Structural Gate Decomposition for Depth-Optimal Technology Mapping in LUT-based FPGA

Transcription

1 Structural Gate Decomposton for Depth-Optmal Technology Mappng n LUT-based FPGA Abstract Jason Cong and Yean-Yow Hwang Department of Computer Scence Unversty of Calforna, Los Angeles Los Angeles, CA 9004 In ths paper, we study the problem of decomposng gates n fann-unbounded or K-bounded networks such that the mappng solutons computed by a depth-optmal mapper have mnmum depth. We present several theoretcal results: () any further decomposton of a K-bounded network wll lead to an optmal mappng depth smaller than or equal to that of the orgnal network, regardless of the decomposton algorthm used, and () the problem of gate decomposton for depth-optmal technology mappng s NP-hard for fann-unbounded networks when K! and remans NP-hard for K-bounded networks when K! 5. We propose a novel gate decomposton algorthm, named DOGMA, whch combnes level-drven node packng technque (Chortle-d) and the network flow based optmal labelng technque (FlowMap) for depth-optmal technology mappng. Expermental results show that the networks produced by DOGMA allow depth-optmal technology mappers to mprove the mappng solutons by up to % n depth and up to 5% n area comparng to the mappng results of networks decomposed by other exstng decomposton algorthms.. Introducton The feld programmable gate array (FPGA) s a popular technology for VLSI ASIC desgn and system prototypng due to ts short desgn cycle and low manufacturng cost. The lookup-table (LUT) based FPGAs have been wdely used. A K-nput LUT (K-LUT) conssts of K SRAM cell to store a truth table whch can mplement an arbtrary functon of up to K nputs. The goal of LUTbased FPGA technology mappng s to cover a gven network usng K-LUTs such that ether area or delay s mnmzed or routablty s maxmzed n the fnal LUT network. The delay of a network can be estmated by the number of levels (.e. depth) under the unt delay model before placement and routng are performed. Several LUT-mappng algorthms have been proposed for depth mnmzaton [FrRV9c, MuSB9a, SaTh9, ChCD9, CoD94a] and reported good results. In partcular, the FlowMap algorthm [CoD94a] guarantees a depth-optmal mappng soluton for any K-bounded network. However, t can not be appled drectly to a fann-unbounded network and depth-optmal result may not hold when further gate decomposton of a K-bounded network s allowed pror to the mappng. Gate decomposton of the ntal network can consderably mpact the depth of mappng solutons. For example, assume K =. The network N shown n Fgure (a) s not K-bounded snce

2 node v has 5 fanns. Assumng that the nodes u, u and u have two nputs each, f v s decomposed n such a way that PI and PI are merged nto one node, as shown n Fgure (b), there s no way to obtan a mappng soluton of depth less than. However, f a dfferent decomposton s carred out for v, as shown n Fgure (c), a mappng soluton of depth equal to s obtaned. Even for a K- bounded network, the depth of the optmal mappng soluton may decrease f ts gates are further decomposed before technology mappng. For example, the network N shown n Fgure (a) s K- bounded for K =. The mnmum depth n any mappng soluton of N s. But f the network s further decomposed nto Fg. (b), a mappng soluton wth depth can be obtaned. Whle FlowMap guarantees depth-optmal mappng solutons for K-bounded networks, t does not explot the freedom ntroduced by further gate decomposton. Nevertheless, we would lke to emphasze that the optmalty result by FlowMap for K-bounded networks enables the study and evaluaton of varous gate decomposton algorthms and ther mpact on technology mappng. Otherwse, when a gate decomposton algorthm leads to a poor mappng soluton, t would be dffcult to determne whether the gate decomposton algorthm, or the mappng heurstc, or a combnaton of both result n the poor mappng result. To our knowledge, ths work s the frst n-depth study and evaluaton of the mpact of gate decomposton on the depth of the fnal mappng solutons, based on the depth-optmal mappng algorthm for K-bounded networks, whch was avalable only recently. Gate decomposton methods can be classfed as () structural decomposton and () Boolean decomposton. In structural decomposton, a mult-fann gate s replaced by a fann tree. Structural decomposton can be appled only to networks wth gates whch are assocatve wth respect to ther fanns, such as networks of smple gates (e.g. AND, OR, NAND, NOR, XOR). Fgure and Fgure show examples of ths type. On the other hand, Boolean decomposton decomposes a complex gate accordng to ts functonalty and representaton. Examples of Boolean decomposton ncludes AND-OR decomposton [BrRS87], co-factorng [SaTh9], cube-packng [MuSB9a], and functonal decomposton [LaPP94, WuEA95]. Ths paper focuses on optmal structural decomposton for PI PI PI PI PI PI u u u u u u u u u v? v v (a) (b) (c) Fgure Gate decomposton of an unbounded networks for K-LUT mappng (K = ). (a) ntal network, (b) decomposton yeldng mappng depth =, (c) decomposton yeldng mappng depth =.

3 depth mnmzaton n LUT mappng. Several gate decomposton routnes have been used for LUT-mappng. The tech_decomp routne n SIS [SeSL9] produces a heght-balanced fann tree for each decomposed gate to mnmze the heght of the fann tree. The dmg routne [Wa89, ChCD9] uses a Huffman codng tree to mnmze the level of the decomposed node n the resultng network. It guarantees a mnmum level n the decomposed network but does not guarantee the optmal depth n the fnal LUT network. In fact, Fgure (b) s the outcome of the dmg routne whch s sub-optmal for depth-optmal mappng. The SIS routne speed_up s often used to obtan delay-optmzed -nput networks for technology mappng. It uses Boolean decomposton based on repeated node collapsng and decomposton to speed-up the crtcal path. MIS-pga-delay [MuSB9a] and TechMap-D [SaTh9] used speed_up as well as other logc synthess technques to obtan delay-mnmzed mappng solutons. These decomposton methods, although effectve to some extent, are not desgned to mnmze the depth of LUT networks drectly. Chortle-d [FrRV9c] s the frst algorthm whch computes a depth-optmal gate decomposton and mappng soluton based on a bn packng approach when the nput network s a tree (may be unbounded). For general networks, t consders packng nodes on reconvergent paths nto ether the same or dfferent bns for all pars of reconvergent paths. Because Chortle-d takes a greedy approach n generatng the LUTs to cover nodes, ts gate decomposton and mappng solutons for general networks are suboptmal n term of depth. In ths paper, we study the gate decomposton problem and algorthms for depth-optmal technology mappers so that mnmum depth are obtaned n the fnal mappng solutons. The remander of ths paper s organzed as follows. Secton defnes basc termnology, presents propertes of structural gate decomposton and gves problem formulaton. Secton shows the complexty of the problem of gate decomposton for optmal depth n LUT-based technology mappng. A novel algorthm of gate decomposton for depth-optmal mappng, named DOGMA, s v v (a) (b) Fgure Further gate decomposton of a K-bounded network (K = ). (a) Intal K-bounded network. (b) After further gate decomposton. (The label n each node s the mnmum depth of the node n any mappng soluton.)

