) is the least. By Lemma 7 an Lemma 8, it has probability (1 e r ) 2 such that we can get another L (parallel to L) such

10 10 B. Fu an Z. Chen that F 1 (L, a, ) F 1 (L, a 1, ) + 2δ 1 w j n 1 j for some j k or F 1 (L, a, ) ( ) k b 1 + δ a k j=1 w j n 1 j. The number of points in Q in each sie of L is ( +1 + δ)n Q. 1 We have probability at most 1+α that F a 1,,o(L) (1 + α)e(f a1,,o). If the algorithm repeats z times, let L 1,, L z be the ranom hyper planes selecte for L. With probability (1 ( 1 1+α )z ), one of those L i s has another hyper-plane L i such that L i is parallel to L i an has F a1,,o(l i ) (1 + α)e(f a 1,,o). Therefore, we have probability at least (1 ( 1 α+1 )z )(1 e r ) 2z to fin out such a L with F 1 (L, a, ) (1 + α)e(f a1,,o) + 2δ 1 w j n 1 j for some j ( ) k or F 1 (L, a, ) k b 1 + δ a k j=1 w j n 1 j. Thus, F 1 (L, a, ) ( ) k b 1 + δ a k j=1 w jn 1 j + O(n 2 +s 1 ). Now we give a boun for the probability. Let z = 2r 1 ln(1+α). Then 1 ( 1+α )z > 1 e r. 1 (1 ( 1 + α )z )(1 e r ) 2z > (1 e r ) 2z+1 > 1 (2z + 1)e r > v, where we let r = c 4 v for some constant c 4 large enough. The phase 1 of the algorithm takes O(m 1 + m 2 ) steps. The case 1 of phase 2 takes O(m 2 ) steps. The case 2 of phase 2 takes O(m 2 ) steps. Totally, it takes O(z(m 1 +m 2 )) = O(v (n 2 +2(s1+s2) (log n +v)+v log n Q )) = O(v 2 (n 2 +2(s1+s2) log n + log n Q )) steps. Corollary 1. Let 2 be the imension number an the parameter v > 0. Let a > 0 be a constant an δ > 0 be a small constant. There exists a ranomize O(v 2 n 2 log n) time such that given a set Q of n gri points in R, the algorithm fins a hyper-plane L with probability at least such that each sie of L v has at most ( +1 + δ)n points of Q, an the number of points of Q with istance a to L is (k + δ)an 1. 6 An Application to rotein Sie-Chain acking roblem We follow the escription of Xu [26] for the moel of protein sie chain packing. The sie-chain preiction problem can be formulate as follows. We use a resie interaction graph G = (V, E) to represent a protein resies an their interactions. Each vertex in V represents a resiue of the protein. For each resie i V, D(i) is the set of all possible rotamers of sie chain i. There is an interaction ege (i, j) E if an only if there are l D(i) an k D(j) such that there exist an atom in the rotamer l conflicts with another atom in the rotamer k. Two atoms conflict each other iff their istance is less than the sum of their raii. For each two rotamers l D(i) an k D(j) (i j), there is an associate

11 Sublinear Time With-Boune Separators 11 score i,j (l, k) if resiue i interacts with resiue j. For each rotamer l D(i), there is a score S i (l), which characterizes the interaction energy between l an the backbone of the protein. The preiction problem is to give A(i) D(i) to resiues i V so that the following energy value is minimize. E(G) = i V S i(a(i)) + i j,(i,j) E i,j(a(i), A(j)). For more etaile escription about the protein sie chain packing, see (e.g. [23, 4, 26, 5]). Let u be istance such that there is no interaction between two resies if their istance is u. Let l be the minimal istance between two amino acis. Both u an l are constants. Theorem 3. There exists a r O(n 3 2 ) max -time algorithm to fin the optimal solution for the protein sie chain packing problem, where r max is the maximal number of rotamers of one amino aci. In other wors, r max = max i D(i). roof. Our algorithm is base on the ivie an conquer metho. Let 0 = l be the unit istance. Since l = 2 0, we consier that the minimal istance between two amino acis is l = 2 an the minimal istance for the interaction between two sie chains is u = u 0. For a gri point p = (x, y, z) (x, y, z are integers), efine cube(p) = {(u, v, w) R 3 x 1 2 u < x an y 1 2 v < y an z 1 2 w < z }. The 3D space R3 is partitione into many cubes: R 3 = cube(p 0 ) cube(p 1 ). For ifferent gri points p p, cube(p) cube(p ) =. Each amino aci is represente by the position of its C α. Therefore, no two amino acis can stay at the same cube(p) for any gri point p. Let be the set of all gri points p such that cube(p) contains the C α for an amino aci. Let w = u By Corollary 1, there exists a w-wie separator L plane such that each sie has at most ( δ)n contain amino aci, an the number of gri points (with amino acis in its cube) is boune by 1.209wn 2 3, where δ > 0 is an arbitrary small constant. The w-wie separator partitions the problem into 1, S an 2, where S is the separator area. Clearly, a sie chain whose amino aci C α is in cube(p) with p 1 oes not interact another sie chain in 2 because of the w-wie separator between 1 an 2. The number of ways to arrange the sie chains in the separator area S is boune by r 1.209wn 2 3 max. We only nee O(n) time for computing the separator. We assume that r max 2 (otherwise, it is trivial). Let T (n) is the computational time for the protein sie chain packing problem with n resies. Solving each sub-problem i (i = 1, 2) takes T (( δ)n) steps. We have the recursive T (n) 2(r 1.209wn 3 2 max + O(n))T (( δ)n). This gives that T (n) = ro(n 3 ) max. References 1. T. Akutsu, N-harness results for protein sie-chain packing, In S. Miyano an T. Takagi, eitors, Genome Informatics 8, 1997, pp N. Alon,. Seymour, an R. Thomas, lanar Separator, SIAM J. Discr. Math. 7,2(1990)

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