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## week 2

by: Abhinav Notetaker

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# week 2 EECE 6086C

Abhinav Notetaker
UC
GPA 3.5

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week 2 FM algo
COURSE
VLSI Design automation
PROF.
Ranga Vemuri
TYPE
Class Notes
PAGES
19
WORDS
KARMA
25 ?

## Popular in Electrical Engineering

This 19 page Class Notes was uploaded by Abhinav Notetaker on Monday March 21, 2016. The Class Notes belongs to EECE 6086C at University of Cincinnati taught by Ranga Vemuri in Spring 2016. Since its upload, it has received 21 views. For similar materials see VLSI Design automation in Electrical Engineering at University of Cincinnati.

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Date Created: 03/21/16
Fiduccia-Mattheyses (FM) Algorithm • Modified version of KL • A single vertex is moved across the cut in a single move   Unbalanced partitions • Vertices are weighted • Concept of cutsize extended to hypergraphs • Special data structure to improve time complexity to O(n )2  (Main feature) • Can be extended to multi-way partitioning th C. M. Fiduccia and R. M. MattheyDAC, 1982. The FM Algorithm: Data Structure Ist Partition +pmax v v a1 a2 -pmax Vertex 1 2 . . . . . . . . n List of free vertices 2nd Partition +pmax vb1 vb2 -pmax Vertex 1 2 . . . . . . . . n [©Sherwani] The FM Algorithm: Data Structure • Pmax  Maximum gain  pmax= d max. wmax, where d = max degree of a vertex (# edges incident to it) max wmax is the maximum edge weight  What does it mean intuitively? • -Pmax .. Pmax array  Index i is a pointer to the list of unlocked vertices with gain i. • Limit on size of partition  A maximum defined for the sum of vertex weights in a partition (alternatively, the maximum ratio of partition sizes might be defined) The FM Algorithm • Initialize  Start with a balance partition A, B of G (can be done by sorting vertex weights in decreasing order, placing them in A and B alternatively) • Iterations  Similar to KL  A vertex cannot move if violates the balance condition  Choosing the node to move: pick the max gain in the partitions  Moves are tentative (similar to KL)  When no moves possible or no more unlocked vertices available, the pass ends  When no move can be made in a pass, the algorithm terminates Why Hyperedges?  For multi terminal nets, K-L may decompose them into many 2-terminal nets, but not efficient!  Consider this example:  If A = {1, 2, 3} B = {4, 5, 6}, graph model shows the cutsize = 4 but in the real circuit, only 3 wires cut  Reducing the number of nets cut is more realistic than reducing the number of edges cut q 4 3 q 4 m 3 q 1 m 5 1 m q 5 k m 2 k 2 6 p p 6 [©Kang] Hyperedge to Edge Conversion • A hyperedge can be converted to a “clique”. w 3 1 4 4 3 w w 2 2 “Real” cut=1 “net” cut=2w • w=?  w=2/(n-1) has been used, also w=2/n 2  Best: w=4/(n – mod(n,2)) for n=3, w=4/(9-1)=0.5 • Always necessary to convert hyper-edge to edge? [©Keutzer] FM Gain Calculation: Direct Hyperedge Calc • FM is able to calculate gain directly using hyperedges ( not necessary to convert hyperedges to edges) • Definition:  Given a partition (A|B), we define the terminal distribution of n as an ordered pair of integers (A(n),B(n)), which represents the number of cells net n has in blocks A and B respectively (how fast can be computed?)  Net is critical if there exists a cell on it such that if it were moved it would change the net’s cut state (whether it is cut or not).  