Distributed Decomposition Over Hyperspherical Domains

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1 Distribute Decomposition Over Hyperspherical Domains Aron Ahmaia 1, Davi Keyes 1, Davi Melville 2, Alan Rosenbluth 2, Kehan Tian 2 1 Department of Applie Physics an Applie Mathematics, Columbia University, New York, NY 10027, USA 2 IBM T.J. Watson Research Center, Yorktown Heights, New York 10598, USA Summary. We are motivate by an optimization problem arising in computational scaling for optical lithography that reuces to fining the point of minimum raius that lies outsie of the union of a set of iamons centere at the origin of Eucliean space of arbitrary imension. A ecomposition of the feasible region into convex regions suggests a heuristic sampling approach to fining the global minimum. We escribe a technique for ecomposing the surface of a hypersphere of arbitrary imension, both exactly an approximately, into a specific number of regions of equal area an small iameter. The ecomposition generalizes to any problem pose on a spherical omain where regularity of the ecomposition is an important concern. We specifically consier a storage-optimize ecomposition an analyze its performance. We also show how the ecomposition can parallelize the sampling process by assigning each processor a subset of points on the hypersphere to sample. Finally, we escribe a freely available C++ software package that implements the storage-optimize ecomposition. 1 Global Optimization for Semiconuctor Lithography Mask Design In the newly herale fiel of computational scaling [7], inustrial scientists are now investigating a global minimization formulation of the problem of mask esign for optimizing process winows in semiconuctor lithography [5, 6]. We are consiering the problem of forming an optimal mask esign for the optical printing of a given 2-D target image, which is consiere as a set of sample target points that must be sufficiently illuminate for the image to correctly print. We attempt to minimize the total intensity of a set of exposure moes, with each axis x i corresponing to an intensity for moe i. In the space of the exposure moes, each of the n samples of image features are represente as a -imensional iamon of infeasible space. Each sample image feature is sufficiently illuminate by the set of exposure moes if their representative coorinate x R lies outsie the iamon. Sufficient illumination of all sample target features in an exposure is achieve in all points outsie of the union of the set of iamons. Although a global minimum is ieal, the goal of the problem is to fin goo solutions in a reasonable amount of time. Aitionally, global

2 252 A.J. Ahmaia, D.E. Keyes, D.O. Melville, A.E. Rosenbluth, K. Tian minima that satisfy all the constraints may still be rejecte ue to manufacturability consierations, so a goo solution metho will provie the global minima an may provie a set of the best available local minima within each orthant of the search space. 1.1 Convex Partitions of the Feasible Domain The semiconuctor lithography problem can be consiere as the search for the global minima of a linear optimization problem subject to nonconvex constraints: minimize x 1 subject to A i (x) x b i i = 1...n (1) We efine the j th normalize principal axis of iamon A i as l i, j c i, j, with c i, j representing the irection an l i, j the magnitue of the principal axis. We then note that each of the 2 planes efining a half-space exterior of a iamon connects the principal axes of the iamon. A i (x) can be consiere as a function of the choices of signs for the vectors representing the principal axes of the iamon, the choice of connection to the positive or negative en of each principal axis uniquely etermines one of the planes. Since the iamon is an intersection of half-spaces, a point x is consiere feasible with respect to the iamon if it lies insie any of the reflecte (pointing outwars from the origin) half-spaces A + i,k arising from the hyperplanes A i,k. Each combination of the positive an negative principal c j axes in a given iamon A i is a set of points that uniquely etermine a constraint half-space A + i,k. For any given k an iamon A i: s i,k ( j) c i, j A i,k j = {1...} (2) Since all of the constraint planes are linear an o not contain the origin: In particular: if x 1,x 2,...,x A k, an θ R, θ j 0, (θ 1 x 1 + θ 2 x 2 + +θ x ) A + k θ R,θ j 0, j j θ j 1 θ j 1 (3) (θ 1 s k (1)c 1 + θ 2 s k (2)c 2 + +θ s k ()c ) A + k (4) Because the set of principal axes l j c j for each iamon forms an orthogonal basis on R, we can express: x = j l j θ j c j, θ j = c j x (5)

