EFFICIENT SOLUTION OF SYSTEMS OF ORIENTATION CONSTRAINTS

Transcription

1 EFFICIENT SOLUTION OF SSTEMS OF OIENTATION CONSTAINTS Joseh L. Ganley Cadence Design Systems, Inc. ABSTACT One sbtask in constraint-drien lacement is enforcing a set of orientation constraints on the deices being laced. Sch constraints are created in order to, for examle, imlement matching constraints or enforce reglarity among members of an array of deices. Here we resent an efficient algorithm for soling systems of discrete orientation constraints. The algorithm handles oerconstraints by selectiely relaxing constraints ntil the remaining set can all be simltaneosly enforced. The algorithm rns in linear time in the absence of oerconstraints. 1. INTODUCTION Part of the constraint-drien lacement roblem is finding a set of orientations for the deices being laced that satisfies certain orientation constraints. In this aer we resent an efficient algorithm that soles this roblem. It has been imlemented in the rectilinear metric for for tyes of orientation constraints, and can be easily extended to other metrics and other tyes of orientation constraints. For clarity of exosition, we focs on the rectilinear model, in which only horizontal and ertical lines are allowed, since this is the model we se in or imlementation and nder which most integrated circits are fabricated. (Note, howeer, that or algorithm is alicable to any metric in which the orientation of a deice can be described by a finite seqence of indeendent rotations and/or reflections.) In the rectilinear model, there are eight ossible orientations of a deice, which are shown in Figre 1 (we se the standard Cadence nomenclatre to describe them). With resect to these orientations, the following constraints are defined: AsingledeicehasFIED OIENTATION set to one of the eight orientations. Eery deice in a set of two or more mst hae the SAME OIENTATION. A air of deices mst hae MIOED OIENTATION abot either the x-ory-axis. Eery deice in a set of two or more mst hae either the SAME O MIOED OIENTATION (if mirrored, then abot either the x-ory-axis). 0 M M M90 M90 Figre 1. The eight ossible orientations. Here we resent an efficient algorithm that ses grah coloring to sole the following roblem: gien a set of deices and a set of constraints on these deices, where each constraint is one of the for described aboe, choose an orientation for each deice sch that all the constraints are satisfied (if ossible; see below). If one is only concerned with the first three of these tyes of constraints, then a simle iteratie algorithm that roagates orientations ot from the deices with fixed orientations sffices to sole this roblem. Howeer, not only is the coloring algorithm more efficient than sch an iteratie algorithm, bt it also handles SAME O MIOED OIENTATION constraints. The iteratie techniqe is insfficient to handle sch constraints; details are gien in the next section. Sometimes all constraints cannot be satisfied simltaneosly. For examle, sose two deices are members of both a SAME OIENTATION and a MIOED OIENTATION constraint; clearly these two constraints cannot be simltaneosly satisfied. Sch a condition is called an oerconstraint, and it is handled by relaxing (i.e. not enforcing) one or more of the constraints inoled in the oerconstraint, sch that the remaining constraints can be simltaneosly satisfied. The coloring algorithm, nlike the reios iteratie algorithm, allows total control oer the selection of which constraints to relax in order to resole oerconstraints. 2. PEVIOUS WOK To or knowledge, the literatre contains no reios work on this roblem. In the first release of or lacement rodct, soling orientation constraints was accomlished sing a simle iteratie algorithm. This algorithm roceeds as follows: Initially the orientation of eery deice is set to ndefined. Next the orientations of those deices on which there is a FIED OIENTATION constraint are set aroriately. The algorithm then reeats the following orientation roagation oeration: find an as-yet-nrocessed constraint, some of whose members orientations hae already been set, and set the orientations of the remaining deices in the constraint aroriately. If no sch constraint exists, then ick a deice whose orientation is ndefined and set its orientation arbitrarily, and reeat the orientation roagation loo. As mentioned aboe, this algorithm is sfficient to handle the