Net is critical if A(n)=0,1 or B(n)=0,1 [©Keutzer] FM Gain Calc: Direct Hyperedge Calc (cont.) • Gain of cell depends only on its critical nets:  If a net is not critical, its cutstate cannot be affected by the move  A net which is not critical either before or after a move cannot influence the gains of its cells • Let F be the “from” partition of cell i and T the “to”: • g(i) = FS(i) - TE(i), where:  FS(i) = # of nets which have cell i as their only F cell  TE(i) = # of nets connected to i and have an empty T side [©Keutzer] Hyperedge Gain Calculation Example • If node “a” moves to the other partition… h 1 b h2 a h3 c d e h4 f i g j l k m n FM Example +6 · +6 · : : : : · v3 · · v1 · · v0 · +4 · +1 · : : 0 · -1 · · v4 · -1 · · v2 · : : : : -6 · -6 · · · · · · · · · · · v0 v1 v2 v3 v4 A 0 1 2 3 4 B v0 B v0 · v1 · v3 · v2 · v1 B v2 v0 v1 · v0 · v3 · v2 B 3 v2 · v0 · v4 v3 v3 A 2 v1 v3 · v4 · v1 · v0 · v4 A v4 · v3 · FM Example +6 · +6 · : : : : · v3 · · v1 · · v0 · +4 · +1 · : : 0 · -1 · · v4 · -1 · · v2 · : : : : -6 · -6 · · · · · · · · · · · v0 v1 v2 v3 v4 A 0 1 2 3 4 B v0 B v0 · 11 · v3 · v2 · v1 B v2 v0 v1 · v0 · v3 · v2 B 3 v2 · v0 · v4 v3 v3 A 2 v1 v3 · v4 · v1 · v0 · v4 A v4 · v3 · FM Example +6 · +6 · : : : : · v3 · · v1 · · v0 · +4 · +1 · : : 0 · -1 · · v4 · -1 · · v2 · : : : : -6 · -6 · · · · · · · · · · · v0 v1 v2 v3 v4 A 0 1 2 3 4 B v0 B v0 · 11 · v3 · v2 · v1 B v2 v0 v1 · v0 · v3 · v2 B 3 v2 · v0 · B v4 v3 v3 A 2 v1 v3 · v4 · v1 · v0 · v4 A v4 · v3 · FM Example +6 · +6 · : : : : · v1 · · v0 · +1 · : : 0 · -1 · · v4 · -1 · · v2 · : : : : -6 · -6 · · · · · · · · · · · v0 v1 v2 v3 v4 A 0 1 2 3 4 B v0 B v0 · 11 · v3 · v2 · v1 B v2 v0 v1 · v0 · v3 · v2 B 3 v2 · v0 · v4 v3 v3 B 2 v1 v3 · v4 · v1 · v0 · v4 A v4 · v3 · FM Example +6 · +6 · : : : : · v1 · · v0 · +1 · : : 0 · -1 · · v4 · -1 · · v2 · : : : : -6 · -6 · · · · · · · · · · · v0 v1 v2 v3 v4 A 0 1 2 3 4 B v0 B v0 · 11 · v3 · v2 · v1 B v2 v0 v1 · v0 · v3 · v2 B 3 v2 · v0 · v4 v3 v3 B 2 v1 v3 · v4 · v1 · v0 · v4 A v4 · v3 · FM Example +6 · +6 · : : : : · v1 · · v0 · +1 · +1 · : : 0 · -1 · · v4 · -1 · · v2 · : : : : -6 · -6 · · · · · · · · · · · v0 v1 v2 v3 v4 A 0 1 2 3 4 B v0 B v0 · v1 · v3 · v2 · v1 B v2 v0 v1 · v0 · v3 · v2 B 3 v2 · v0 · v4 v3 v3 B 2 v1 v3 · v4 · v1 · v0 · v4 A v4 · v3 · FM Example +6 · +6 · : : : : · v1 · · v0 · +1 · +1 · : : 0 · -1 · · v4 · -1 · · v2 · : : : : -6 · -6 · · · · · · · · · · · v0 v1 v2 v3 v4 A 0 1 2 3 4 B v0 B v0 · v1 · v3 · v2 · v1 B v2 v0 v1 · v0 · v3 · v2 B 3 v2 · v0 · v4 v3 v3 B 2 v1 v3 · v4 · v1 · v0 · v4 A v4 · v3 · FM Example +6 · +6 · : : : : · v1 · · v0 · +1 · +1 · : : 0 · -1 · · v4 · -1 · · v2 · : : : : -6 · -3 · · · · · · : : · · · · · -6 · v0 v1 v2 v3 v4 A 0 1 2 3 4 B v0 B v0 · 11 · v3 · v2 · v1 B v2 v0 v1 · v0 · v3 · v2 B 3 v2 · v0 · v4 v3 v3 B 2 v1 v3 · v4 · v1 · v0 · v4 A v4 · v3 · FM Example +6 · +6 · : : : : · v1 · · v0 · +1 · +1 · : : 0 · -1 · · v4 · -1 · · v2 · : : : : -6 · -3 · · v1 · · · · · · : : · · · · · -6 · v0 v1 v2 v3 v4 A 0 1 2 3 4 B v0 B v0 · 11 · v3 · v2 · v1 B v2 v0 v1 · v0 · v3 · v2 B 3 v2 · v0 · v4 v3 v3 B 2 v1 v3 · v4 · v1 · v0 · v4 A v4 · v3 · FM Example +6 · +6 · : : : : · v0 · +1 · +1 · : : 0 · -1 · · v4 · -1 · · v2 · : : : : -6 · -3 · · v1 · · · · · · : : · · · · · -6 · v0 v1 v2 v3 v4 A 0 1 2 3 4 B v0 B v0 · 11 · v3 · v2 · v1 B v2 v0 v1 · v0 · v3 · v2 B 3 v2 · v0 · v4 v3 v3 B 2 v1 v3 · v4 · v1 · v0 · v4 A v4 · v3 ·

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