4 presented n Secton 4 based on the mn-heght K-feasble bn packng formulaton. Expermental results are presented n Secton 5 and Secton 6 concludes the paper. Snce we use the concepts and technques n [CoD94a] extensvely, t s benefcal for the reader to be famlar wth the results n [CoD94a].. Problem Formulaton.. Notatons and Prelmnares A combnatonal Boolean network can be represented by a drected acyclc graph (DAG) where a node represents a logc gate and a drected edge (u,v) represents a connecton from the output of node u to the nput of node v. A prmary nput (PI) node s a node of n-degree zero and a prmary output (PO) node s a node wth no outgong edge. Other nodes are nternal nodes. The depth of a node v s the number of edges on the longest path from any PI to v. A PI node has a depth of zero. The depth of a network s the largest node depth among POs. Let nput (v) be the set of fann nodes of node v. Let K be the nput sze of an LUT. A network s K-bounded f every node v n the network satsfes nput (v) " K. Otherwse, t s an unbounded network. Gven a subgraph H of a network, we use nput (H) to denote the set of dstnct nodes outsde H whch supply nputs to nodes n H. Gven a node v n network N, let N v denote the subnetwork consstng of node v and all the predecessors of v. The mnmum mappng depth of v, denoted MMD N (v), s defned as the mnmum depth among all possble K-LUT mappng solutons of N v. Note that MMD N (v) s the same as the label (v) computed by the FlowMap algorthm. PI nodes have a mappng depth of 0. The mnmum mappng depth of a network N, denoted MMD (N), s the maxmum mappng depth among all PO nodes. Gven a K-bounded network N, the FlowMap [CoD94a] algorthm computes MMD N (v) for every node v # N n polynomal tme. The computaton s based on the max-flow mn-cut computaton. A cut n N v s a partton (X v,x v ) of nodes n N v such that PI nodes are n X v and v # X v. The cutset of a cut, denoted n (X v,x v ), s defned as nput (X v ). A cut s K-feasble f n (X v,x v ) " K. The heght of a cut, denoted heght (X v,x v ), s the maxmum mappng depth for nodes n n (X v,x v ). We have the followng lemmas based on results n [CoD94a]. Lemma A node v has MMD N (v) = p f there s a K-feasble cut of heght of p $ n N v but no K-feasble cut of heght of p $ or smaller exsts. Lemma If u # nput (v) then MMD N (u) " MMD N (v)... Propertes of Structural Decomposton Let N be a network and node v # N satsfes nput (v) >. Gven a decomposton algorthm D, we defne a decomposton step of D at v, denoted D v, as follows: Decomposton step D v () chooses two nodes u and u from nput (v), () removes edges (u,v) and (u,v), () ntroduces a new node w and new edges (u,w), (u,w), (w,v) and adds them to N. The resultng network s 4

5 denoted as D v (N). For example, Fgure (a) shows a node v wth fann nodes u,u,u,u 4,u 5. Fgure (b) s the network after one decomposton step at v. Snce gates are assocatve, the ntroduced node w have the same gate type as v. We present two propertes of the structural decomposton. () Any decomposton method wll not ncrease the mnmum mappng depth of a network. It can only decrease the depth. Ths phenomena has been observed n [CoD94a] wthout an explanaton. () Under certan crcumstance, further decomposton of a gate wll not decrease the mnmum mappng depth. Under ths crcumstance, t s rrelevant whch decomposton algorthm s used for those gates. The crcumstance occurs n the packng-based decomposton methods. These two propertes are gven n Lemma and Lemma 4 respectvely. Lemma Gven a K-bounded network N, any decomposton algorthm D, and any node v # N, MMD (D v (N)) " MMD (N). Proof Because the decomposton step D v does not remove nodes, every node n N s stll n D v (N). Let u # N be an arbtrary node and u% # D v (N) be the node correspondng to u n the new network. Every K-feasble cut for u n N s stll a K-feasble cut for u% n D v (N) (but not vce versa as the new node w ntroduced by D v may be n some K-feasble cut n D v (N), but w does not exst n N.) Hence MMD Dv (N)(u%) " MMD N (u). Snce u s arbtrary, MMD (D v (N)) " MMD (N). Lemma 4 If MMD N (u) = MMD N (v) for all u # nput (v) n a K-bounded network N, then MMD (N) = MMD (D v (N)) for any decomposton algorthm D. Proof Let N% = D v (N). Let w be the ntermedate node ntroduced n D v and u # nput (w). Accordng to Lemma, MMD N% (u) " MMD N% (w) " MMD N% (v). Snce MMD N (u) = MMD N (v)! MMD N% (v) (Lemma ) and MMD N (u) = MMD N% (u) (as N u = N% u ), we have MMD N% (u) = MMD N% (w) = MMD N% (v). We now use mathematcal nducton accordng to topologcal u u u u 4 u 5 u u u u 4 u 5 u u u u 4 u 5 w w y x v (a) v (b) v (c) Fgure Decomposton of node v. (a) Before decomposton, (b) D v (N) after one decomposton step of D v, (c) after a sequence of decomposton steps. 5

