Two NP-hard Interchangeable Terminal Problems*

Transcription

1 o NP-hard Interchangeable erminal Problems* Sartaj Sahni and San-Yuan Wu Universit of Minnesota ABSRAC o subproblems that arise hen routing channels ith interchangeable terminals are shon to be NP-hard hese problems are: P1: Is there a net to terminal assignment that results in an acclic vertical constraint graph? P2: or instances ith acclic vertical constraint graphs, obtain net to terminal assignments for hich the length of the longest path in the vertical constraint graph is minimum Ke ords and phrases Interchangeable terminals, NP-hard, PLA-routing, Compleit *his research as supported, in part, b the National Science oundation under grant MCS

2 1 Introduction he problem of routing channels ith interchangeable terminals arises, for eample, hen the cells on either side of the channel are programmable logic cells (eg ROMs and PLAs) An instance of this problem is shon in igure 1(a) his instance consists of eight cells A H Cells A D are on one side (top/upper) of the channel hile cells E H are on the other (bottom/loer) On the top side, terminals 1 4 are in cell A While terminals on the bottom side of the channel are in cell H he nets to be assigned to the terminals of a cell are given in braces hus, the nets for cell C s five terminals are NES(C)={1,1,3,4,5} Since the terminals in a cell are interchangeable, all net to terminal assignments are permissible igures 1(b) and 1(c) sho to possible assignments of nets to terminals With an assignment of nets to terminals, e ma associate a vertical constraint graph (VCG) his graph has one verte for each net In addition, the directed edge <i, j > is an edge of the VCG iff there is a k such that net i is assigned to terminal k on the top side of the channel and net j is assigned to terminal k on the bottom side of channel he VCGs for the assignments of igures 1(b) and (c) are shon in igures 1(d) and (e), respectivel Man channel routing algorithms require the VCG to be acclic uther, the height of the VCG (ie, the length of the longest path in the VCG) is a loer bound on the number of horiontal tracks needed to route the channel hen to routing laers ( one for horiontal and the other for vertical routes ) are available Keeping these factors in mind, Kobaashi and Drod [KOBA85] have proposed a three step method to assign nets to terminals he three steps are: 1 Permute the terminals in each cell so as to maimie the number of aligned terminal pairs 2 Echange terminals that are in nonaligned pairs so as to remove cclic constraints in the resulting vertical constraint graph 3 Echange terminals so as to reduce the height of the vertical constraint graph Lin and Sahni [ LIN86 ] have developed a linear time algorithm for step 1 In this paper, e sho that the remaining to steps are NP-hard We sho this b reducing the knon NPcomplete problem monotone 3SA to each of these Monotone 3SA [ GARE79] INPU:Set V of variables, collection C of clauses over V such that each clause cεc has c = 3 and contains either onl negated variables or onl un-negated variables QUESION:Is there a satisfing truth assignment for C? 2 Basic constructs In this section e develop some interchangeable terminal instances that ill be used in our proofs We sho ho the possible terminal assignments for these instances ma be interpreted as truth assignments for logical variables 21 ruth Assignment Bo A truth assignment bo for a variable consists of to cells A and B on the upper side of the channel and one, C, on the loer side ( see igure 2(a) ) Cells A and B each have one terminal hile cell C has to NES(A)= { }, NES(B)= { }, and NES(C)= {, } here are eactl to as in hich the nets can be assigned to the terminals of A, B, and C hese are

3 2 shon in igures 2(b) and (c) We shall interpret the assignment of igure 2(b) as setting variable to true and to false his assignment is called a true assignment he assignment of igure 2(c), called a false assignment, corresponds to setting to false and to true he VCG corresponding to the assignment of igure 2(b) is shon in igure 2(d) and that corresponding to the assignment of igure 2(c) is shon in igure 2(e) A truth assignment bo for variable, AB( ), ill be dran schematicall as in igure 2(f) We ill refer to the four terminals as AB(), AB(), AB(), and AB() A {} B { } {,} C (a) (b) (c) (d) AB()=1 (e) AB()=0 (f) AB() set to true set to false igure 2: ruth assignment bo A {} B { } {,} C (a) (b) (c) (d)iab()=1 (e)iab()=0 (f)iab() set to true set to false igure 3:Inverted truth assignment bo 22 Inverted ruth Assignment Bo An inverted truth assignment bo, IAB, is identical to a AB ecept that the schematic is dran upside don ( see igure 3 ) he upside don draing of the schematic results in cleaner figures

4 23 AB Connection Bo A AB connection bo is used to connect the VCGs of to ABs Consider the to ABs 3 Cell A {1,2,3,4} Cell B {2,3,1,4} Cell C {1,1,3,4,5} Cell D {1,4} {1,2,3,3,4} Cell E {2,4,5} Cell {1,2,3} Cell G {1,2,4,5} Cell H (a) (b) (c) (d) VCG for (b) (e) VCG for (c) igure 1: An instance of the interchangeable terminal routing problem

