Global Flow Optimization (GFO) in Automatic Logic Design “ TCAD91 ” by C. Leonard Berman &...

Global Flow Optimization (GFO) in Automatic Logic Design

“TCAD91”by

C. Leonard Berman&

Louise H. Trevillyan

Global Flow Optimization (GFO) in Automatic Logic Design

“TCAD91”by

C. Leonard Berman&

Louise H. TrevillyanCAD Group Meeting

Prepared by Ray Cheung

Global Flow Optimization in Automatic Logic Design

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OutlineOutline

Introduction

Background and Definitions

Min-Cut

Main Results

Discussion & Conclusions


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IntroductionIntroduction

Machine becomes more complex more logic Use local pattern matching to improve a circuit. Moving signal connections from one gate to another

or by deleting connection. Advantage – Efficiency, Effectiveness Disadvantage – Miss global information (only view at

small logic windows)

Main spirit of the GFO paper Use global flow analysis Gather and utilize information from the entire circuit. Especially faster than local transformation for large

circuits. Present how to use this global information to

simplify logic. Supply a firm theoretical foundation.


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Background & DefinitionsBackground & Definitions

Given a circuit C which have function F, which associate with every wire w, F(w).

Rearrange the connection points of w such that F(w) does not change.

Use data flow analysis, compute a quick approx. to F(w) and then construct a weighted, directed graph. Every cut of the graph corresponds to a set of

connection points for w. The cost of cut equals to the number of connection

points of w in the new circuit.


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DefinitionsDefinitions

All the logic is combinational and all functional nodes are single-output NOR’s.

If a signal i is an input to node j, we write i j

If there is a path from node i to node j, we write i j which means j is reachable from i.


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The set of immediate successors of a signal s as sink(s) = {t: s t}

The set of immediate predecessors of s as input(s) = {t: t s}


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If n is a node in C and S is a subset of nodes of C, then “n is blocked by S” if every path beginning at a sink of n and ending at an output contains a node of S.


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For each wire, x C, there are four sets of wires, forcing sets. Fij(x) = {s: if x = i then s = j}, i,j {0,1} Below shows F10 = {n1, n2, n3, n4, n8, n9, n12, n13, n18} There are F00, F01, F10 and F11


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Lemma 1Lemma 1

If i F10(s) and there is no path i s, then s may be connected as an input to node i without changing the function.

No path from i to s, therefore still DAG after reconnect.

If s = 0, no problem, since they are all NOR gate, 0 is non-controlling value of NOR gate.

If s = 1, since i F10(s), therefore no change.


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Lemma 2Lemma 2

If s is connected to node j in C and j is blocked by sinks(s) then the connection of s at j can be replaced by the constant 0 without changing the function of C.

If s = 0, two circuit equivalent.

If s = 1, since all NOR gate which s is connected. Therefore, there is no difference.


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A frontier node is defined as from this node to any primary output, there exists a path which does not contain a node in F10 or F11. i.e. the far-most node close to PO. E.g. {n8, n9, n12, n13}


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S-frontier of iS-frontier of i

S = { S1, …, Sn } be a set indexed by the nodes of C, and let Si be a set of nodes of C. This is summary information.

S-frontier of i denoted as c(i,S) Element of c(i,S) are nodes j such that

j Si There is a path j j1 j2 … OUTPUT such that for no jl Si; j is reachable in the circuit from i.


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Theorem 3Theorem 3

Let C be a circuit, and s a wire in C. Let F10 be the forcing set defined earlier, and let D be a set for which sink(s) D F10(s) and such that for no d D is there a path d s. If C’ is identical to C except that in C’, signal s is an input only to those nodes in c(s,D), then C C’.

Figure on the left side is C, and C’ on the right side.

Use Lemma 1 and Lemma 2 and the definition of c


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Corollary 4Corollary 4

Let C and C’ be two circuits which are identical except in the set of gates to which one wire s, is an input. Let F10 and F’10 be the two forcing sets for C and C’, respectively. Assume sink(s) D F10(s) and sink(s) D’ F’10(s). If c(s,D) = c’(s,D’) then C C’.

In the figure below, C C’.


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Alter connection of signal sAlter connection of signal s

Determine a set Ns of nodes of C such that n Ns with three properties. There is no path n s. Node n is blocked by c(s,D). If C is changed to C’ in which s is connected only at the nodes of

Ns, c(s,D) = c’(s,D)

Properties 1 and 2 are easy to determine.

But finding the minimal set of nodes which satisfies property 3 is NP-Hard – we call it Ns problem.

Efficient method (min-cut) to find an approximate solution to the Ns problem.

Other approach is Linear programming.