6 orderng from PIs n N% to show that for any node t # N%, MMD N% (t) = MMD N (t). Base: when node t s a PI, t s obvously true. Inducton hypothess: Assume that MMD N% (t%) = MMD N (t%) for any t% # N% t before t. For the mn-heght K-feasble cut (X t,x t ) n N% t, () f w s not n the cutset n (X t,x t ), the cut also exsts n N t. Accordng to the hypothess, every node n the cutset has the same depth n N and N%. As a result MMD N% (t) = MMD N (t); () f w s n the cutset n (X t,x t ), snce the decomposton tree of v s a fanout-free cone, we can replace w by v to form a new cut of equal or smaller sze n N% t. Snce MMD N% (w) = MMD N% (v), the new cut s of the same heght as (X t,x t ), whch also corresponds to a K-feasble cut n N of the same heght. Therefore, MMD N% (t) = MMD N (t). Snce the result holds for any t # N% = D v (N), we have MMD (N) = MMD (D v (N)) for any decomposton algorthm D... Problem Defnton Accordng to Lemma, the further a network s decomposed, the smaller the mappng depth mght be. Therefore, we decompose every node v such that ts fanns form a bnary tree n the decomposed network. For example, Fgure (c) s a complete decomposton of node v. Every decomposton step ntroduces one ntermedate node and nput (v) $ steps are requred to decompose v. A network s fully decomposed f every mult-fann node n the network s decomposed nto a bnary tree. In ths paper, we study the problem of decomposng an unbounded or K-bounded network nto -nput networks such that the mnmum mappng depth can be obtaned by a depth-optmal technology mapper such as FlowMap. We formulate the followng problems. Structural Gate Decomposton for K-LUT Mappng (SGD/K) Gven a smple-gate unbounded network N &, decompose N & nto a -nput network N such that for any other -nput network decomposton N% of N, MMD (N ) " MMD (N% ). Structural Gate Decomposton for K-LUT Mappng of K-bounded Network (K-SGD/K) Gven a smple-gate K-bounded network N K, decompose N K nto a -nput network N such that for any other -nput network decomposton N% of N, MMD (N ) " MMD (N% ). There are two ssues related to the problem formulaton of gate decomposton. () Independent Versus Smultaneous Gate Decomposton: A smaller depth mght be obtaned when several gates are decomposed together nstead of ndependently. For example, assume both u and v are AND-type gates n Fgure 4(a) and K =. We have MMD N (u) = MMD N (v) =. When u and v are decomposed ndependently to obtan N n Fgure 4(b), MMD N (x)! because the mn-cut of heght for x has at least 4 nodes and therefore s not -feasble. However, f u and v are decomposed together to obtan N n Fgure 4(c), we have MMD N (x) =. Smultaneous gate decomposton, however, must take nto account the node functonalty (e.g. requres nodes u and v are of the same or compatble gate types), and s no longer a pure structural gate decomposton. We study only ndependent gate decomposton n ths paper. 6

7 a b c d e f a b c d e f a b c d e f u v u v u v x? x x N N N (a) (b) Fgure 4 Computng MMD N (x) when K =.(a) orgnal network N, (b) N obtaned by ndependent gate decomposton, (c) N obtaned by smultaneous gate decomposton. () Decomposton Before Mappng Versus Decomposton Durng Mappng: Gate decomposton can be performed before the mappng phase n a -step approach or embedded nto the mappng process beng part of an ntegrated approach. For example, the dmg algorthm was used n a -step approach followed by DAG-MAP or FlowMap whle Chortle-d s an ntegrated approach (snce t decomposes gates and generates LUTs n an nterleavng manner). It may appears that the ntegrated approach can produce better mappng depth because t can choose any decomposton technques durng technology mappng. However, ths s not the case. For an nput network N, assume that the ntegrated approach produces a depth-optmal mappng soluton M N. Then M N also nduces a decomposed K-bounded network N%. A depth-optmal mapper for K-bounded networks (such as FlowMap) can take N% as nput and generate a mappng soluton M N%. Snce M N% s depthoptmal wth respect to N%, we have depth (M N% ) " depth (M N ). However, snce M N s optmal wth respect to N, we have depth (M N ) " depth (M N% ). Therefore, depth (M N ) = depth (M N% ). Ths mples the best ntegrated approach and the best -step approach result n the same optmal depth. In ths paper, we take the -step approach,.e., gates are decomposed before a depth-optmal mappng algorthm (such as FlowMap [CoD94a] or ts enhancement CutMap [CoHw95]) s appled to obtan an LUT mappng soluton. Nevertheless, our gate decomposton algorthm consders the depth optmzaton technques used n FlowMap and the effect of gate decomposton n the mappng phase of FlowMap n order to obtan the decomposed network whch s most sutable for FlowMap to acheve optmal mappng depth.. Complexty of SGD and K-SGD Problems In ths secton we shall show the followng results of the SGD/K and the K-SGD/K problems: () The SGD/K problem s NP-hard for K! ; and () The K-SGD/K problem s NP-hard for K! 5. We only gve the constructon of the NP-Complete reducton n ths secton. Complete proofs of the results can be found n the Appendx. For readers not nterested n the NP-Complete reducton, they (c) 7

8 may proceed drectly to Secton 4 to follow the novel gate decomposton algorthms proposed n ths paper for depth-optmal mappng. Our proof s based on polynomal tme transformatons from the SAT problem to the decson verson of the SGD/K problem and the K-SGD/K problem. We frst defne the SAT problem, whch s a well-known NP-Complete problem [GaJo79]. Problem: -Satsfablty (SAT) Instance: A set of Boolean varables X = { x,x,..., x n } and collecton of m clauses C = { C,C,...,C m }, where () each clause s the dsuncton (OR) of lterals of the varables and () each clause contans at most one of x and x for any varable x. Queston: Is there a truth assgnment of the varables n X such that C = for " " m? We shall transform each nstance of SAT to an nstance of SGD n polynomal tme. The basc dea s to relate the decson of the truth assgnment n SAT to the decson of node decomposton n SGD/K. Snce determnng the truth assgnment s dffcult, the transformaton shows that determnng the decomposton s also dffcult. problem. Problem: Smple-Gate Decomposton(SGD/K-D) We defne the decson verson of the SGD/K Instance: A constant K!, a smple-gate unbounded network N & and B. Queston: Is there a way to decompose N & nto a -nput network N such that the depthoptmal K-LUT mappng soluton of N has a depth no more than B? Gven an nstance F of the SAT wth n varables x,x,...,x n and m clauses C,C,...,C m, we construct an unbounded network N (F) correspondng to the nstance F as follows. Frst, for each varable x, we construct a subnetwork N (x ) whch conssts of the followng nodes: (a) Two output nodes denoted as x and x ; (b) (K $ K) PI nodes n whch two of them are denoted as PI and PI ; (c) (K $ ) nternal nodes, denoted as v,...,v K $, u,...,u K $, w, w and s respectvely; The nodes are connected as shown n Fgure 5. Nodes w and w each have K $ PI fanns. Node s has 4 fanns from w, w, PI and PI. Every other nternal node has K PI fanns. Note that N (x ) s a well-defned subnetwork for K! and s K-bounded for K! 4. Next, for each clause C wth lterals l,l,l, we construct a subnetwork N (C ) whch conssts of the followng nodes: (a) One output node denoted C ; (b) Three lteral nodes denoted as l,l,l ; (c) (K $ 5) nternal nodes q,...,q K $5 each of them s the root of a complete -level K- ary tree wth PI nodes as leaves; (d) (K $ ) nternal nodes r,...,r K $ each of them s the root of a complete -level K-ary tree wth PI nodes as leaves. The connecton s shown n Fgure 6(a). The output node C has all nternal nodes as ts fanns n N (C ). Note that N (C ) s a well-defned subnetwork for K!. However, the output node C s not K-bounded. Fnally, we connect the subnetworks N (C ), =,,...,m, wth the subnetworks N (x ), =,,...,n, to form the network N (F). Let l k ( " k " ) be a lteral n C. If for some varable x 8