5 4 of igure 4(a) he connection bo for these consists of four cells D, E,, and G as shon in igure 4(b) here are four possible VCGs for the interchangeable terminal instance that consists of the to ABs and the connection bo hese correspond to the four terminal assignments ( and are true ),,, and and are shon in igure 5 igure 5(e) shos the schematic for to ABs and a connection bo >rom igure 5, e observe that the terminal assignment results in a VCG ith a ccle hile the VCG for the remaining three terminal assignments are acclic D { } E { } (a) { } (b) {} G igure 4: AB Connection Bo (a) (b) (c) (d) igure 5 (e) schematic 24 AB Chain A AB chain consists of k, k 1, ABs, AB( 1 ),, AB( k ) ith a connection bo for each pair of ABs, ( AB( i ), AB( +1 ) ), 1 i < k he schematic for this is shon in igure 6 he folloing propert is an immediate consequence of our discussion of the preceding section

6 5 Propert 1:he VCG for a AB chain is acclic iff one of the folloing is true: a) he terminal assignment for all ABs is true b) he terminal assignment for all ABs is false c) here eists a j, 1 j<k, such that the terminal assignment for AB( i ), 1 i j is false and that for the remaning ABs is true k 6 ABs igure 6: AB Chain 25 IAB connection bo he connection bo for to inverted truth assignment boes IAB() and IAB() is similar to that for to ABs and is shon in igure 7 he schematic for the to IABs and their connection bo is shon in igure 8(e) igures 8(a)-(d) sho the VCGs for the four possible terminal assignments he VCG for the assignment contains a ccle he remaning three VCGs are acclic D { } E { } (a) { } (b) {} G igure 7: IAB Connection Bo 26 IAB Chain his is just k IABs and k 1 IAB connection boes he schematic is shon in igure 9 he folloing propert is readil verified Propert 2:he VCG for an IAB chain is acclic iff one of the folloing is true: a) All IABs are assigned true b) All IABs are assigned false

7 6 (a) (b) (c) (d) igure 8 (e) schematic c) here eists a j, 1 j < k, such that the terminal assignment for IAB( i ), 1 i j, is false and that for the remaining IABs is true k 6 IABs igure 9: IAB Chain 27 AB to IAB Isolation Connection An isolation connection is used to connect together a truth assignment bo AB() and an inverted truth assignment bo IAB() he connection is established using si cells as shon in igure 10(a) As can be seen from igures 10(b)-(e), if the AB and IAB have different truth assignments, then the VCG has a ccle igure 11(a) shos the schematic for a AB chain and an IAB chain he rightmost AB is connected to the leftmost IAB using an isolation connection Propert 3: Assume that a AB and IAB chain are connected using an isolation connection as in igure 11(a) If the VCG for the resulting chain has no ccles, then the folloing are true: (i) (ii) he VCG has no path that begins in a AB other than the rightmost one and ends in an IAB other than the leftmost IAB he VCG has no path that begins in an IAB other than the leftmost one and ends in a AB other than the rightmost one In other ords, an isolation connection isolates the ABs from the IABs and vice versa

8 7 Proof: >rom igures 10(b)-(e), it follos that if the VCG of igure 11(a) has no ccle, then the rightmost AB and leftmost IAB have the same truth assignment Hence there are onl to possibilities for the VCG for the rightmost AB, the leftmost IAB, and the isolation connection hus, this part of the VCG has one of the forms given in igures 11(b) and (c) As can be seen, there can be no path that violates (i) or (ii) C 1 { } C 2 { } C 3 { } { } { } C 4 C 5 {} C 6 (a) (b) (c) (d) (e) IGURE 10: Isolation connection (f) schematic 28 IAB to AB Isolation Connection his is shon in igure 12 It has the same properties as a AB to IAB isolaion connection 3 Acclic VCG ( Problem P1 ) is NP-hard We sho this b reducing monotone 3SA to P1 Specificall, e sho ho to construct an instance P (S) of P1 for an given instance S of monotone 3SA P (S) has the folloing properties: (i) (ii) P (S) ma be constructed from S in polnomial time P (S) has a terminal assignment ith an acclic VCG iff S is satisfiable Properties (i) and (ii) impl that P1 is NP-hard Actuall, P1 is readil seen to be solvable

9 8 (a) schematic (b) (c) IGURE 11: Isolation from AB chain to IAB chain (a) schematic (b) (c) IGURE 12: Isolation from IAB chain to AB chain in nondeterministic polnomial time Hence this observation together ith our NP-hard proof impl that P1 is NP-complete

