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Global Flow Optimization (GFO) in Automatic Logic Design “ TCAD91 ” by C. Leonard Berman & Louise H. Trevillyan CAD Group Meeting Prepared by Ray Cheung

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CAD Group Global Flow Optimization in Automatic Logic Design 2 Outline Introduction Background and Definitions Min-Cut Main Results Discussion & Conclusions

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CAD Group Global Flow Optimization in Automatic Logic Design 3 Introduction 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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CAD Group Global Flow Optimization in Automatic Logic Design 4 Background & 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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CAD Group Global Flow Optimization in Automatic Logic Design 5 Definitions 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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CAD Group Global Flow Optimization in Automatic Logic Design 6 Definitions 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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CAD Group Global Flow Optimization in Automatic Logic Design 7 Definitions 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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CAD Group Global Flow Optimization in Automatic Logic Design 8 Definitions For each wire, x C, there are four sets of wires, forcing sets. F ij (x) = {s: if x = i then s = j}, i,j {0,1} Below shows F 10 = {n 1, n 2, n 3, n 4, n 8, n 9, n 12, n 13, n 18 } There are F 00, F 01, F 10 and F 11

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CAD Group Global Flow Optimization in Automatic Logic Design 9 Lemma 1 If i F 10 (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 F 10 (s), therefore no change.

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CAD Group Global Flow Optimization in Automatic Logic Design 10 Lemma 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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CAD Group Global Flow Optimization in Automatic Logic Design 11 Definitions A frontier node is defined as from this node to any primary output, there exists a path which does not contain a node in F 10 or F 11. i.e. the far-most node close to PO. E.g. {n 8, n 9, n 12, n 13 }

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CAD Group Global Flow Optimization in Automatic Logic Design 12 S-frontier of i S = { S 1, …, S n } be a set indexed by the nodes of C, and let S i 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 S i There is a path j j 1 j 2 … OUTPUT such that for no j l S i ; j is reachable in the circuit from i.

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CAD Group Global Flow Optimization in Automatic Logic Design 13 Theorem 3 Let C be a circuit, and s a wire in C. Let F 10 be the forcing set defined earlier, and let D be a set for which sink(s) D F 10 (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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CAD Group Global Flow Optimization in Automatic Logic Design 14 Corollary 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 F 10 and F’ 10 be the two forcing sets for C and C’, respectively. Assume sink(s) D F 10 (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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CAD Group Global Flow Optimization in Automatic Logic Design 15 Alter connection of signal s Determine a set N s of nodes of C such that n N s 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 N s, 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 N s problem. Efficient method (min-cut) to find an approximate solution to the N s problem. Other approach is Linear programming.

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CAD Group Global Flow Optimization in Automatic Logic Design 16 Construction of G s Add Nodes Source node labeled SOURCE with inf weight. Sink node labeled SINK with inf weight. For each i C 10 (s) which is reachable from s, add node(i) with weight = 1. For each node i, blocked by (s,C 10 ), from which s is not reachable, add node(i) with weight = 1. Add Edges SET = { (s,C 10 )} 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 C 10 –node(k) G s – l {inputs(k)} node(l) G s B. For k found in A above –Remove node(k) from G s – l {inputs(k)}: add edge(l,I)to G s, add node(l) to SET. If no such k exists, add edge (SOURCE, i) End DO For each node(k) with k (s,C 10 ), add an edge from k to the sink.

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CAD Group Global Flow Optimization in Automatic Logic Design 17 Construction of G s Add Nodes Source node labeled SOURCE with inf weight. Sink node labeled SINK with inf weight. For each i C 10 (s) which is reachable from s, add node(i) with weight = 1. For each node i, blocked by (s,C 10 ), from which s is not reachable, add node(i) with weight = 1.

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CAD Group Global Flow Optimization in Automatic Logic Design 18 Construction of G s Add Edges SET = { (s,C 10 )} 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 C 10 –node(k) G s – l {inputs(k)} node(l) G s B. For k found in A above –Remove node(k) from G s – l {inputs(k)}: add edge(l,I)to G s, add node(l) to SET. If no such k exists, add edge (SOURCE, i) End DO

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CAD Group Global Flow Optimization in Automatic Logic Design 19 Construction of G s Add Edges SET = { (s,C 10 )} 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 C 10 –node(k) G s – l {inputs(k)} node(l) G s B. For k found in A above –Remove node(k) from G s – l {inputs(k)}: add edge(l,I)to G s, add node(l) to SET. If no such k exists, add edge (SOURCE, i) End DO