9 v K- nodes K- nodes PI PI K PI s K PI s K- PI s K- PI s K PI s K PI s... w w u... v K- u K- s? x x? Fgure 5 Constructon of network N (x ) for each Boolean varable x. K-5 nodes K- nodes l q K-5 l l r q r K- q K-5 nodes K- nodes q K-5 r l l l r K ? (a) C 4 (b) C Fgure 6 (a) Constructon of network N (C ) for each clause C. (b) Exact K nodes of depth appear when MMD (l ) =. l k = x, we connect the output node x of subnetwork N (x ) as the fann to the nternal node l k of subnetwork N (C ). Smlarly, f l k = x, we connect node x of N (x ) as the fann to node l k of N (C ). We call x the varable node of l k. N (F) wll have m output nodes C,...,C m. We llustrate the constructon of N (F) by an example. Assume F = (x + x + x )(x + x + x 4 )(x + x + x 4 ). The network N (F) s shown n Fgure 7. Every subnetwork N (x ) provdes two output sgnals from nodes x and x. Every subnetwork N (C ) requres three nput sgnals l,l,l. Because the frst clause C n F s (x + x + x ), we connect nodes x to l, x to l and x to l, respectvely. We have the followng lemma and theorem. The proofs are detaled n the Appendx. Lemma 5 The SAT nstance F s satsfable f and only f there s a decomposton D (N (F)) of N (F) such that MMD (D (N (F))) = 4. 9

10 N(X ) X X X l l l N(C ) C N(X ) N(X ) N(X ) 4 X X X X4 X4 l l l l l l N(C ) N(C ) C C Fgure 7 The network N (F) for F = (x + x + x )(x + x + x 4 )(x + x + x 4 ). Theorem The SGD/K problem s NP-hard for K!. We now show the complexty of the K-SGD/K problem. In Fgure 5, the network N (x ) s K- bounded for K! 4. Hence the constructons of N (x ) remans the same. In Fgure 6(a), the network N (C ) s unbounded. Consder an alternatve constructon N K (C ) whch conssts of the followng nodes: (a) One output node denoted C ; (b) Three lteral nodes denoted as l,l,l ; (c) (K $ 5) nternal nodes q,... K,q $5 each of them s the root of a complete -level K-ary tree wth PI nodes as leaves. The connecton s shown n Fgure 8(a). In ths constructon, N K (C ) s a K-bounded subnetwork for K! 5. We connect N (x ) and N K (C ) as before and denote the resultng network as N K (F). We have the followng lemma and theorem. The proofs are detaled n the Appendx. Lemma 6 The SAT nstance F s satsfable f and only f there s a decomposton D (N K (F)) of N K (F) such that MMD (D (N K (F))) =. Theorem The K-SGD/K problem s NP-hard for K! Gate Decomposton Algorthm for Depth-Optmal LUT Mappng 4.. Algorthm Overvew Intutvely speakng, our decomposton algorthm, named DOGMA (Depth-Optmal Gate decomposton for MAppng), combnes the level-drven node packng technque n Chortle-d and the network flow based optmal labelng technque used n FlowMap. Gven an unbounded network N, nodes are vsted from PIs to POs n a topologcal order. For each node v, we decompose v (the decomposed network s denoted as N (v)) and label v wth the mnmum mappng depth MMD N (v) (v). When v s vsted, ts fanns have already been decomposed and labeled accordngly. We partton the nodes n nput (v) nto groups such that each group conssts of nodes wth the same label (.e. 0

11 q K-5 K-5 nodes q... l l l K-5 nodes q... q K-5 l l l? (a) C (b) C Fgure 8 Constructon of K-bounded subnetwork N K (C ) for each clause C. mnmum mappng depth). We process these groups n an ascendng order wth respect to ther labels. For a group of nodes labeled p, we pack the nodes nto a mnmum number of bns such that there exsts a K-feasble cut of heght p $ for the nodes n each bn. Such a bn s called a mnheght K-feasble bn. A node u s created for each bn B wth fanns from nodes n B and a fanout to v. Node u has a mappng depth of p and wll be gven a label p. If more than nodes are n the bn B, node u needs to be further decomposed to form a bnary tree. Note that accordng to Lemma 4, no matter how u s decomposed, the mnmum mappng depth of the network s always the same. We then proceed to the group of a next hgher label p +. For each bn node u created n the prevous step, a sngle-fann node w s created wth fann from u and s labeled p +. These snglefann nodes together wth orgnal nodes n the group (all of them are labeled p + ) are agan packed nto a mnmal number of mn-heght K-feasble bns. Ths process contnues untl all nodes are packed nto one bn whch corresponds to the node v. We llustrate the algorthm n Fgure 9. Assume K =. Fgure 9(a) shows a network N n whch every gate has been decomposed nto a -fann node except the output gate v. Every decomposed gate u has been gven a label MMD N (u). Gate v has 5 fanns, among whch nodes b,c,d are labeled and nodes a,e are labeled. Accordng to the algorthm, we frst pack b,c,d nto a mnmal number of mn-heght K-feasble bns. Snce a K-feasble cut of heght exsts for b and c together, we put b,c nto one bn and d nto another bn, and ntroduce nodes f and g for each bn respectvely. Note that f and g are labeled. Then we proceed to the group of label. Nodes h and are created and gven a label wth fann from f and g respectvely. Fgure 9(b) shows the network at ths pont. Shaded nodes represent nodes created durng the decomposton. We try to pack a,h,,e nto a mnmal number of mn-heght K-feasble bns. By the mn-cut computaton, we fnd mn-heght K-feasble cuts of heght for { a,h} and {,e} respectvely. As a result we merge a,h nto and,e nto k respectvely. We contnue and create nodes l and m for and k respectvely, and pack l and m nto a bn whch corresponds to v. The decomposed network (denoted N%) at ths pont s shown n Fgure 9(c). Node v has a mnmum mappng depth MMD N% (v) = 4 and s gven a label 4. In the last step,