10 9 31 Construction of P (S) Let S be an instance of MONOONE 3SA Let m and k, respectivel, be the number of variables and clauses in S he clauses in S ma be partitioned into to sets U and N U contains all clauses that contain onl unnegated variables and N contains all clauses that contain onl negated variables Let u and n, respectivel, be the number of clauses in U and N Clearl, u +n = k As an eample, consider the case: S = ( A+B+C ) ( A+C+D ) ( B+C+D ) or this, U = { (A+B+C), (A+C+D) }, N = {(B+C+D) }, u = 2, and n = 1 or each of the m variables in S, construct a ro of ABs and IABs Each such ro consists of k ( note k is the number of clauses in S ) AB and IAB chains connected b AB to IAB or IAB to AB isolation connections as appropriate Specificall, each ro is a AB chain ith to ABs folloed b an IAB chain ith 3 IABs; folloed b a AB chain ith 3 ABs; folloed b an IAB chain ith 3 IABs; etc or our eample S, m = 4 and k = 3 So e ill have 4 ros of ABs and IABs ith each ro consisting of a AB chain ith 2 ABs; folloed b an IAB chain ith 3 IABs; folloed b a AB chain ith 2 ABs his is shon in igure 13 Note that AB to IAB isolation connections are used to connect the first AB chain of each ro to the IAB chain in that ro and that IAB to AB isolation connections are used to connect the IAB chain in each ro to the second AB chain in that ro or each i, 1 i k, the truth boes in column 3i 2 of the above construction ill be used to represent a distinct clause of S he first u of these columns ( ie, columns 1, 4, 7,, 3u 2 ) ill represent the u unnegated clauses of S hile the last n of these columns ( ie, columns 3u +1, 3u +4,, 3k 2 ) ill represent the n negated clauses Let U 1, U 2,, U u be the u unnegated clauses Let U i = ( i 1 + i 2 + i 3 ) Without loss of generalit, e ma assume that the AB/IAB ro for variable i 1 is above that for i 2, hich in turn is above that for i 3 Column 3i 2 of the truth bo construction of igure 13 is used to represent U i If i is odd, this column contains ABs Otherise, it contains IABs We introduce the three edges < AB( i 1 ), AB( i 2 ) i 2 >, < AB( i 2 ), AB( i 3 ) i 3 >, < AB( i 3 ), AB( i 1 ) i 1 > in case this is a column of ABs and the three edges < IAB( i 1 ), IAB( i 3 ) i 3 >, < IAB( i 3 ), IAB( i 2 ) i 2 >, < IAB( i 2 ), IAB( i 1 ) i 1 > in case this is a column of IABs Let N 1, N 2,, N n be the n negated clauses Let N i = ( i 1 + i 2 + i 3 ) Once again, e assume that the ro for i 1 is above that for i 2 hich is above that for i 3 Column 3(u +i) 2 is used to represent N i Into this column, the three edges < AB( i 1 ), AB( i 2 ) i 2 >, < AB( i 2 ), AB( i 3 ) i 3 >, < AB( i 3 ), AB( i 1 ) i 1 > are introduced if this is a column of ABs he three edges < IAB( i 1 ), IAB( i 3 ) i 3 >, < IAB( i 3 ), IAB( i 2 ) i 2 >, < IAB( i 2 ), IAB( i 1 ) i 1 > are introduced if this is a column of IABs he resulting edges for our eample are shon in igure 13 Recall that to introduce an edge <a, b> into the VCG, e need merel add to cells to the terminal assignment instance Each of these has eactl one terminal; one cell is above the other; the upper cell has net a; the loer cell has net b uther, note that hile e have used the smbols and for the loer nets of all ABs and the upper nets of all IABs, the intent is that all these nets are different his

11 10 A B C D IGURE 13: = (A +B +C) (D +C +A) (B +C +D) completes the construction of P (S) Propert 4:If all the literals in clause i are false, then column 3i 2, 1 i k of P (S) contains a ccle 32 Correctness Proof We need to sho that P (S) as constructed in Section 31 has the to properties listed in Section 3 It is obvious that propert (i) ( ie, P (S) ma be constructed in polnomial time ) is satisfied or propert (ii), first suppose that S is satisfiable Consider an truth assignment that satisfies S Use this truth assignment for all the truth boes ( ABs and IABs ) in the construction >rom properties 1-4, it follos that the resulting VCG contains no ccles Note that because of the isolation propert (propert 3) there can be no path that includes boes from to or more clause columns Net, suppose that P (S) has a terminal assignment that contains no ccles >rom Propert

