1. October, 1998DARPA / Melamed / Singh1 Parallelization of Search Algorithms for Modeling QTES Processes Joshua Kramer and Santokh Singh Rutgers University Faculty of Management Dept. of MSIS 94 Rockafeller Rd. Piscataway, NJ 08854 Benjamin Melamed Rutgers University Faculty of Management Dept. of MSIS 94 Rockafeller Rd. Piscataway, NJ 08854 DARPA/ITO BAA 97-04 AON F316

2. October, 1998DARPA / Melamed / Singh2 Minimize over the space of probability vectors where : the maximal autocorrelation lag : the fixed weighting coefficients : QTES model autocorrelation function : empirical autocorrelation function. MODELING OPTIMIZATION PROBLEM

3. October, 1998DARPA / Melamed / Singh3 Parallelization of computations by partitioning the vector space into subspces Assume m processors with a designated processor to initiate the computation distribute the task of minimization to the m processors compute in parallel the minimization within each subspace combine the results by the designated processor COMPUTATIONAL SPEEDUP

4. October, 1998DARPA / Melamed / Singh4 ALGORITHM PARALLELIZATION H1H1 H2H2 H3H3 H4H4 SPACE H argmin f (p) p є H Processor 4Processor 3Processor1 argmin f (p) p є H 4 argmin f (p) p є H 3 argmin f (p) p є H 1 argmin f (p) p є H 2 Processor2

5. October, 1998DARPA / Melamed / Singh5 MATHEMATICAL PROBLEM FORMULATION objective function : quantization factor : non-negative integer in position i quantized probability vectors where : quantization level

6. October, 1998DARPA / Melamed / Singh6 Partition the quantized vector space of cardinality into subspaces, such that cardinality of is Two partitioning methods interleaving method segmentation method WORK PARTITIONING METHODS for all

7. October, 1998DARPA / Melamed / Singh7 Facts it is known how to enumerate recursively all vectors in it is, however, hard to map an index i to the associated vector Solution ( interleaving ) let be the index set of define example : m = 2 Features very simple to implement by skipping skipping is wasted work that slows down the algorithm OVERVIEW OF INTERLEAVING METHOD

8. October, 1998DARPA / Melamed / Singh8 Notify each processor j of the starting vector for its subspace OVERVIEW OF SEGMENTATION METHOD H1H1 H2H2 HmHm A mapping connects each vector index to the corresponding vector for enumeration and computation of optimal vectors This converts operations in the vector domain to operations in the integer domain can easily find the indices of first and last vectors in each subspace can readily enumerate vectors via their indices

9. October, 1998DARPA / Melamed / Singh9 Definition a circulant matrix is one composed of vectors in which each row is obtained by circular right shift of the previous row Properties - Columns can also be formed by circular shifts - every row can be generated from the first row or first column Examples 0 1 0 0 0 3 2 0 0 0 0 0 1 0 0 0 3 2 0 0 basic circulant = 0 0 0 1 0, 0 0 3 2 0 0 0 0 0 1 0 0 0 3 2 1 0 0 0 0 2 0 0 0 3 DIGRESSION: CIRCULANT MATRICES

10. October, 1998DARPA / Melamed / Singh10 Definition a similarity class is a set of circulant matrices, which have the same elements, irrespective of their positions Example: n = 5, k = 5 3 2 0 0 0 3 0 2 0 0 3 0 0 2 0 3 0 0 0 2 0 3 2 0 0 0 3 0 2 0 0 3 0 0 2 2 3 0 0 0 S 1 = 0 0 3 2 0 0 0 3 0 2 2 0 3 0 0 0 2 3 0 0 0 0 0 3 2 2 0 0 3 0 0 2 0 3 0 0 0 2 3 0 2 0 0 0 3 0 2 0 0 3 0 0 2 0 3 0 0 0 2 3 3 1 1 0 0 3 0 1 1 0 3 0 0 1 1 3 0 1 0 1 3 1 0 1 0 3 1 0 0 1 0 3 1 1 0 0 3 0 1 1 1 3 0 0 1 1 3 0 1 0 0 3 1 0 1 1 3 1 0 0 S 2 = 0 0 3 1 1 1 0 3 0 1 1 1 3 0 0 0 1 3 0 1 1 0 3 1 0 0 1 3 1 0 1 0 0 3 1 1 1 0 3 0 0 1 1 3 0 1 0 1 3 0 0 1 0 3 1 0 0 1 3 1 1 1 0 0 3 0 1 1 0 3 0 0 1 1 3 0 1 0 1 3 1 0 1 0 3 1 0 0 1 3 DIGRESSION: SIMILARITY CLASSES

11. October, 1998DARPA / Melamed / Singh11 SEGMENTATION ALGORITHM mapping vector index i similarity class S t circulant C s assignment of vectors v i to subspace H j assignment enumeration enumeration of vector v i associated with circulant C s circulant C s

12. October, 1998DARPA / Melamed / Singh12 Number of vectors in similarity class is calculated via the multinomial formula MAPPING INDICES TO CIRCULANTS where : multiplicity of the element j in vector : number of distinct elements in vector Example: for n = 5, k = 5,

13. October, 1998DARPA / Melamed / Singh13 Vector is identified as belonging to circulant matrix ( ) MAPPING INDICES TO CIRCULANTS (Cont.) Vector is classified into similarity class if by the mapping

14. October, 1998DARPA / Melamed / Singh14 Let be a decomposition vector at quantization level k, such that Example: k = 5 d 0 = { 5 }; d 1 = { 4,1 }; d 2 = { 3, 2 }; d 3 = { 3, 1, 1 }; d 4 = { 2, 2, 1 }; d 5 = { 2, 1, 1, 1 }; d 6 = { 1, 1, 1, 1 }; VECTOR ENUMERATION WITHIN CIRCULANTS

15. October, 1998DARPA / Melamed / Singh15 The j -th element of the vector is defined by : number of elements in the t -th decomposition vector VECTOR ENUMERATION WITHIN CIRCULANTS : sum of the nonzero off-diagonal elements : = 1, if i = j ; = 0, otherwise ( Kronecker’s delta ) : u -th element of the t -th decomposition vector : (i,j) -th element of the basic circulant matrix

16. October, 1998DARPA / Melamed / Singh16 SUMMARY OF SEGMENTATION METHOD Decomposition for parameters n and k Construction of circulants C s Construction of Similarity Classes S t Formation of subspaces H j H1 H1 H2H2 H3H3 H4H4 S1S1 S2S2 S3S3 S4S4 S5S5 S6S6 S7S7

17. October, 1998DARPA / Melamed / Singh17 FEATURES OF SEGMENTATION ALGORITHM Versatile arithmetic on enumeration of vector indices i permits fast enumeration of vectors in each subspace may also be effectively used as an alternative interleaving method Fast algorithm makes it possible to quickly enumerate the first and the subsequent vectors belonging to each subspace, directly from vector indices Space saving since a vector belonging to a subspace can be enumerated as and when needed, no storage is required