13 May 1983, Volume 220, Number 4598

Optimization bySimulated Annealing

S. Kirkpatrick, C. D. Gelatt, Jr., M. P. Vecchi

In this article we briefly review thecentral constructs in combinatorial opti-mization and in statistical mechanics andthen develop the similarities between thetwo fields. We show how the Metropolisalgorithm for approximate numericalsimulation of the behavior of a many-body system at a finite temperature pro-vides a natural tool for bringing the tech-niques of statistical mechanics to bear onoptimization.We have applied this point of view to a

number of problems arising in optimaldesign of computers. Applications topartitioning, component placement, andwiring of electronic systems are de-scribed in this article. In each context,we introduce the problem and discussthe improvements available from optimi-zation.Of classic optimization problems, the

traveling salesman problem has receivedthe most intensive study. To test thepower of simulated annealing, we usedthe algorithm on traveling salesmanproblems with as many as several thou-sand cities. This work is described in afinal section, followed by our conclu-sions.

Combinatorial Optimization

The subject of combinatorial optimiza-tion (I) consists of a set of problems thatare central to the disciplines of computerscience and engineering. Research in thisarea aims at developing efficient tech-niques for finding minimum or maximumvalues of a function of very many inde-pendent variables (2). This function, usu-ally called the cost function or objectivefunction, represents a quantitative mea-13 MAY 1983

sure of the "goodness" of sorrsystem. The cost function dthe detailed configuration ofparts of that system. We are riar with optimization problemrin the physical design of conexamples used below are di

CIENCE

with N, so that in practice exact solu-tions can be attempted only on problemsinvolving a few hundred cities or less.The traveling salesman belongs to thelarge class of NP-complete (nondeter-ministic polynomial time complete)problems, which has received extensivestudy in the past 10 years (3). No methodfor exact solution with a computing ef-fort bounded by a power of N has beenfound for any of these problems, but ifsuch a solution were found, it could bemapped into a procedure for solving allmembers of the class. It is not known

ne complex what features of the individual problemslepends on in the NP-complete class are the cause ofthe many their difficulty.most famil- Since the NP-complete class of prob-s occurring lems contains many situations of practi-nputers, so cal interest, heuristic methods have beenrawn from developed with computational require-

Summary. There is a deep and useful connection between statistical mechanics(the behavior of systems with many degrees of freedom in thermal equilibrium at afinite temperature) and multivariate or combinatorial optimization (finding the mini-mum of a given function depending on many parameters). A detailed analogy withannealing in solids provides a framework for optimization of the properties of verylarge and complex systems. This connection to statistical mechanics exposes newinformation and provides an unfamiliar perspective on traditional optimization prob-lems and methods.

that context. The number of variablesinvolved may range up into the tens ofthousands.The classic example, because it is so

simply stated, of a combinatorial optimi-zation problem is the traveling salesmanproblem. Given a list of N cities and ameans of calculating the cost of travelingbetween any two cities, one must planthe salesman's route, which will passthrough each city once and return finallyto the starting point, minimizing the totalcost. Problems with this flavor arise inall areas of scheduling and design. Twosubsidiary problems are of general inter-est: predicting the expected cost of thesalesman's optimal route, averaged oversome class of typical arrangements ofcities, and estimating or obtainingbounds for the computing effort neces-sary to determine that route.

All exact methods known for deter-mining an optimal route require a com-puting effort that increases exponentially

ments proportional to small powers of N.Heuristics are rather problem-specific:there is no guarantee that a heuristicprocedure for finding near-optimal solu-tions for one NP-complete problem willbe effective for another.There are two basic strategies for

heuristics: "divide-and-conquer" and it-erative improvement. In the first, onedivides the problem into subproblems ofmanageable size, then solves the sub-problems. The solutions to the subprob-lems must then be patched back togeth-er. For this method to produce very goodsolutions, the subproblems must be natu-rally disjoint, and the division made mustbe an appropriate one, so that errorsmade in patching do not offset the gains

S. Kirkpatrick and C. D. Gelatt, Jr., are researchstaff members and M. P. Vecchi was a visitingscientist at IBM Thomas J. Watson Research Cen-ter, Yorktown Heights, New York 10598. M. P.Vecchi's present address is Instituto Venezolano deInvestigaciones Cientificas, Caracas IOIOA, Vene-zuela.

671

obtained in applying more powerfulmethods to the subproblems (4).

In iterative improvement (5, 6), onestarts with the system in a known config-uration. A standard rearrangement oper-ation is applied to all parts of the systemin turn, until a rearranged configurationthat improves the cost function is discov-ered. The rearranged configuration thenbecomes the new configuration of thesystem, and the process is continueduntil no further improvements can befound. Iterative improvement consists ofa search in this coordinate space forrearrangement steps which lead down-hill. Since this search usually gets stuckin a local but not a global optimum, it iscustomary to carry out the process sev-eral times, starting from different ran-domly generated configurations, andsave the best result.There is a body of literature analyzing

the results to be expected and the com-puting requirements of common heuris-tic methods when applied to the mostpopular problems (1-3). This analysisusually focuses on the worst-case situa-tion-for instance, attempts to boundfrom above the ratio between the costobtained by a heuristic method and theexact minimum cost for any member of afamily of similarly structured problems.There are relatively few discussions ofthe average performance of heuristic al-gorithms, because the analysis is usuallymore difficult and the nature of the ap-propriate average to study is not alwaysclear. We will argue that as the size ofoptimization problems increases, theworst-case analysis of a problem willbecome increasingly irrelevant, and theaverage performance of algorithms willdominate the analysis of practical appli-cations. This large number limit is thedomain of statistical mechanics.