12 sngle-fann nodes g, h,, l and m are removed. The fnal decomposed network has a mnmum mappng depth of 4. Our algorthm s smlar to Chortle-d n that decomposton s done by packng nodes nto a mnmal number of bns accordng to ther node labels (.e. mnmum mappng depth). However, there are two maor dfferences. Frst, Chortle-d assgns LUTs durng the decomposton whle our algorthm smply computes mnmum mappng depth for every node. Our approach s optmal n computng mappng depth (usng FlowMap algorthm) whle Chortle-d computes the node mappng depth based on the partally generated LUT network of the node. Snce the fann constrant s not a monotone clusterng constrant [CoD94a], Chortle-d mght obtan naccurate labels for nodes. Secondly, Chortle-d uses bn-packng heurstcs together wth enumeratve consderaton for nodes on reconvergent paths, whle we use network flow based mn-cut packng methods (to be descrbed n detals n Secton 4.) for packng nodes nto mn-heght K-feasble bns. Sometme t s not obvous to decde locally whether two nodes can be packed nto one mn-heght K-feasble bn unless a global mn-cut computaton s used. For example, nodes e and can be packed nto a mn-heght K-feasble bn but the Chortle-d algorthm may fal to do so. Of course, we obtan ths packng qualty at an expense of extra computaton tme. However, the runnng tme s never a problem n practce n our experments. c d c d c d b b b u f g u f g u a e a h e a h e k v v 4 l 4 m (a) (b) 4 v (c) Fgure 9 Decomposton of gate v by the DOGMA algorthm. (a) Before decomposton, (b) b,c,d are packed nto f,g, (c) a,h,,e are packed nto,k.

13 4.. Network Flow Based Packng Methods The central problem remans to be solved n our approach s the mn-heght K-feasble bn packng problem defned as follows. We shall gve three heurstcs n ths secton and one exact method n the next secton to solve the problem. Mn-Heght K-Feasble Bn Packng Problem Gven a set A of nodes of mnmum mappng depth p, partton A nto a mnmal number of bns such that there s a K-feasble cut of heght p $ for nodes n each bn. Such a bn s called a mn-heght K-feasble bn. Our heurstcs are based on the max-flow mn-cut algorthm and bn-packng heurstcs. The frst-ft-decreasng (FFD) and best-ft-decreasng (BFD) are two heurstcs for the bn packng problem [HoSa78]. Items are sorted nto a decreasng order accordng to ther sze and then assgned to bns. FFD assgns each tem n the sorted order to the frst-ft bn whle BFD assgns tems to the the best-ft bn (.e. the bn wth the least resdual capacty after assgnment). The total sze of tems n any bn must not exceed the bn capacty. For the mn-heght K-feasble bn packng problem, we propose two mn-cut based heurstcs MC-FFD and MC-BFD, whch are analogous to FFD and BFD except that the total tem sze of a group of nodes S wth label p s now defned as the sze of the mn-cut of heght p $ for S. Nodes n S can be packed nto one bn as long as the mn-cut of heght p $ for these nodes s stll K-feasble. To determne f there s a K-feasble cut of heght p $ for a set of nodes n S = { v,v,..., v m } wth label p, we construct the flow network and apply the max-flow algorthm [CoD94a] as follows. () We ntroduce a snk node t wth fanns from every node n S. () We ntroduce a source node s connected to all PIs n the network rooted at t. () Every edge s gven an nfnte flow capacty. (4) Every node u n the flow network except s and t s replaced by two nodes q and r connected by a drected edge e = (q,r ) such that nput (q ) = nput (u ) and output (r ) = output (u ). If u s labeled p, e s assgned an nfnte flow capacty. Otherwse, e s assgned a unt flow capacty. (5) Fnally, we compute a max-flow to obtan a mn-cut n the flow network, whch can be shown to be a cut of heght p $ [CoD94a]. By examnng the sze of ths cut, we can determne f there s a K-feasble cut of heght p $ for nodes n S. The thrd heurstc s the network flow based maxmal-sharng-decreasng (MC-MSD) heurstc whch emphaszes on sharng nodes among cuts. Assume both u and v are fanns of a node beng decomposed and are labeled p. Let n u, n v and n { u,v} represent the sze of mn-cuts of heght p $ for node u, v and the nodes { u,v}, respectvely. The sharng between u and v s defned as share (u,v) = n u + n v $ n { u,v}. When share (u,v) > 0, the sze of mn-cut of heght p $ for u and v together s smaller than the sum of ther ndvdual mn-cut sze, whch s benefcal to the packng. The MC-MSD heurstc s smlar to the MC-BFD heurstc n that the best-ft bn s now chosen to be the bn that produces a maxmal sharng.

14 4.. The Optmal Mn-Heght K-Feasble Bn Packng Method We compute the optmal mn-heght K-feasble bn packng by a dynamc programmng approach whch s smlar to the one for the number parttonng problem. In the number parttonng problem we want to fnd a way to dvde an nteger set nto two subsets of equal sum. Ths problem s NP-complete but can be solved n pseudo-polynomal tme [GaJo79]. We frst formulate a general number parttonng problem and an analogous node parttonng problem. Then we gve an algorthm for the number parttonng problem. The method for the node parttonng problem s smlar and hence skpped. Based on the soluton, we gve an algorthm whch solves the mn-heght K-feasble bn packng problem optmally. The k-subset Number Parttonng Problem (k-snp) Gven an nteger set A and a bn sze b, s there a way to partton A nto subsets (bns) A, A,..., A k such that ' a " b for " " k? The Mn-Heght K-Feasble k-bn Packng Problem ((k,k)-mbp) Gven a set A of nodes wth label p and a LUT nput sze K, partton A nto k subsets such that there s a K-feasble cut of heght p $ for nodes n each bn. When k = and b s half of the sum of numbers n A, the k-snp problem becomes the number parttonng problem. a # A We solve the k-snp problem by dynamc programmng as follows. Let A = { a,a,...,a n } and S be the ordered set of vectors (t,t,...,t k $ ) where 0 " t " b. Clearly S = (b +) k $. Vectors n S are ordered lexcographcally. Let u represent the unt vector (u,u,...,u k $ ) where u l = 0 f ( l and u =. Create an n rows by S columns table T. Rows of T are assocated wth ndexes of ntegers a,a,...,a n respectvely. Columns of T are assocated wth vectors n S lexcographcally. An entry T (, t) s marked TRUE, denoted T, f there exsts dsont sets A%,A%,...,A% k $ ) { a,a,...,a } such that ' a l = t. u for " " k $. We compute T accordng to the followng rules. () T (, t 0 ) = T, () T (,a u ) = T for " " k $, () T (, t) = T f T ( $,t) = T, (4) T (, t) = T f T ( $,s) = T where t = s + a u for " " k $. a l # A% A soluton to the k-snp problem exsts f and only f for some t = (t,..., t k $ ), T (n, t) = T and ' a k $ $ t. u " b. The computaton takes O (n S. k) tme where n S s from the sze of table T a # A ' = and k s from the length of vector. The tme may decrease f the table symmetry s exploted. It s clear the (k,k)-mbp problem can be solved smlarly. The dfference s that the sze of the mn-cut of heght p $ s now consdered as the total sze of ths subset durng bn packng. Agan, the cut sze can be found n lnear tme usng the network flow computaton. Startng from k =, we can do a lnear search or a bnary search on k to fnd the mnmum k such that there s a soluton for the (k,k)-mbp problem. Ths mnmum k leads to an optmal soluton for the mn-heght 4