2 A Fixed (0) Same or? B C? Same or D Fixed (0) Figre 2. A Test Case for the Iteratie Algorithm. first three tyes of constraints, bt it faces a roblem when resented with a SAME O MIOED OIENTATION constraint. Within the local context of the constraints and deices being considered in each orientation roagation ste, there is insfficient information with which to choose whether to enforce a SAME O MIOED OIENTATION constraint as a SAME OIENTATION constraint or as a MIOED OIENTATION constraint. In or imlementation, one of the two was arbitrarily chosen; howeer, this can lead to sitations where no soltion is fond to a system of constraints that does, in fact, hae a soltion satisfying all constraints. Consider Figre 2, which illstrates a set of constraints on for deices A, B, C, andd. The algorithm begins by setting deices A and D to orientation 0. Howeer, it mst then decide whether to enforce each of the SAME O MIOED OIENTATION constraints as a SAME OI- ENTATION constraint or as a MIOED OIENTATION constraint. De to the MIOED OIENTATIONconstraint between B and C, the two SAME O MIOED OIENTATION constraints mst be enforced oositely one as a SAME OIENTATIONconstraint and the other as a MIOED OIENTATION constraint bt there is no way to determine this while considering only the constraints inoling deices whose orientations hae already been determined. One can easily deise a strategy to deal with this articlar test case, bt we can constrct a similar bt longer chain of SAME OI- ENTATION, MIOED OIENTATION,andSAME O MIOED OIENTATION constraints, terminated at the ends by FIED OI- ENTATION constraints, to defeat any sch ad hoc strategy. The iteratie algorithm also deals oorly with oerconstraints. The way they are handled in or imlementation was that if the orientation of a deice is already set, and a constraint now being considered wold force that deice to hae an orientation different from its crrent one, then there is an oerconstraint and the crrent constraint is relaxed. The iteratie algorithm has time comlexity O((n + m) 2 ), where n is the nmber of deices and m is the nmber of constraints. This rnning time cold robably be imroed by the se of sitable data strctres, bt in light of or new algorithm this hardly seems worthwhile. 3. THE COLOING ALGOITHM The key obseration leading to or coloring algorithm is that the orientation of a deice in the rectilinear metric can be reresented by the indeendent alication or nonalication of three oerations: otate the deice 90 degrees conterclockwise (90). Mirror the deice abot the x-axis (M). Mirror the deice abot the y-axis (M). If two or more of these oerations are alied, then they mst be alied in this order; i.e., 90 is alied before M or M, and M is alied before M. These oerations, if alied, start from the defalt orientation 0. Ths, each of the eight ossible orientations can be recisely and niqely described by choosing whether 90? M? M? 0 No No No M No es No M No No es 180 No es es 90 es No No 270 es es es M90 es es No M90 es No es Table 1. Oerations alied to rodce each orientation. or not to aly each of these oerations. Table 1 shows, for each of the orientations shown in Figre 1, which of these oerations are alied to rodce that orientation. We will now constrct three grahs. A grah G =(V; E) consists of a collection V of ertices and a collection E of edges, where each edge has two ertices and as endoints. Sch an edge is denoted (; ). Edges are ndirected, so (; ) and (; ) denote the same edge between ertices and. If edge (; ) is in E, then we say that and are adjacent and are one another s neighbors. The grahs will be colored sing two colors: red and green. A coloring is alid if, for eery edge (; ), ertices and hae different colors. Note that this imlies that if two ertices 1 and 2 are adjacent to a ertex i.e. ( 1;)and ( 2;)are both edges in E then 1 and 2 mst hae the same color. We will now constrct three grahs, denoted,, and, which corresond to the three orientation comonents 90, M, and M. In each grah, there is initially a ertex for each deice, and there are initially no edges. For a gien deice, the corresonding ertices in,, and are,,and, resectiely (if there is no danger of confsion, then these three ertices are collectiely referred to as ). We are going to color each of these three grahs sing two colors, red and green, sch that adjacent ertices hae different colors. After the grahs hae been sccessflly colored, if a ertex in is green, then the 90 oeration is to be alied to the corresonding deice; if is red, then the 90 oeration is not to be alied. Similarly, the colors of the ertices in and indicate whether the M and M oerations, resectiely, are to be alied to the deice corresonding to each ertex. Initially, the color of eery ertex is ndefined. A FIED OI- ENTATION constraint is imlemented for the deice corresonding to ertex by setting the colors of ertices,,and to the aroriate ales to imlement the orientation as rescribed in Table 1. We will now add edges and intermediate ertices between the ertices corresonding to deices inoled in