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Construction of GsConstruction of Gs

Add Nodes Source node labeled SOURCE with inf weight. Sink node labeled SINK with inf weight. For each iC10(s) which is reachable from s, add node(i) with weight = 1. For each node i, blocked by (s,C10), from which s is not reachable, add node(i)

with weight = 1. Add Edges

SET = {(s,C10)} DO WHILE SET is not EMPTY

In reverse breadth first order, choose and remove i from SET A. Choose k{inputs(i)} for which

– k C10

– node(k) Gs

l {inputs(k)} node(l) Gs B. For k found in A above

– Remove node(k) from Gs

l {inputs(k)}: add edge(l,I)to Gs, add node(l) to SET. If no such k exists, add edge (SOURCE, i)

End DO

For each node(k) with k (s,C10), add an edge from k to the sink.


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Add Nodes Source node labeled SOURCE with inf weight. Sink node labeled SINK with inf weight. For each iC10(s) which is reachable from s, add node(i) with weight = 1. For each node i, blocked by (s,C10), from which s is not reachable, add

node(i) with weight = 1.


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Add Edges SET = {(s,C10)} DO WHILE SET is not EMPTY


– k C10

– node(k) Gs

l {inputs(k)} node(l) Gs

B. For k found in A above



End DO


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– k C10

– node(k) Gs





End DO


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– k C10

– node(k) Gs





End DO


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– k C10

– node(k) Gs





End DO


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– k C10

– node(k) Gs





End DO


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– k C10

– node(k) Gs





End DO


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– k C10

– node(k) Gs





End DO


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– k C10

– node(k) Gs





End DO


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For each node(k) with k (s,C10), add an edge from k to the sink.


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Derived Flow graphDerived Flow graph

Add a source node (src) and sink node (snk).

For each node in F10, add node in flow graph with weight 1.


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Max-Flow Min-Cut TechniqueMax-Flow Min-Cut Technique

Edmonds-Karp: O(|V||E|2)

Ford & Fulkerson: O(|E||f|)Note: For unit capacity, |f| |V|, so O(|V||E|) time.

s t

a b

c d

16

13

10 4 9 7

12

20

4

14

s t

a b

c d

11/16

12/13

10 1/4 9 7/7

12/12

19/20

4/4

11/14

min-cut = max-flow


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Theorem 5Theorem 5

Let K be any node cut separating the source and sink in Gs, and let C’ be a circuit which is identical to C except that the sinks of signal s are precisely those nodes which are cut by K, then c(s,C10) = c’(s,C10).

Proof Theorem 5 needs Lemma 6, 7, 8.

Theorem 5 allows us to rearrange the connections of signal s without changing the circuit functionality. Given a circuit C, and wire s, construct Gs

Use min-cut algorithm to find a node cut of Gs. Rearrange the connections of s based on the cut-set & Theorem 3.


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Lemma 6Lemma 6

Every gate in C corresponding to a node in Gs is blocked by (s,C10).


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Lemma 7Lemma 7

In Gs, every node corresponding to a gate in (s,C10(s)) is reachable from the SOURCE.


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Lemma 8Lemma 8

Every gate in C’ which corresponds to a node in Gs which is reachable from K is in C10(s). If the longest path from K to a node is 0, then s will be the direct

input of this node, there it is in C10(s). Assume it is true for the nodes of distance less than i+1 When the node n of distance is i+1,

By the construction of Gs, it implies One of the input to n is in C11(s) Therefore, n C10(s)


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Theorem 9Theorem 9

Let K be any cut separating SOURCE and SINK in Gs, then let C and C’ be related as in Theorem 5, then C C’.

Theorem 5 shows two Frontier are the same.

Theorem 9 shows two circuit are the same.

We have the following - Connection Optimization Procedure

PROC: OPTCON; /* OPTimize CONnections */ FOR EACH SIGNAL

CONSTRUCT Gs; FIND MINIMAL CUT, K; CREATE C’ AS IN THEOREM 9; C = C’;

END;

END OPTCON;


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ResultsResults

The flow graph has the following property. If the source node is re-connected to the corresponding nodes in the cut-set, the circuit is unchanged.

Therefore, the problem finding minimum number of nodes (fan-outs of s), can be solved by finding a minimum cut-set in the flow graph.

Cut-set reduce the fanouts from 6 to 3.


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ConclusionsConclusions

Present a new approach to logic optimization.

Use global flow analysis to gather information about the relationships between wires.

Use network flow to optimize the circuit.

Decreasing the number of fan-out nodes.

Is there any better approx. than recurrences approx.?

We can use Linear programming technique to replace min-cut technique.

Another ICCAD2000 paper introduced “Implication Flow Graph” based on GFO.

Because it has a larger solution space than GFO, and for the previous example, IFG can reduce the fanout of s from 6 to 2.


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The EndThe End

Please feel free to ask any question !

Global Flow Optimization (GFO) in Automatic Logic Design “ TCAD91 ” by C. Leonard Berman &...

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