Applying Edge Partitioning to SPFD's 1 Applying Edge Partitioning to SPFD’s 219B Project Presentation Trevor Meyerowitz Mentor: Subarna Sinha Professor:

Applying Edge Partitioning to SPFD's

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Applying Edge Partitioning to SPFD’s

219B Project PresentationTrevor Meyerowitz

Mentor: Subarna SinhaProfessor: Robert Brayton

5/16/2000


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Outline of Presentation

• A Refresher on SPFD’s

• Motivation of the Program

• Example Partitioning

• Explanation of Different Algorithms

• Results

• Future Work

• Conclusion


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SPFD’s• Sets of Pairs of Functions to

be Distinguished• Can be represented as graph

– Each node represents a care value

– Missing nodes are don’t cares– Nodes with different values

have an edge between them– For Binary functions this results

in a bi-partite graph

000

010

001

100

101

111

ABC F(A,B,C)0 0 0 00 0 1 00 1 0 00 1 1 *1 0 0 11 0 1 11 1 0 *1 1 1 1


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SPFD’s• When Applied to a

multivalued function with k values, a k-partite graph results

• Sample example– Make 0’s red, 1’s green,

2’s yellow– Add edges between different

colors

000

001

100

101

110

ABC F(A,B,C)0 0 0 00 0 1 00 1 0 10 1 1 11 0 0 21 0 1 21 1 0 21 1 1 *

010 011


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Motivation• SPFD’s provide great flexibility but…

– This is a non-polynomial problem– Current implementations (bdd, sat) are slow and

memory intensive• More Heuristics are needed

• So let’s try formulating this as a graph partitioning and coloring problem– both have many heuristics, but they haven’t really

been studied together


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SPFD Partitioning

• Initial SPFD’s are propagated back to the local inputs to form their SPFD’s– This can also be viewed as taking the edges

from the main and placing them into the graphs of the local inputs

• Each edge must be propagated back to a subgraph that can distinguish between its vertices


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SPFD Partitioning

• The number of colors in each sub-graph represents complexity

• We want the minimal coloring over all of the sub-graphs

• Can’t assume k-partite topology of a graph because this is only true for the output nodes


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SPFD Partitioning ExampleABC F(A,B,C)0 0 0 00 0 1 *0 1 0 10 1 1 *1 0 0 21 0 1 1 1 1 0 *1 1 1 *

F(a,b,c)

G(a,b) H(b,c)

a b c

Primary Output F(a,b,c) has two local inputs G(a,b) and H(b,c)

000 100

010 101

Truth Table for F(A,B,C)Truth Table for F(A,B,C)

SPFD GraphSPFD Graph

Observation: If a local function doesn’t contain any distinguishing primary inputs between the vertices for the given edge, then the edge can’t be placed in the SPFD graph of that function.


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000 100

010 101


000 100

010 101

SPFDSPFD Graph for Graph for G(a,b)G(a,b)

000 100

010 101

SPFDSPFD Graph for Graph for H(b,c)H(b,c)


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000 100

010 101


000 100

010 101


000 100

010 101


Edge From 000 to 001. Only distinguished by A so it must be placed in G.Color graph appropiately.


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000 100

010 101


000 100

010 101


000 100

010 101


Edge From 100 to 101. Only distinguished by C so it must be placed in H.Color graph appropriately.


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000 100

010 101


000 100

010 101


000 100

010 101


Choose Edge From 000 to 010. It can go into either graph without increasing coloring. So add it to H.


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000 100

010 101


000 100

010 101


000 100

010 101


Choose Edge From 100 to 010. It can be placed in either w/o increasing coloring. So add it to G.


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000 100

010 101


000 100

010 101


000 100

010 101


Final Edge From 100 to 010. It can be placed in either w/o increasing coloring. So add it to H.


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SPFD Partitioning Example (Lessons Learned)

• With no restrictions the minimally colored partition is obtained by placing all edges into 1 sub-graph

• Edge Ordering is key to the result– Choose forced edges first– Next choose edges that don’t increase coloring– Finally place the edges that do increase

coloring


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Outline of Presentation

• A Refresher on SPFD’s

• Motivation of the Program

• Example Partitioning

• Explanation of Different Algorithms

• Results

• Future Work

• Conclusion


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The Algorithm

• The Family of Algorithms outlined here can be broken down into these groups.– Pre-processing– Edge Ordering– Edge Placement– Post-processing

Currently varied

To be varied in the future


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The Algorithm: Pre-processing• Read in initial graph data + build SPFD

graph

• Color the initial SPFD graph using favorite graph coloring heuristic (DSATUR)– This gives information about graph topology

that can be used in future steps

• Read in edge restrictions for each sub-graph and store using hash tables


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The Algorithm: Edge Ordering• One of the most important steps

• Different ordering techniques explored– Random (the control)– Vertices based on their number of edges and their

frequencies– Vertices based on their edges + colors and their

frequencies– Edge based on the colors of the 2 vertices + their

frequencies


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The Algorithm: Edge Placement

• This is the same for all algorithms• The steps are:

– Edges are traversed based on order obtained– Place all edges that have up no options– Place all edges that don’t add to the coloring of

any of the sub-graphs– Place edge(s) that increase colorings by

minimal amount. – Repeat steps until all edges are placed


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The Algorithm: Post-Processing

• Given greediness of coloring in edge placement, recolor the sub-graphs

• Then try edge swapping among different subgraphs using below techniques– breaking up maximal cliques– removing edges from minimally used colored

vertices– random annealing for ripup and recoloring


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Results

• Implementation (2 separate programs)– A program integrated into MVSIS that derives

SPFD graphs and sub-graph constraints for a given output on a network and outputs them to 2 output files

– Another program that reads in the SPFD graph and constraints and performs the partitioning to get results using the techniques previously described


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Results (output numbers)

• Due to Memory required by large graphs only trivial examples have been presented here

• Graphs with >3000 nodes can’t be generated due to O(n2) space requirements where n is the number of vertices.

• Bottom 2 examples should be executed before the week is upGraph node #pi's#fanins# colors #vertices #edges ord 0 clrsord 0 run ord 1 clrsord 1 run ord 1 clrsord 1 rundecod_40_20 n_0 5 5 5 5 10 7 <.05s 7 <.05s 7 <.05sdecod_40_20 n_1 5 5 5 5 10 6 <.05s 6 <.05s 6 <.05sdecod_40_20 n_2 5 5 5 5 10 7 <.05s 7 <.05s 7 <.05scount_40_15 n_0 10 10 5 2064 1605632 * * * * * *pcle_40_15 n_0 11 11 4 2368 1714112 * * * * * *

Ord 0 - no ordering

Ord 1 – vertex ordering

Ord 2 – edge-color ordering


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Conclusions• This is a good proof of concept• Does edge ordering matter?• This code uses way too much memory

– can only handle small examples– has some memory leaks– Is edge level partitioning too fine a grain?

• Needs to be compared with other SPFD techniques

• Probably map-able to other problems


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Future Work

• Optimization of current code to handle larger examples

• Consideration of global SPFD’s + global topology

• Benchmark vs. current SPFD code

• Explore more heuristics

• Get a better idea of runtimes/memory usage

Applying Edge Partitioning to SPFD's 1 Applying Edge Partitioning to SPFD’s 219B Project Presentation Trevor Meyerowitz Mentor: Subarna Sinha Professor:

Documents

edge partitioning

c edge

c slide

c spfd graph observation

spfds motivation

motivation spfds

initial spfds

given edge