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Meshes of Trees (MoT) and Applications in Integer Arithmetic

Meshes of Trees (MoT) and Applications in Integer Arithmetic. Panagiotis Voulgaris Petros Mol Course: Parallel Algorithms. Outline of the talk. The two-Dimensional Mesh of Trees Definitions Properties Variations Integer Arithmetic Applications Multiplication Division Powering

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Meshes of Trees (MoT) and Applications in Integer Arithmetic

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  1. Meshes of Trees (MoT) and Applications in Integer Arithmetic Panagiotis Voulgaris Petros Mol Course: Parallel Algorithms

  2. Outline of the talk • The two-Dimensional Mesh of Trees • Definitions • Properties • Variations • Integer Arithmetic Applications • Multiplication • Division • Powering • Root Finding

  3. Definition Construction: grid Mesh of Trees Nodes

  4. u u v v Properties 1)Diameter (maximum distance between any pair of processors): 4logN Proof Case 1: u belongs to a row tree and v to a column tree Dist<=2logN +2logN

  5. u v u v u Properties (cont.) Case 2:u,v belong only to row trees (or only to column trees) Dist=logN –r +2logN + logN + s<=4logN since r>=s

  6. Properties (cont) • 2)Bisection Width(the minimum number of wires that have to be removed in order to disconnect the network into two halves with “almost” identical number of processors) :N (Proof omitted) Thus meshes of trees enjoy both small diameter and large bisection width. This fact makes them a more efficient structure than arrays and simple trees

  7. Recursive Decomposition Mesh of trees Four disjoint copies of Mesh of trees Importance: This property makes mesh of trees appropriate for recursive algorithms for parallel computation

  8. “Ideal” Parallel Computer P+M P+M P: Processor M: Memory P+M P+M Every processor is linked to every other processor. Advantage: Speed !! Drawback: Cost P+M P+M P+M P+M

  9. P M P M P M … … P M “Ideal” Parallel Computer Process/Memory separation Every Processor has direct access to a memory register Again here the degree of each node becomes large as the number of processor increases Idea: Why not “simulate” the complete bipartite graph?

  10. “Ideal” Parallel Computer

  11. “Ideal” Parallel Computer

  12. “Ideal” Parallel Computer

  13. Benefits and Drawbacks +Simulation of any step of in 2logN steps +Bounded degree graph with essentially the computational power as +We have actually constructed the NxN mesh of Trees - The mesh of Trees has nearly nodes whereas the initial complete bipartite graph had only 2N Solution: Later

  14. Transformation to mesh of Trees Back

  15. Variations 1)

  16. Variations (cont)

  17. Variations (cont) 2)

  18. Variations (cont.) 3)

  19. Variations (cont.) 4)

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