each constraint, sch that a alid 2-coloring of each grah translates to an orientation for each deice sch that all constraints are satisfied. This is accomlished sing the fact that adjacent ertices mst hae different colors and airs of ertices each connected to an intermediate ertex mst hae the same color. A SAME OIENTATION constraint is imlemented as follows. If the constraint inoles more than two deices, then this oeration is alied airwise between an arbitrarily chosen deice A in the constraint and each additional deice in the constraint (clearly if all deices hae the same orientation as A, then they all hae the same orientation). For each sch air, let and be the ertices corresonding to the air of deices. Intermediate ertices,, and are added to,, and, resectiely. Edges ( ; ) and ( ; ) are added to, and similarly edges ( ; ) and ( ; ) are added to and edges ( ; ) and ( ; ) are added to.inthisway, and are forced to be colored the same color, as are and as well as and. Ths, the

3 Figre 3. The ertices and edges added for a SAME OIENTA- TION constraint. Figre 5. The ertices and edges added for a SAME O MI- OED OIENTATION constraint abot the x-axis. Figre 4. The ertices and edges added for a MIOED OI- ENTATION constraint abot the x-axis. deices corresonding to and are forced to hae the same orientation. This configration is shown in Figre 3. A MIOED OIENTATION constraint is imlemented as follows. Assme withot loss of generality that the constraint is with resect tothe x-axis. A MIOED OIENTATION constraint mst inclde exactly two deices; let and be the ertices corresonding to these two deices. Intermediate ertices and are added to and, resectiely. Edges ( ; ) and ( ; ) are added to, and edges ( ; )and ( ; )are added to. A single edge ( ; )withot an intermediate ertex is added to. Inthisway, and are forced to hae the same color, as are and. The ertices and are forced to hae different colors. Insection of Figre 1 and Table 1 shows that this forces the deices corresonding to and to hae mirrored orientation abot the x-axis. This configration is illstrated in Figre 4. The imlementation of a MIOED OIENTATION constraint with resect to the y-axis is similar, bt with and reersed throghot. That is, we add ertex and edges ( ; ) and ( ; ) to, weaddertex and edges ( ; ) and ( ; )to, and we add edge ( ; )to. The final constraint is the SAME O MIOED OIENTATION constraint. Again, assme withot loss of generality that the constraint is abot the x-axis. This tye of constraint can inclde more than two deices; as with the SAME OIENTATION constraint, the following oeration is alied airwise between an arbitrarily cho- sen deice A in the constraint and each additional deice in the constraint (again, if all deices satisfy the constraint with resect to A, then they mst satisfy it with resect to one another). For each sch air, let and be the corresonding air of ertices. We again add intermediate ertex and edges ( ; ) and ( ; )to and add intermediate ertex and edges ( ; ) and ( ; )to. No edge is added between and.inthis way, ertices and mst hae the same color, as mst ertices and. Vertices and may either hae the same color, in which case the deices corresonding to and will hae the same orientation, or they may hae different colors, in which case the deices corresonding to and will hae mirrored orientation abot the x-axis (again, this can be erified by examining Figre 1 and Table 1). This configration is illstrated in Figre 5. Again, a SAME O MIOED OIENTATION constraint abot the y-axis is similar, bt with and reersed throghot. That is, we add ertex and edges ( ; )and ( ; )to, and we add ertex and edges ( ; )and ( ; )to. We hae shown how to constrct grahs,,and sch that a alid 2-coloring of each grah translates to an orientation for each deice sch that all constraints are satisfied. What remains to be shown is how to comte sch a coloring, and how to deal with oerconstraints. In the absence of oerconstraints, a 2-coloring of each grah can be easily comted in time linear in the nmber of ertices and edges in the grah. One way to do so is by sing a dethfirst search (see Cormen, Leiserson, and iest [1, ], for a descrition of deth-first search, and see Kozen [2,. 119], for an exlanation of 2-coloring sing deth-first search). First, choose a ertex whose color has already been set de to a FIED OIENTATION constraint. If no sch ertex exists, then choose an arbitrary ertex and assign it the color green (in the latter case, there are at least two alid soltions with oosite colors, so the choice of color for the starting ertex is irreleant). We add afieldarent() to each ertex ; the contents of this field will be the ertex that was isited before. Now, aly the following rocedre starting from.