Statistical Mechanics

Statistical mechanics is the central dis-cipline of condensed matter physics, abody of methods for analyzing aggregateproperties of the large numbers of atomsto be found in samples of liquid or solidmatter (7). Because the number of atomsis of order 1023 per cubic centimeter,only the most probable behavior of thesystem in thermal equilibrium at a giventemperature is observed in experiments.This can be characterized by the averageand small fluctuations about the averagebehavior of the system, when the aver-age is taken over the ensemble of identi-cal systems introduced by Gibbs. In thisensemble, each configuration, definedby the set of atomic positions, {ri}, of the

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system is weighted by its Boltzmannprobability factor, exp(-E({ri})/kB7),where E({r,}) is the energy of the config-uration, kB is Boltzmann's constant, andT is temperature.A fundamental question in statistical

mechanics concerns what happens to thesystem in the limit of low temperature-for example, whether the atoms remainfluid or solidify, and if they solidify,whether they form a crystalline solid or aglass. Ground states and configurationsclose to them in energy are extremelyrare among all the configurations of amacroscopic body, yet they dominate itsproperties at low temperatures becauseas T is lowered the Boltzmann distribu-tion collapses into the lowest energystate or states.As a simplified example, consider the

magnetic properties of a chain of atomswhose magnetic moments, Ri, are al-lowed to point only "up" or "down,"states denoted by p.i = + 1. The interac-tion energy between two such adjacentspins can be written JRiui+ l. Interactionbetween each adjacent pair of spins con-tributes ±J to the total energy of thechain. For an N-spin chain, if all configu-rations are equally likely the interactionenergy has a binomial distribution, withthe maximum and minimum energies giv-en by ±NJ and the most probable statehaving zero energy. In this view, theground state configurations have statisti-cal weight exp(-N/2) smaller than thezero-energy configurations. A Boltz-mann factor, exp(-E/kB7), can offsetthis if kBT is smaller than J. If we focuson the problem of finding empirically thesystem's ground state, this factor is seento drastically increase the efficiency ofsuch a search.

In practical contexts, low temperatureis not a sufficient condition for findingground states of matter. Experimentsthat determine the low-temperature stateof a material-for example, by growing asingle crystal from a melt-are done bycareful annealing, first melting the sub-stance, then lowering the temperatureslowly, and spending a long time at tem-peratures in the vicinity of the freezingpoint. If this is not done, and the sub-stance is allowed to get out of equilibri-um, the resulting crystal will have manydefects, or the substance may form aglass, with no crystalline order and onlymetastable, locally optimal structures.

Finding the low-temperature state of asystem when a prescription for calculat-ing its energy is given is an optimizationproblem not unlike those encountered incombinatorial optimization. However,the concept of the temperature of a phys-ical system has no obvious equivalent in

the systems being optimized. We willintroduce an effective temperature foroptimization, and show how one cancarry out a simulated annealing processin order to obtain better heuristic solu-tions to combinatorial optimization prob-lems.

Iterative improvement, commonly ap-plied to such problems, is much like themicroscopic rearrangement processesmodeled by statistical mechanics, withthe cost function playing the role ofenergy. However, accepting only rear-rangements that lower the cost functionof the system is like extremely rapidquenching from high temperatures toT = 0, so it should not be surprising thatresulting solutions are usually metasta-ble. The Metropolis procedure from sta-tistical mechanics provides a generaliza-tion of iterative improvement in whichcontrolled uphill steps can also be incor-porated in the search for a better solu-tion.

Metropolis et al. (8), in the earliestdays of scientific computing, introduceda simple algorithm that can be used toprovide an efficient simulation of a col-lection of atoms in equilibrium at a giventemperature. In each step of this algo-rithm, an atom is given a small randomdisplacement and the resulting change,AE, in the energy of the system is com-puted. If AE s 0, the displacement isaccepted, and the configuration with thedisplaced atom is used as the startingpoint of the next step. The case AE > 0is treated probabilistically: the probabili-ty that the configuration is accepted isP(AE) = exp(-AE/kB1). Random num-bers uniformly distributed in the interval(0,1) are a convenient means of imple-menting the random part of the algo-rithm. One such number is selected andcompared with P(AE). If it is less thanP(AE), the new configuration is retained;if not, the original configuration is usedto start the next step. By repeating thebasic step many times, one simulates thethermal motion of atoms in thermal con-tact with a heat bath at temperature T.This choice of P(AE) has the conse-quence that the system evolves into aBoltzmann distribution.