15 K-feasble bn packng problem. We refer to ths algorthm as the MC-DP algorthm. 5. Expermental Results We have mplemented the DOGMA algorthm wth MC-FFD, MC-BFD, MC-MSD heurstcs and the optmal MC-DP method usng the C language and ncorporated our mplementaton nto the SIS package [SeSL9]. We apply the DOGMA algorthm on MCNC combnatonal benchmarks n our experments. These benchmark crcuts are frst optmzed for area and then decomposed nto smple-gate networks usng standard SIS routnes. We apply varous gate decomposton routnes, ncludng DOGMA, to obtan -nput networks whch are fed as nputs to a depth-optmal technology mapper to obtan fnal LUT networks so that we can compare the effects of varous gate decomposton algorthms on the fnal depth of mappng solutons. We choose the LUT nput sze to be K = 5 n the experments. We frst compare the performance of the heurstcs and the optmal method for mn-heght K- feasble bn packng n DOGMA. The test crcuts are flattened (-level) area-optmzed MCNC benchmarks. We decompose them by MC-FFD, MC-BFD, MC-MSD and MC-DP respectvely then use the FlowMap (FM) technology mapper to get mappng solutons. Note that FlowMap obtans optmal depth for the decomposed networks. The mappng results are shown n Table. Some entres under MC-DP+FM are mssng due to the long computaton tme usng pseudo-polynomal tme dynamc programmng. We observe the four methods perform almost equally well. Because MC-FFD s faster than other three methods, the DOGMA algorthm employs the MC-FFD heurstc to solve the mn-heght K-feasble bn packng problem. We now compare the DOGMA algorthm wth two other decomposton routnes: the tech_decomp (td) routne n the SIS package and the dmg routne developed n DAG-Map [ChCD9]. The tech_decomp routne s based on a balanced-tree heurstc whch only mnmzes the gate level locally. The dmg routne mnmzes the gate level of the decomposed networks. In ths experment, the flattened -level MCNC crcuts are decomposed and then fed to CutMap (CM) [CoHw95] to obtan fnal LUT networks. The mappng results are shown n Table. From the results, we see CutMap produces the same or better depth when crcuts are decomposed by the DOGMA algorthm. On average, DOGMA acheves 0% and 4% depth reducton compared to tech_decomp and dmg, respectvely. In the case where the mappng soluton has the same depth, CutMap often obtans a smaller area for crcuts decomposed by DOGMA. Note that CutMap not only guarantees optmal depth (as FlowMap) but also mnmzes area durng the mappng phase. One of the technques used n CutMap for area mnmzaton s explotng tmng slack on non-crtcal paths. We beleve the smaller area n mappng solutons of crcuts decomposed by DOGMA s from the addtonal tmng slacks created by DOGMA and then exploted by CutMap. We now compare our mappng solutons wth prevous results by varous gate decomposton and technology mappng algorthms. Chortle-crf [FrRV9a] uses a set of (mult-level) area- 5

16 MC-FFD+FM MC-BFD+FM MC-MSD+FM MC-DP+FM Crcut LUT depth LUT depth LUT depth LUT depth 5xp sym apex apex clp con duke e msex msex msex rd rd sao vg Total Table Comparson of MC-FFD, MC-BFD, MC-MSD and MC-DP methods n the DOGAM algorthm. td+cm dmg+cm DOGMA+CM Crcut LUT depth LUT depth LUT depth 5xp 4 4 9sym apex apex clp con duke e msex msex 6 msex rd7 5 7 rd sao vg Total % +% +% +4% Table Comparson of tech_decomp, dmg and MC-FFD decomposton methods. optmzed MCNC benchmarks as nput networks n ther experments. These benchmarks, after decomposed by the SIS speed_up routne or the dmg routne, are used by Chortle-d [FrRV9c], FlowMap [CoD94a] and CutMap [CoHw95] as nputs to obtan LUT networks. MIS-pga-delay [MuSB9a] and TechMap-D [SaTh9] also start wth area-optmzed MCNC benchmarks (dfferent 6

17 from those used by Chortle-crf) followed by a delay reducton scrpt and the speed_up routne to get nput networks n ther experments. Both MIS-pga-delay and TechMap-D combne logc synthess wth technology mappng n ther approaches. In our experments, we take the benchmarks used by Chortle-crf as ntal networks, whch are subsequently decomposed by DOGMA and then fed to CutMap to obtan LUT networks. It has been shown n [CoD94a] that FlowMap outperforms Chortle-d and MIS-pga-delay by 5% to 7% n depth and 9% to 50% n area. Therefore we compare our results wth results by FlowMap, TechMap-D and CutMap respectvely n Table. The correspondng gate decomposton and technology mappng methods taken by these approaches are lsted n the second row. Comparng our results wth results n [CoHw95], we can see that speed_up and DOGMA decompose gates equally well n terms of depth. However, 5% less LUTs are generated by CutMap for crcuts decomposed by DOGMA than by speed_up. It ndcates DOGMA s an area-effcent gate decomposton algorthm for depth-optmal technology mappng. Comparng wth the results n [CoD94a], we see dmg performs farly well even though ts decomposton goal s to mnmze the level n the decomposed network rather than the fnal LUT network. Among four data sets n Table, TechMap-D obtans the smallest overall depth (partly due to resynthess) at an expense of a large area overhead. On average, speed_up + TechMap-D generates 5% more LUTs than DOGMA + CM. [SaTh9] [CoHw95] [CoD94a] Ours speed_up+tm-d speed_up+cm dmg+fm DOGMA+CM Crcut LUT depth LUT depth LUT depth LUT depth 5xp 7 4 9sym symml C C alu apex apex count des duke msex rd rot vg z4ml Total % -7% +6% -% +% +% Table Comparson wth recent prevous results. 7

