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2
Outline of the talk
The two-Dimensional Mesh of Trees
- Definitions
- Properties
- Variations
Integer Arithmetic Applications
- Multiplication
- Division
- Powering
- Root Finding
Meshes of Trees (MoT) and Applications in Integer Arithmetic - - PowerPoint PPT Presentation
Meshes of Trees (MoT) and Applications in Integer Arithmetic - - PowerPoint PPT Presentation
Meshes of Trees (MoT) and Applications in Integer Arithmetic Panagiotis Voulgaris Petros Mol Course: Parallel Algorithms 1 Outline of the talk The two-Dimensional Mesh of Trees Definitions Properties Variations Integer
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Definition
Construction:
N N ×
grid Mesh of Trees
2
3 2 N N −
Nodes
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Properties
1)Diameter (maximum distance between any pair of processors): 4logN
Proof u v u v Case 1: u belongs to a row tree and v to a column tree Dist<=2logN +2logN
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Properties (cont.)
Case 2: u,v belong only to row trees (or only to column trees)
u v u u u v u u Dist=logN –r +2logN + logN + s<=4logN since r>=s
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Properties (cont)
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
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)
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Recursive Decomposition
N N ×
Mesh of trees
2 2 N N ×
Four disjoint copies of Mesh of trees Importance: This property makes mesh of trees appropriate for recursive algorithms for parallel computation
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“Ideal” Parallel Computer
P+M P+M P+M P+M P+M P+M P+M P+M P+M P+M P+M P+M P+M P+M P+M P+M P: Processor M: Memory
Every processor is linked to every
- ther processor.
Advantage: Speed !! Drawback: Cost
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“Ideal” Parallel Computer
…
P P P P P P P P M M M M M M M M
…
Process/Memory separation
Again here the degree of each node becomes large as the number of processor increases Idea: Why not “simulate” the complete bipartite graph? Every Processor has direct access to a memory register
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“Ideal” Parallel Computer
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“Ideal” Parallel Computer
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“Ideal” Parallel Computer
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Benefits and Drawbacks
+ Simulation of any step of in 2logN steps
, N N
K
, N N
K + 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
- nly 2N
Solution: Later
2
3N
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Back
Transformation to mesh of Trees
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Variations
1)
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Variations (cont)
≡
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Variations (cont)
2)
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Variations (cont.)
3)
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