Parallelization of Search Algorithms for Modeling QTES Processes

1. DARPA/ITO BAA 97-04 AON F316 Parallelization of Search Algorithms forModeling QTES Processes Benjamin Melamed Rutgers UniversityFaculty of Management Dept. of MSIS 94 Rockafeller Rd. Piscataway, NJ 08854 Joshua Kramer and Santokh Singh Rutgers UniversityFaculty of Management Dept. of MSIS 94 Rockafeller Rd. Piscataway, NJ 08854 DARPA / Melamed / Singh

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

3. COMPUTATIONAL SPEEDUP • Parallelizationof computations bypartitioningthe 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 DARPA / Melamed / Singh

4. ALGORITHM PARALLELIZATION SPACE H H2 H1 argmin f (p) p є H H3 H4 Processor1 Processor2 Processor 3 Processor 4 argmin f (p) p є H2 argmin f (p) p є H3 argmin f (p) p є H4 argmin f (p) p є H1 DARPA / Melamed / Singh

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

6. for all WORK PARTITIONING METHODS • Partition the quantized vector space of cardinality into • subspaces , such that • cardinality of is • Two partitioning methods • interleavingmethod • segmentationmethod DARPA / Melamed / Singh

7. OVERVIEW OF INTERLEAVING METHOD • 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 DARPA / Melamed / Singh

8. H1 Hm OVERVIEW OF SEGMENTATION METHOD • Notify each processor j of the starting vector for its subspace H2 • 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 DARPA / Melamed / Singh

9. DIGRESSION: CIRCULANT MATRICES • Definition • a circulant matrixis onecomposed of vectors in which each row is obtained by circular right shiftof 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 03 2 0 0 0 • 0 0 1 0 0 0 3 20 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 02 0 0 0 3 DARPA / Melamed / Singh

10. DIGRESSION: SIMILARITY CLASSES • Definition • a similarity class is a set of circulant matrices, which have thesameelements,irrespective of their positions • Example: n =5, k =5 • 320 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 • S1 = 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 10 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 • S2 = 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 DARPA / Melamed / Singh

11. SEGMENTATION ALGORITHM vector indexi similarity classSt circulantCs mapping circulantCs enumeration enumeration of vectorvi associated with circulant Cs assignment of vectors vito subspaceHj assignment DARPA / Melamed / Singh

12. MAPPING INDICES TO CIRCULANTS • Number of vectors in similarity class is calculated via the • multinomial formula • where • : multiplicity of theelement jin vector • : number of distinct elements in vector • Example: • for n = 5, k = 5, DARPA / Melamed / Singh

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

14. VECTOR ENUMERATION WITHIN CIRCULANTS • Let be a decomposition vector at • quantization level k, such that • Example:k = 5d0 = { 5 }; d1 = { 4,1 }; d2 = { 3, 2 }; d3 = { 3, 1, 1 }; • d4= { 2, 2, 1 }; d5 = { 2, 1, 1, 1 }; d6 = { 1, 1, 1, 1 }; DARPA / Melamed / Singh

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

16. SUMMARY OF SEGMENTATION METHOD Decomposition for parameters n and k Construction of circulantsCs Construction of Similarity ClassesSt Formation of subspaces Hj S6 S3 S5 S1 H1 H2 H4 H3 S2 S7 S4 DARPA / Melamed / Singh

17. 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 DARPA / Melamed / Singh