12 11 4, it follos that this assignment corresponds to a truth assignment in hich ever clause has at least one literal set to true Hoever, since the truth assignment along a AB or IAB chain ma change from false to true ithout introducing a ccle ( Properties 1 and 2 ), it is possible that the terminal assignment that results in an acclic VCG results in an inconsistent truth assignment to the variables of S ( ie, a variable is false in some clauses and true in others ) If this is the case, the truth assignment ma be made consistent in the folloing a: Suppose j is false in the first q of the truth boes in columns 3i 2, 1 i k ( onl those boes in the ro for j are considered ) and true in the remaining k q boes in columns 3i 2, 1 i k (a) (b) If q u, then set j to true and ( j to false ) in all the clauses in hich j or j appear his makes the truth assignment for j the same in all clauses and does not reduce the number of true literals in an clause If q > u, then set j to false ( and j to true ) his does not reduce the number of true literals in an clause and results in a consistent truth assignment for j Hence, if there is a terminal assignment for P (S) that results in an acclic VCG, S is satisfiable A t 1 b 2 ẉ B C t 3 b 1 D IGURE 14: = (A +B +C) (D +C +A) (B t 2 ẉ +C+D) b 3

13 12 4 Minimum Height VCG (Problem P2) Is NP-hard We actuall prove a stronger result than this in this section We sho that the corresponding decision problem (P3) is NP-complete: P3: Let Q be a terminal assignment instance ith an acclic VCG and r an integer Does Q have a terminal assignment for hich the length of the longest path in the VCG is < r? P3 is easil seen to be solvable in nondeterministic polnomial time So e need onl sho that P3 is NP-hard or this, e sho ho to construct an instance Q (S) of P3 corresponding to an instance S of monotone 3SA his instance Q (S) satisfies properties (i) and (ii) of Section 3 Once e kno that the decision problem P3 is NP-complete, e can conclude that the optimiation problem P2 is NP-hard 41 Construction of Q (S) or an instance S of monotone 3SA, e first construct P (S) as in Section 3 Net, e replace the edges introduced for clauses b edges ith eight as shon in igure 14 he cclic clause constructs of igure 13 are replaced b path constructs Each clause is represented b a ( possible ) path from a nel introduced top verte t i to a nel introduced bottom verte b i Each of the eighted edges of igure 14 is reall a path of length and is realied b the construct of igure 15 Let Q (S) be the instance constructed in igure 14 {A} {v 1 } {v 2 } {v 2 }{v 1 } {v 1 } {v 2 } {v 3 } {v 1 } {B} (a) instance (b) notation IGURE 15: Weighted edge ith eight = 42 Correctness Proof Consider a segment of one ro of Q (S) his segement consists of five truth boes as shon in igure 16 ( the case hen the middle bo is a AB is similar ) As can be seen, no path confined to a ro of truth boes can have length larger than 9 (unless the ro contains a ccle ) Suppose that 9 If P (S) has no acclic terminal assignment, then ever terminal assignment for Q (S) has at least one path from a t i to a b i his path has length 4 +3 If P (S) has an acclic terminal assignment, then the corresponding assignment in Q (S) results in a VCG for hich the maimum path length is no more than = < 4 +3

14 13 Hence, Q (S) has a terminal assignment for hich the length of the longest path in the VCG is < 4 +3 iff P (S) has a terminal assignment for hich the VCG is acclic his, in turn, is possible if and onl if S is satisfiable

15 14 (a) schematic (b) (c) (d) (e) IGURE 16

16 15 5 Conclusions We have shon that both problems P1 and P2 are NP-hard Consequentl, it is ver unlikel that either of these is solvable in polnomial deterministic time Our results motivate the stud of fast heuristic methods for these to problems 6 References [BURS83] M Burstein and R Pelavin, "Hierarchical ire routing", IEEE rans on Computer Aided Design of Integrated Circuits and Sstems, Vol CAD-2, No 4, Oct 1983 [DEU76] D N Deutsh, "A Dogleg channel router", Proc 13th Design automation conference, pp (1976) [GARE79] M R Gare and D S Johnson, "Computers and Intractabilit", W H reeman and Compan, 1979 [KOBA85]H Kobaashi and C E Drod, "Efficient algorithms for routing interchangeable terminals", IEEE rans on Computer Aided Design of Integrated Circuits and Sstems, Vol CAD-4, No 3, pp , (1985) [LIN86] L Lin and S Sahni, "Maimum alignment of interchangeable terminals", Universit of Minnesota, echnical Report # 86-13, 1986 [RIVE82] R L Rivest and C M iduccia, "A greed channel router", Proc 19th Design automation conference, pp (1982) [YOSH82] Yoshimura and E S Kuh, "Efficient algorithms for channel routing", IEEE rans on Computer Aided Design of Integrated Circuits and Sstems, Vol CAD-1, No 1, pp 25-35, Jan