3 Distribute Decomposition Over Hyperspherical Domains 253 Any combination of positive an negative θ i, j can be converte to their absolute values with an appropriate choice of k, an we can say that x lies outsie iamon i (by satisfying constraint half-space i,k) if: θ i, j 1 (6) j=1 l i, j We may enumerate a single plane A i,k with a tuple, or orere list of signs s i,k, with the sign of the jth element s i,k ( j) corresponing to the ens of the principal axes c i,j it connects. We represent the concatenation of all -tuples s i,k corresponing to a choice of plane for each of the n iamons into a single n l -tuple: s, with n l = n 2. Similarly, we concatenate the list of all principal axes into a list {c i }, with i = 1 n l. We efine a set of nonoverlapping regions R s that collectively exhaust R, with a total of 2 n l potential tuples s an corresponing regions R s. R s : {x R sign(c i x) = s i, i = 1,...,n l } (7) Each s efines a convex region containing R s : s i, j (c i,j x) 1 i (8) j=1 l i, j If we enumerate the set of convex regions R s, the global minima is the best local minima from each of the regions. 2 Partitions of n-space We expect the true number of convex regions p in the space to be less than 2 n l, as some intersections of iamon half-spaces will be empty. We seek an upper boun on p as a function of, the imensionality of the problem, an n, the number of iamons. We consier each of the iamons as a set of cutting planes, with c i,j represente as a cutting plane that intersects the origin. Upper bouns for the number of regions generate by origin-centere cutting planes in general position were establishe inepenently by [1], Perkins, Willis, an Whitmore (unpublishe), an [8]. Partitions of N-Space by Hyperplanes, [9] allows tighter bouns base on the egeneracy of the planes. We consier the iamons to be non-egenerate, so the upper bouns for cutting planes in general position are sufficient: p B n = 2 1 i=0 ( ) n 1 i (9) 3 Incomplete Search Heuristics When applie to the semiconuctor lithography problem, the coorinate axes represent linear combinations of the unerlying physical variables that are chosen for their

4 254 A.J. Ahmaia, D.E. Keyes, D.O. Melville, A.E. Rosenbluth, K. Tian efficient average coupling to the features represente by the constraint iamons. This selection process causes the moes to be strongly couple to iniviual constraint features, which in turn causes the axes of the iamons to be preferentially aligne with the coorinate axes. As a result the principal axes of the iamons have a tenency to cluster together irectionally, making the size istribution of the regions between intersecting ellipsois very non-uniform, with the largest an eepest regions comprising a small fraction of the total. Empirically it is foun that solutions which are aequately close to the global optimum can be foun by searching only the largest regions. In [5], the authors suggest that irectly sampling points on a hypersphere is a useful heuristic. Fig. 1. Sampling reuces the number of regions to search by skipping small-angle regions, unsearche regions are arkly shae. The number of sample regions n s using a ecomposition of the ( 1)-sphere embee in R can be consiere as a function of the search ensity per imension ρ: n s (ρ) = ρ. (10) The number of sample points is much more managable than the actual number of potential convex regions. We accept the currently unquantifie risk of missing the global minimum in exchange for a more computationally tractable approximation to the problem. 4 Hypersphere Decomposition 4.1 Previous Work We are now face with the task of partitioning a ( 1)-sphere into regions of approximately equal area an small iameter. Fortunately, this topic has been wellstuie in [3], base on a construction for 2-spheres introuce earlier in Zhou s

5 Distribute Decomposition Over Hyperspherical Domains PhD Thesis as well as unpublishe work by E.B. Saff an I.H. Sloan. A omain ecomposition metho for particle methos on the 2-sphere was also introuce in [2]. A full introuction to the general algorithm for -spheres into n s partitions is impossible here, but we generalize the algorithm as a partitioning of m-spheres into regions which are then recursively partitione as (m 1)-spheres. 4.2 A Memory-Efficient Tree Storage Scheme for Equal Area Hypersphere Regions We are motivate by the large value of n s to seek a compresse storage partition of n s points. At each level j of the ecomposition, starting at j =, an ening at j = 1, we have some number of j-spheres embee into ( j+1)-space. We enote the number of j-spheres in the ecomposition at level j by k j. We propose using the proceure s recursive tree as a storage scheme for the points, avoiing the costs of storing full coorinate information for each point. This iea was originally propose in [4] in section in 2.5 for spherical coing, though its potential for compresse storage was not fully explore. We are intereste in forming a loose boun on the total number of noes require for the storage of the tree. Since the number of noes on the tree at level j is base on the number of spheres at level k j, we seek an upper boun on k j. At each level j, we ecompose each j-sphere into some finite number of ( j 1)-spheres an 2 polar caps. The ( j 1)-spheres correspon to collars of the j- spheres, an are assigne a number of regions proportional to their fractional area of the sphere. We impose an inexing on all the spheres for a given level j, such that if there are k j spheres at level j, then the spheres are inexe from i = 1,...,k j, an let k i, j equal the number of ( j 1)-spheres to ecompose the i, j-th sphere into. Finally, we also affix z i, j to every sphere in the system, enoting the number of regions containe by sphere i, j. If we fix = 1 an ecompose the -sphere, we have k = 1. We can now recursively buil an estimate for k l 1 from k l for l = 0,..., 1. We claim that k l = O(n l/ ). This is true for the base case l = 0, an is true for all l if we can show that k l 1 = O(n l+1 ). We use the fact that for a -sphere being ecompose into s regions, the number k of ( 1)-spheres/collars it contains is: k = π 2θ,n σ(s ) s Where θ,n is the polar cap angle for a -sphere ecompose into n s regions an σ(s ) is the measure of surface area of a -sphere. At each level j, we can account for all regions containe by summing over the assigne regions for each j-sphere an the polar caps for all m-spheres, where m > j: k j i=1 z i, j m= j+1 (11) k m = n (12)