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CAD Group Global Flow Optimization in Automatic Logic Design 20 Construction of G s Add Edges SET = { (s,C 10 )} 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 C 10 –node(k) G s – l {inputs(k)} node(l) G s B. For k found in A above –Remove node(k) from G s – l {inputs(k)}: add edge(l,I)to G s, add node(l) to SET. If no such k exists, add edge (SOURCE, i) End DO

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CAD Group Global Flow Optimization in Automatic Logic Design 21 Construction of G s Add Edges SET = { (s,C 10 )} 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 C 10 –node(k) G s – l {inputs(k)} node(l) G s B. For k found in A above –Remove node(k) from G s – l {inputs(k)}: add edge(l,I)to G s, add node(l) to SET. If no such k exists, add edge (SOURCE, i) End DO

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CAD Group Global Flow Optimization in Automatic Logic Design 22 Construction of G s Add Edges SET = { (s,C 10 )} 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 C 10 –node(k) G s – l {inputs(k)} node(l) G s B. For k found in A above –Remove node(k) from G s – l {inputs(k)}: add edge(l,I)to G s, add node(l) to SET. If no such k exists, add edge (SOURCE, i) End DO

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CAD Group Global Flow Optimization in Automatic Logic Design 23 Construction of G s Add Edges SET = { (s,C 10 )} 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 C 10 –node(k) G s – l {inputs(k)} node(l) G s B. For k found in A above –Remove node(k) from G s – l {inputs(k)}: add edge(l,I)to G s, add node(l) to SET. If no such k exists, add edge (SOURCE, i) End DO

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CAD Group Global Flow Optimization in Automatic Logic Design 24 Construction of G s Add Edges SET = { (s,C 10 )} 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 C 10 –node(k) G s – l {inputs(k)} node(l) G s B. For k found in A above –Remove node(k) from G s – l {inputs(k)}: add edge(l,I)to G s, add node(l) to SET. If no such k exists, add edge (SOURCE, i) End DO

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CAD Group Global Flow Optimization in Automatic Logic Design 25 Construction of G s Add Edges SET = { (s,C 10 )} 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 C 10 –node(k) G s – l {inputs(k)} node(l) G s B. For k found in A above –Remove node(k) from G s – l {inputs(k)}: add edge(l,I)to G s, add node(l) to SET. If no such k exists, add edge (SOURCE, i) End DO

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CAD Group Global Flow Optimization in Automatic Logic Design 26 Construction of G s For each node(k) with k (s,C 10 ), add an edge from k to the sink.

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CAD Group Global Flow Optimization in Automatic Logic Design 27 Derived Flow graph Add a source node (src) and sink node (snk). For each node in F 10, add node in flow graph with weight 1.

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CAD Group Global Flow Optimization in Automatic Logic Design 28 Max-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. st a b cd 16 13 10497 12 20 4 14 st a b cd 11/16 12/13 101/497/7 12/12 19/20 4/4 11/14 min-cut = max-flow

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CAD Group Global Flow Optimization in Automatic Logic Design 29 Theorem 5 Let K be any node cut separating the source and sink in G s, 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,C 10 ) = c’ (s,C 10 ). 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 G s Use min-cut algorithm to find a node cut of G s. Rearrange the connections of s based on the cut-set & Theorem 3.

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CAD Group Global Flow Optimization in Automatic Logic Design 30 Lemma 6 Every gate in C corresponding to a node in G s is blocked by (s,C 10 ).

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CAD Group Global Flow Optimization in Automatic Logic Design 31 Lemma 7 In G s, every node corresponding to a gate in (s,C 10 (s)) is reachable from the SOURCE.

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CAD Group Global Flow Optimization in Automatic Logic Design 32 Lemma 8 Every gate in C’ which corresponds to a node in G s which is reachable from K is in C 10 (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 C 10 (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 G s, it implies One of the input to n is in C 11 (s) Therefore, n C 10 (s)

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CAD Group Global Flow Optimization in Automatic Logic Design 33 Theorem 9 Let K be any cut separating SOURCE and SINK in G s, 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 G s ; FIND MINIMAL CUT, K; CREATE C’ AS IN THEOREM 9; C = C’; END; END OPTCON;

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CAD Group Global Flow Optimization in Automatic Logic Design 34 Results 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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CAD Group Global Flow Optimization in Automatic Logic Design 35 Conclusions 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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CAD Group Global Flow Optimization in Automatic Logic Design 36 The End Please feel free to ask any question !

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