4 Color(): For each neighbor of not already colored: 1. Assign the oosite color from 2. Set arent()to 3. Color() In the absence of oerconstraints, when this rocedre comletes it will hae rodced a alid 2-coloring of the grah. An oerconstraint can be recognized if, in Ste (1) aboe, a neighbor of has already been assigned the same color as. In this case, the grah edges corresonding to the constraints in the oerconstraint either form a cycle, or else they form a ath whose endoints are ertices whose colors were assigned de to a FIED OIENTATION constraint. Using the arent field, we trace backward from ntil we either enconter again or enconter a ertex whose color was assigned de to a FIED OIENTATION constraint. One of these two eents mst occr, since if we do not enconter a ertex whose color was assigned de to a FIED OIENTATION constraint, then we mst find a cycle starting and ending at,orelsearent() wold be eqal to. Haing fond the oerconstraint, we form a set of constraints corresonding to the edges that the algorithm traced, ls the edge (; ). We ass this set of constraints to a rocedre that relaxes one of the constraints, and then we start all oer, bild the grahs,, and, and attemt to 2-color them again. This rocedre is reeated once for each oerconstraint ntil all oerconstraints hae been resoled and we rodce a alid 2-coloring of each grah. The criteria sed by the rocedre to decide which constraint to relax are indeendent of the algorithm; i.e., the algorithm works correctly regardless of which constraint the rocedre chooses to relax. Note that this differs from the way oerconstraints are handled by the iteratie algorithm, which simly relaxes the constraint it is crrently considering at the time the oerconstraint is detected. Once a alid 2-coloring has been comted, it is simle to translate the colors of each trio of ertices back into an orientation for the corresonding deice, sing the translations gien in Table 1. When this is done, each deice will hae been assigned an orientation sch that all constraints that were not relaxed de to oerconstraints are satisfied. Each coloring attemt reqires O(n + m) time [2], where n is the nmber of deices and m is the nmber of constraints (which is within a constant mltile of the nmber of edges in the grahs). This rocess, as well as the rocedre that decides which constraint in an oerconstraint to relax, is reeated once each time an oerconstraint is fond. Ths, if there are c oerconstraints and f (n; m) is the time reqired by the oerconstraint resoltion rocedre, then the total time reqired is O(c(n + m + f (n; m))). A tyical oerconstraint resoltion rocedre reqires at most O(n + m) time itself, which makes the total time reqired eqal to O(c(n + m)). In the actal imlementation, there is no need to constrct three searate grahs. Instead, one can comte a single grah, each of whose ertices contains three searate colors and three searate sets of edges (one for each of,, and ). In addition, there is no need to exlicitly add the intermediate ertices; rather, one can simly set a flag on the aroriate edges sch that they are treated as thogh were resent. In addition, it might be ossible to reconstrct only the ortions of the grahs inoled in an oerconstraint when one is fond, rather than reconstrcting the grahs entirely and 2-coloring them from scratch. Howeer, there are tyically few or no oerconstraints, and the algorithm rns in linear time for each oerconstraint, so this otimization is robably not cost-effectie. Time (seconds) Deices Constraints Figre 6. nning times for the gien nmbers of deices and constraints. 4. EPEIMENTAL ESULTS Or coloring algorithm has been imlemented as art of a Cadence lacement rodct. Exerimental reslts of rnning the algorithm on a nmber of randomly generated test cases are shown in Figre 6 (detailed nmerical data aailable on reqest). The test cases are randomly generated so as to aoid the most degenerate tyes of oerconstraints: constraints between two deices that both hae fixed orientation, and mltile constraints of different tyes between the same air of deices. Each tye of constraint is eqally robable. Each data oint is the reslt of rnning 1000 tests. Note that becase or test cases are randomly generated, desite being generated as to aoid simle tyes of oerconstraints, they still contain far more oerconstraints than wold a tyical real design. The majority of the rnning time of the algorithm on this test data is sent dealing with these oerconstraints, as eidenced by the fact that for large nmbers of constraints the rnning time actally decreases with increasing nmbers of deices, de to fewer oerconstraints. We can get more realistic measrements by rnning on test cases with the nmber of constraints eqal to half the nmber of deices. In sch test cases, we see aroximately one oerconstraint for eery 50 constraints, whereas for the first set of data we often see one or more oerconstraints for eery two constraints. The reslts of rnning on these more realistic test cases are shown in Figre CONCLUSIONS We hae resented an efficient algorithm that ses grah coloring to sole an imortant sbroblem in constraint-drien lacement: soling systems of orientation constraints. This algorithm is more efficient than its redecessor, handles oerconstraints more effectiely, and finds soltions in difficlt cases that defeated the reios algorithm. While we hae discssed only the for constraints that were interesting in or alication, this techniqe can enforce any arbitrary constraint that inoles a fixed, same, or different relation among the three orientation comonents 90, M, and M. For examle, one alication that arose after the initial imlementation of this algorithm was comlete was in the lacer s handling of fences. One can define a rectanglar region called a fence, and then reqire that certain deices be laced within the fence. In

5 nning time (seconds) Deices (= Constraints * 2) Figre 7. nning times with the gien nmbers of deices and half that many constraints. The bars indicate the actal rnning time, and the line is a cre of O(c(n+m)) fit to the actal data. some cases, if the fence and a deice that it will contain are both long and thin, then the deice may only fit if it is oriented sch that its long dimension matches that of the fence. In other words, we wold like to enforce for this deice that the 90 comonent of its orientation be fixed, while the M and M comonents are free to ary. This is easily handled by or algorithm. In addition, the techniqe can be easily generalized to handle more general sets of orientations, as long as they can be niqely described by a discrete set of indeendent rotation and reflection oerations. 6. ACKNOWLEDGMENTS Thanks to Enrico Malaasi and Kathy Jones for many helfl comments. The algorithm described here is atent ending. EFEENCES [1] COMEN, T.H.,LEISESON, C.E., AND IVEST,.L. Introdction to Algorithms. MIT Press, Cambridge, Massachsetts, [2] KOZEN, D. The Design and Analysis of Algorithms. Sringer- Verlag, 1992.