6 256 A.J. Ahmaia, D.E. Keyes, D.O. Melville, A.E. Rosenbluth, K. Tian 10 7 Number of noes in compresse storage tree Actual number of noes Estimate upper boun for number of noes n = 10 n = 5 n = Number of regions Fig. 2. Actual vs. Estimate Number of Noes for = 2,5,10 an between 1000 an 10,000,000 sample points Trivially: k j i=1 z i, j n. We now make the observation that the solution to max k i=1 (p i) 1/ j, subject to (p i ) n, is equal to k( n k )1/ j, an substitute: k l 1 = O(n j/ ) l 1 l n 1 l (13) = O(n l l2 l+ ( l) ) (14) = O(n l+1 ) (15) We substitute in equation (12) to obtain an estimate for the upper boun of the total noes of the storage tree. We assume a constant C = 1, an: K = j= j= 1 k j = 2 j=2 j=2 s j/ + s 1 (16) The estimates were compute for = 2,5,10 an n s = 10 3 to n s = 10 7, then compare against actual tree structures in Fig Parallel Decomposition Finally, we introuce an algorithm for two-level ecomposition suitable for massively parallel istribute sampling of the n-sphere. This algorithm successfully istribute a parallel search over 2,048 Blue Gene noes. 1. Apply Leopari s algorithm to generate a ecomposition along some subspace of the original space.

7 Distribute Decomposition Over Hyperspherical Domains 257 Fig. 3. Two-level istribute ecomposition of a 2-sphere 2. For each ecompose region, apply Leopari s algorithm again to sample points, along ecompose imensions, the algorithm operates on region bounaries establishe in the first ecomposition. 5 Ongoing Work We wish to consier compressions of the tree structure by enforcing symmetry in the hypersphere ecomposition. We are also intereste in improving the performance of the tree structure coe. A C++ implementation of the tree structure sampling coe is available from Acknowlegement. The authors gratefully acknowlege Braxton Osting an Matias Coururier for their insightful contributions to several sections of this paper. References [1] Cameron, S.H.: An estimate of the complexity requisite in a universal ecision network. Bionics Symposium, WADD Report, pages , [2] Egecioglu, Ö., Srinivasan, A.: Domain Decomposition for Particle Methos on the Sphere. In Proceeings of the Thir International Workshop on Parallel Algorithms for Irregularly Structure Problems, pages Lecture Notes in Computer Science, 117. Springer, Lonon, [3] Leopari, P.: A partition of the unit sphere into regions of equal area an small iameter. Electron. Trans. Numer. Anal., 25: , 2006.

8 258 A.J. Ahmaia, D.E. Keyes, D.O. Melville, A.E. Rosenbluth, K. Tian [4] Leopari, P.: Distributing points on the sphere: Partitions, separation, quarature an energy. PhD thesis, University of New South Wales, [5] Rosenbluth, A.E., Bukofsky, S., Fonseca, C., Hibbs, M., Lai, K., Molless, A.F., Singh, R.N., Wong, A.K.K.: Optimum mask an source patterns to print a given shape. J. Microlithogr. Microfabrication, Microsyst., 1:13-30, [6] Rosenbluth, A.E., Melville, D., Tian, K., Lai, K., Seong, N., Pfeiffer, D., Colburn, M.: Global optimization of masks, incluing film stack esign to restore TM contrast in high NA TCC s. Proc. SPIE, 6520:65200P, [7] Singh, V.: Computational lithography: the new enabler of Moore s Law. Proc. SPIE, Q, [8] Winer, R.O.:. Single stage threshol logic. Switching Circuit Theory an Logical Design, AIEE Special Publ. S-134, pages , [9] Winer, R.O.: Partitions of N-Space by Hyperplanes. SIAM J. Appl. Math., , 1966.