18 6. Concluson In ths paper, we present an n-depth study of the structural gate decomposton problem for depth-optmal technology mappng n LUT-based FPGA. We show that further decomposton of a K-bounded network often decreases the depth n the fnal mappng solutons. Therefore t s always benefcal to decompose a K-bounded network nto a -nput network for depth-optmal mappng. We also show the problem of gate decomposton for depth-optmal technology mappng s NP-hard for fann-unbounded networks when K! and remans NP-hard for K-bounded networks when K! 5. We present a novel algorthm, named DOGMA, for decomposng gates n a gven unbounded or K- bounded network such that a depth-optmal technology mapper can acheve a mnmal depth n the fnal mappng soluton. A key step n DOGMA s to solve the mn-heght K-feasble bn packng problem, for whch we have developed both effcent heurstcs and an optmal algorthm. The expermental results show that DOGMA outperforms some exstng gate decomposton methods, such as tech_decomp and dmg, by an average of 4% to 0% n depth n the fnal LUT mappng solutons wth almost dentcal LUT usage. We compare the mappng results obtaned by DOGMA + CutMap wth results by other gate decomposton and technology mappng algorthms. comparson ndcates that networks decomposed by DOGMA allow the depth-optmal mapper CutMap to produce LUT networks of depth comparable to prevous approaches whle wth an average of 6% to 5% reducton n area. Acknowledgement Ths work s partally supported by NSF Young Investgator (NYI) Award MIP-95758, and grants from Xlnx, Xerox PARC, and AT&T under NSF NYI and Calforna MICRO programs. The authors would lke to thank Dr. Robert Francs for hs helpful dscussons. References [BrRS87] [ChCD9] Brayton, R. K., R. Rudell, and A. L. Sangovann-Vncentell, MIS: A Multple-Level Logc Optmzaton, IEEE Transactons on CAD, pp , Nov Chen, K. C., J. Cong, Y. Dng, A. B. Kahng, and P. Tramar, DAG-Map: Graph-based FPGA Technology Mappng for Delay Optmzaton, IEEE Desgn and Test of Computers, pp. 7-0, Sep. 99. [CoD94a] Cong, J. and Y. Dng, An Optmal Technology Mappng Algorthm for Delay Optmzaton n Lookup-Table Based FPGA Desgns, IEEE Trans. on Computer-Aded Desgn, Vol., pp. -, Jan [CoHw95] Cong, J. and Y.-Y. Hwang, Smultaneous Depth and Area Mnmzaton n LUT-Based FPGA Mappng, Proc. ACM rd Int l Symp. on Feld Programmable Gate Arrays, pp , Feb [FrRV9a] Francs, R. J., J. Rose, and Z. Vranesc, Chortle-crf: Fast Technology Mappng for Lookup Table-Based FPGAs, Proc. 8th ACM/IEEE Desgn Automaton Conference, pp. 6-69, June 99. The 8

19 [FrRV9c] Francs, R. J., J. Rose, and Z. Vranesc, Technology Mappng of Lookup Table-Based FPGAs for Performance, Proc. IEEE Int l Conf. on Computer-Aded Desgn, pp , Nov. 99. [GaJo79] [HoSa78] Garey, M. and D. Johnson, Computer and Intractablty: A Gude to the Theory of NP- Completeness, Freeman, San Francsco (979). Horowtz, E. and S. Sahn, Fundamentals of Computer Algorthms, Computer Scence Press, Potomac, Maryland (978). [LaPP94] La, Y.-T., K.-R. R. Pan, and M. Pedram, FPGA Synthess usng Functon Decomposton, Proc. Int l Conf. on Computer Desgn: VLSI n Computers, pp. 0-5, Oct [MuSB9a] [SaTh9] [SaTh9] Murga, R., N. Shenoy, R. K. Brayton, and A. Sangovann-Vncentell, Performance Drected Synthess for Table Look Up Programmable Gate Arrays, Proc. IEEE Int l Conf. on Computer-Aded Desgn, pp , Nov. 99. Sawkar, P. and D. Thomas, Technology Mappng for Table-Look-Up Based Feld Programmable Gate Arrays, ACM/SIGDA Workshop on Feld Programmable Gate Arrays, pp. 8-88, Feb. 99. Sawkar, P. and D. Thomas, Performance Drected Technology Mappng for Look-Up Table Based FPGAs, Proc. 0th ACM/IEEE Desgn Automaton Conf., pp. 08-, June 99. [SeSL9] Sentovch, E., K. Sngh, L. Lavagno, C. Moon, R. Murga, A. Saldanha, H. Savo, P. Stephen, R. Brayton, and A. Sangovann-Vncentell, SIS: A System for Sequental Crcut Synthess, U.C.Berkeley Techncal Report UCB/ERL M9/4, May, 99. [Wa89] Wang, A., Algorthms for Mult-level Logc Optmzaton, U.C.Berkeley Memorandum No. UCB/ERL M89/50, Aprl 989. [WuEA95] Wurth, B., K. Eckl, and K. Antrech, Functonal Multple-Output Decomposton: Theory and an Implct Algorthm, Proc. ACM/IEEE Desgn Automaton Conf., pp , Jun

