Graph partition in PCB and VLSI physical synthesis

Lin ZhongELEC424, Fall 2010

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Outline

• Graph representation• Partition

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Graphs

• A graph G = (V, E)– V = set of nodes– E = set of edges = subset of V V

• Variations:– A connected graph has a path from every node to every other– In an undirected graph:

• Edge (u,v) = edge (v,u)• No self-loops

– In a directed graph:• Edge (u,v) goes from node u to node v, notated uv

– A weighted graph associates weights with either the edges or the nodes

• E.g., a road map: edges might be weighted w/ distance

David Luebke

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Representing Graphs

• Assume V = {1, 2, …, n}• An adjacency matrix represents the graph as a

n x n matrix A:– A[i, j] = 1 if edge (i, j) E (or weight of edge)

= 0 if edge (i, j) E

David Luebke

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Graphs: adjacency matrix

• Example:

1

2 4

3

a

d

b c

A 1 2 3 4

1 0 a d 0

2 0 0 b 0

3 0 0 0 0

4 0 0 c 0

David Luebke

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Graphs: adjacency matrix

• The adjacency matrix is a dense representation– Usually too much storage for large graphs– But can be very efficient for small graphs

• Most large interesting graphs are sparse– E.g., planar graphs, in which no edges cross, have

|E| = O(|V|) by Euler’s formula– adjacency list

WWW

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Graphs in embedded system design

• Algorithm & Software– Node: Computation– Edge: Input/output (data dependencies)

– Embedded system synthesis– Parallel computing

• Hardware– Node: component– Edge: wire/bus

– PCB & IC

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PCB & IC physical synthesis

A. Kahng

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Embedded system synthesis

• Node: Computation task• Edge: Data communication/dependency

FPGA ASIC

ARMCore DSP

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Load balancing in parallel computing

• Node: Process• Edge: Inter-process communication

Core 1 Core 2

Core 3 Core 4

Process 1 Process 2

Process 3Process 4

Process 5 Process 6

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Operational research

• Node: Personnel• Edge: Collaboration

Building 1 Building 2

Building 3 Building 4

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Partition

• Constraints– # of partitions– The capability of each partition

• Objective– Edge cut

• Wire/interconnection cost

– Delay in computation (Software)• Too many computations assigned to a core

Bipartitioning: 2-way partitioning.Bisectioning: Bipartitioning such that the two partitions have the same size.

NP-Complete Problem

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Kernighan-Lin (KL) Algorithm

• Bisectioning

• Input: A graph with – Set nodes V (|V| = 2n)– Set of edges E (|E| = m) – Cost cAB for each edge {A, B} in E

• Output: Two partitions X & Y s.t.– Total cost of edges cut is minimized.– Each partition has n nodes

NP-Complete

D. Pan

Kernighan, B. W.; Lin, Shen (1970). "An efficient heuristic procedure for partitioning graphs". Bell Systems Technical Journal 49: 291-307.

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Complexity of graph bisectioning

• Brute-force method: n out of 2n nodes– (2n)!/(n!)2 = [2n*(2n-1)*…*(n+1)]/n!– =Σr=0,…,n-1[(2n-r)/(n-r)]

• (n+1)≥(2n-r)/(n-r)≥2• The complexity is therefore between O(2n) and

O(nn)

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Idea of KL Algorithm

• DA = Decrease in cut value if moving A– External cost (connection) EA – Internal cost IA

– Moving A from partition X to Y would increase the value of the cut set by EA and decrease it by IA

A

BC

D

X Y

A

B

C

D

X Y

DA = 2-1 = 1DB = 1-1 = 0

D. Pan

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Idea of KL Algorithm

• Note that we want to balance two partitions• If switch A & B, gain(A,B) = DA+DB-2cAB

– cAB : edge cost for AB

A

B

C

D

X Y

A

B

CD

X Y

gain(A,B) = 1+0-2 = -1

D. Pan

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Idea of KL Algorithm

• Start with any initial legal partitions X and Y.• A PASS (exchanging each node exactly once) is described

below:1. For i := 1 to n do From the unlocked (unexchanged) nodes, choose a pair (A,B) s.t. gain(A,B) is largest Exchange A and B. Lock A and B for this pass Let gi = gain(A,B).

2. Find the k s.t. G=g1+...+gk is maximized

3. Switch the first k pairs.• Repeat the PASS until no improvement (G=0).

D. Pan

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Time complexity of KL

• For each pass,– O(n2) time to find the best pair to exchange.– n pairs exchanged.– Total time is O(n3) per pass.

• Better implementation can get O(n2log n) time per pass.

• Number of passes is usually small.

D. Pan

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Fiduccia-Mattheyses (FM) Algorithm• Modification of KL Algorithm:

– Allow non-uniform vertex weights (areas)– Allow unbalanced partitions– Extended to handle hypergraphs– Clever way to select nodes to move, run much faster.

• Input: A hypergraph with – Set nodes V (|V| = n)– Set of hyperedges E – Area au for each node u in V

– Cost ce for each hyperedge in e– An area ratio r (Unbalanced partition)

• Output: 2 partitions X & Y such that– Total cost of hyperedges cut is minimized– area(X) / (area(X) + area(Y)) is about r

C. M. Fiduccia and R. M. Mattheyses. "A linear-time heuristic for improving network partitions". Proc. DAC, pp 174-181, 1982.

NP-Complete

D. Pan

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Hypergraph vs. graph

• Nodes: A, B, C, D• Hyperedges: {A,B,C}, {B,D}, {C,D}

• Vertex label: Gate size/area• Hyperedge weight: importance/cost of net

AB

C D

AB

C D

D. Pan

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Ideas of FM Algorithm

• Similar to KL– Work in passes.– Lock nodes after moved.– Actually, only move those nodes up to the maximum

partial sum of gain

• Difference from KL– Not exchanging pairs of nodes

Move only one node at each time– The use of bucket data structure for gains

D. Pan

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Time Complexity of FM

• For each pass,– Constant time to find the best node to move.– After each move, time to update gains is proportional to

degree of node moved.– Total time is O(p), where p is total number of pins

• Number of passes is usually small.

D. Pan

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Find the optimal bisection

1

X

2

3

4

5

6

Y

Original Cut Value = 9

KL Algorithm

D. Pan

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4

X

2

3

1

5

6

Y

Optimal Cut Value = 5

D. Pan

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Homework• C. M. Fiduccia and R. M. Mattheyses

, "A linear-time heuristic for improving network partitions,” Proceedings of the Design Automation Conference, pp 174-181, 1982

• B. Krishnamurthy, “An Improved Min-Cut Algonthm for Partitioning VLSI Networks,” IEEE Transactions on Computers, 1984