20 Appendx The nternal nodes n N (x ) (except s ) have a mappng depth of. Node s has a mappng depth of. The mnmum mappng depth for nodes x and x depends on how s s decomposed. The mappng depth for nodes q k ( " k " K $5) n C s snce each of them s the root of a complete - level K-ary tree wth PI nodes as leaves. Smlarly, the mappng depth for nodes r k ( " k " K $) s. The mnmum mappng depth for node C depends on the mappng depth of ts three lteral nodes. The mnmum mappng depth of N (F) after decomposton s the maxmal mappng depth among all decomposed C nodes. Hence MMD (N (F)) depends on how the node s s decomposed n every N (x ). Before we prove Theorems and, we ntroduce the followng lemmas. Lemma 7 For the subnetwork N (x ) n Fgure 5 and ts decomposton D (N (x )), f MMD D (N (x ))(x ) = then MMD D (N (x ))(x ) =, and f MMD D (N (x ))(x ) = then MMD D (N (x ))(x ) =. Proof The mnmum mappng depth for nodes x and x after decomposton depends on how the node s s decomposed n N (x ). We denoted the decomposed N (x ) as D (N (x )). Fgure 0 shows all the sx possble decompostons of s. In Fgure 0(a) and (b), nodes PI and PI are merged nto a node t followed by dfferent subsequent decomposton steps. It creates a mn-cut of heght for node s wth the cutset { w,w,t }. Together wth nodes v through v K $, a K-feasble cut of heght exsts for node x. Hence MMD D (N (x ))(x ) =. On the other hand, the mn-cut of heght for node x has a sze of K +. Hence MMD D (N (x ))(x ) =. In Fgure 0(c), the mn-cut of heght for node s has a sze of 4. As a result both MMD D (N (x ))(x ) = and MMD D (N (x ))(x ) =. In Fgure 0(d) and (e), the mn-cut of heght for node s has a sze of. However, due to the edge connecton (w,x ) shown n Fgure 5, the mn-cut of heght for node x stll has a sze of K +. As a result both MMD D (N (x ))(x ) = and MMD D (N (x ))(x ) =. In Fgure 0(f), node w s merged wth PI and w s merged wth PI whch create a -feasble cut of heght for node s. Together wth nodes u through u K $, node x has a K-feasble cut of heght. Hence MMD D (N (x ))(x ) =. However, both nodes w and w connects to node x. As a result the mn-cut of heght for node x after decomposton has a sze of K +. So MMD D (N (x ))(x) =. From the above enumeraton of decompostons for node s, t s easy to see Lemma 7 holds. Lemma 8 For the subnetwork N (x ) shown n Fgure 5, there exst a -feasble cut of heght for node x after decomposton. Smlarly, there exsts a -feasble cut of heght for node x after decomposton. Proof In nput (x ) shown n Fgure 5, K $ nodes has a mappng depth of whle the node s has a depth of. The K $ nodes can be merged nto a node whch has a mnmum mappng depth of. As a result, there exsts a -feasble cut of heght for node x. Smlarly, there exsts a -feasble cut of heght for node x. The -feasble cut of heght s llustrated n Fgure 6(b) and 8(b) by nodes (labeled ) respectvely. 0

21 K- PI s K- PI s PI PI K- PI s K- PI s PI PI K- PI s K- PI s PI PI w w t w w t w w t t t s (a) (b) (c) s t s K- PI s K- PI s PI PI K- PI s K- PI s PI PI K- PI s K- PI s PI PI w w w w w w t t t t t t s (d) s (e) s (f) Fgure 0 Sx dfferent ways to decompose node s n N (x ). In (a) and (b), MMD (x ) = and MMD (x ) = ; In (c), (d) and (e), MMD (x ) = MMD (x ) = ; Only n (f) MMD (x ) = and MMD (x ) =. Lemma 9 For the network N (C ) n Fgure 6(a) and ts decomposton D (N (C )), MMD D (N (C ))(C ) = 4 f and only f MMD D (N (C ))(l k ) = for some " k ". Proof In the subnetwork N (C ), nput (C ) contans K $ 5 nodes of mnmum mappng depth, K $ nodes of mnmum mappng depth, and lteral nodes l,l,l. Let D (N (C )) denoted the decomposed N (C ). Consder the followng two cases n computng MMD D (N (C ))(C ). Frst, assume MMD D (N (C ))(l ) = MMD D (N (C ))(l ) = MMD D (N (C ))(l ) =. Accordng to Lemma 8, there exst - feasble cuts of heght for nodes l,l,l respectvely. Together wth the K $ 5 nodes of depth n nput (C ), the cut of heght wll have a sze of K +. As a result, durng the decomposton of C, nodes of mappng depth have to be ntroduced n order to merge the K + nodes n the cut of heght. But there are already K $ nodes of mappng depth n nput (C ). Hence the mn-cut of heght for C n D (N (C )) have a sze of K +. Ths wll cause MMD D (N (C ))(C ) = 5. On the other hand, f at least one lteral node has a depth of, as shown n the example n Fgure 6(b)) where the node l has a mappng depth of, only nodes of mappng depth wll be ntroduced. Together wth the K $ nodes of depth n nput (c), MMD D (N (C ))(C ) = 4 can be obtaned.

22 Lemma 0 The decomposed network D (N (F)) of N (F) satsfes MMD (D (N (F))) = 4 f and only f there s a way to decompose the node s n every N (x ) such that for all N (C ) at least one lteral node of N (C ) has a mappng depth of. Proof The mnmum mappng depth of N (F) after decomposton s the maxmal mappng depth among m output nodes C,C,...,C m. In order to have a mappng depth of 4, t must be the case MMD D (N (C ))(C ) = 4 for " " m. Accordng to Lemma 9, at least one of the lteral nodes n each N (C ) must have a mappng depth of. Lteral nodes are from ether node x or node x of some network N (x ). From Lemma 7, only one of them can have a mappng depth of. Whch one has a depth of depends on whether the node s s decomposed n the way specfed by Fgure 0(a) or Fgure 0(f). As a result the mnmum mappng depth of N (F) after decomposton would be 4 f and only f there s a way to decompose the node s n each N (x ) such that at least one lteral node n each N (C ) has a mappng depth of. Ths proves Lemma 0. [Proof of Lemma 5] We lnk the truth assgnment of Boolean varables n an nstance F of SAT to the decomposton of gate s n all N (x ) as follows: a Boolean varable x = n SAT f MMD D (N (x ))(x ) =, and x = n SAT f MMD D (N (x ))(x ) =. It leads to the concluson that the SAT nstance F s satsfable f and only f MMD (N (F)) = 4. Ths proves Lemma 5. [Proof of Theorem ] The transformaton from an nstance F of SAT to the network N (F) takes O (K (n + m)) tme. The network N (F) conssts of subnetworks N (x ) for " " n and subnetworks N (C ) for " " m. Every N (x ) and N (C ) s well-defned for K!. From Lemma 5, F s satsfable f and only f N (F) has a decomposton D (N (F)) such that MMD (D (N (F))) = 4. As a result the SGD/K-D problem s NP-Complete. Hence the SGD/K problem s NP-hard. [Proof of Lemma 6] If all the three lteral nodes l,l,l have a mnmum mappng depth of after decomposton, accordng to Lemma 8, the mn-cut of heght for node C wll have a sze of K $ = K +. Hence node C has a mnmum mappng depth of 4 after decomposton. On the other hand, f at least one of the lteral node has a mappng depth of, a K-feasble cut of heght exsts for node C. An example s shown n Fgure 8(b) where the lteral node l s assumed a mappng depth of after decomposton. As a result node C has a mappng depth of after decomposton f and only f at least one of the three lteral nodes l,l,l has a mappng depth of after decomposton. Ths proves Lemma 6. [Proof of Theorem ] Smlar to the proof of Theorem and omtted.