Using the cost function in place of theenergy and defining configurations by aset of parameters {x,}, it is straightfor-ward with the Metropolis procedure togenerate a population of configurationsof a given optimization problem at someeffective temperature. This temperatureis simply a control parameter in the sameunits as the cost function. The simulatedannealing process consists of first "melt-ing" the system being optimized at ahigh effective temperature, then lower-

SCIENCE, VOL. 220

ing the temperature by slow stages untilthe system "freezes" and no furtherchanges occur. At each temperature, thesimulation must proceed long enough forthe system to reach a steady state. Thesequence of temperatures and the num-ber of rearrangements of the {xi} attempt-ed to reach equilibrium at each tempera-ture can be considered an annealingschedule.

Annealing, as implemented by the Me-tropolis procedure, differs from iterativeimprovement in that the procedure neednot get stuck since transitions out of alocal optimum are always possible atnonzero temperature. A second andmore important feature is that a sort ofadaptive divide-and-conquer occurs.Gross features of the eventual state ofthe system appear at higher tempera-tures; fine details develop at lower tem-peratures. This will be discussed withspecific examples.

Statistical mechanics contains manyuseful tricks for extracting properties ofa macroscopic system from microscopicaverages. Ensemble averages can be ob-tained from a single generating function,the partition function, Z,

Z = Tr exp(kT) (1)

in which the trace symbol, Tr, denotes asum over all possible configurations ofthe atoms in the sample system. Thelogarithm of Z, called the free energy,F(T), contains information about the av-erage energy, <E(7)>, and also the en-tropy, S(1), which is the logarithm of thenumber of configurations contributing tothe ensemble at T:

-kBT In Z = F(7T) = <E(7T)> - TS(2)

Boltzmann-weighted ensemble averagesare easily expressed in terms of deriva-tives of F. Thus the average energy isgiven by

<E(7)> =-dlnZ (3)

=d(l/kBT) (3

and the rate of change of the energy withrespect to the control parameter, T, isrelated to the size of typical variations inthe energy by

CMd <E(T)>

C <(TdT

_[<EU7)2> - <E(T)>2]4

kBT2

In statistical mechanics C(7) is called thespecific heat. A large value of C signals achange in the state of order of a system,and can be used in the optimizationcontext to indicate that freezing has be-13 MAY 1983

gun and hence that very slow cooling isrequired. It can also be used to deter-mine the entropy by the thermodynamicrelation

dS(T) 7C(T)dT T (5)

Integrating Eq. 5 gives

Tr Q(T) dTS(7) = S(T1) - CT9 l (6)

where T, is a temperature at which S isknown, usually by an approximation val-id at high temperatures.The analogy between cooling a flui(

and optimization may fail in one impor-tant respect. In ideal fluids all the atomsare alike and the ground state is a regularcrystal. A typical optimization problemwill contain many distinct, noninter-changeable elements, so a regular solu-tion is unlikely. However, much re-search in condensed matter physics isdirected at systems with quenched-inrandomness, in which the atoms are notall alike. An important feature of suchsystems, termed "frustration," is thatinteractions favoring different and in-compatible kinds of ordering may besimultaneously present (9). The magnet-ic alloys known as "spin glasses," whichexhibit competition between ferromag-netic and antiferromagnetic spin order-ing, are the best understood example offrustration (10). It is now believed thathighly frustrated systems like spin glass-es have many nearly degenerate randomground states rather than a single groundstate with a high degree of symmetry.These systems stand in the same relationto conventional magnets as glasses do tocrystals, hence the name.The physical properties of spin glasses

at low temperatures provide a possibleguide for understanding the possibilitiesof optimizing complex systems subjectto conflicting (frustrating) constraints.

Physical Design of Computers

The physical design of electronic sys-tems and the methods and simplifica-tions employed to automate this processhave been reviewed (11, 12). We firstprovide some background and defini-tions related to applications of the simu-lated annealing framework to specificproblems that arise in optimal design ofcomputer systems and subsystems.Physical design follows logical design.After the detailed specification of thelogic of a system is complete, it is neces-sary to specify the precise physical real-ization of the system in a particular tech-nology.

This process is usually divided intoseveral stages. First, the design must bepartitioned into groups small enough tofit the available packages, for example,into groups of circuits small enough to fitinto a single chip, or into groups of chipsand associated discrete components thatcan fit onto a card or other higher levelpackage. Second, the circuits are as-signed specific locations on the chip.This stage is usually called placement.Finally, the circuits are connected bywires formed photolithographically outof a thin metal film, often in severallayers. Assigning paths, or routes, to thewires is usually done in two stages. Inrough or global wiring, the wires areassigned to regions that represent sche-matically the capacity of the intendedpackage. In detailed wiring (also calledexact embedding), each wire is given aunique complete path. From the detailedwiring results, masks can be generatedand chips made.At each stage of design one wants to

optimize the eventual performance of thesystem without compromising the feasi-bility of the subsequent design stages.Thus partitioning must be done in such away that the number of circuits in eachpartition is small enough to fit easily intothe available package, yet the number ofsignals that must cross partition bound-aries (each requiring slow, power-con-suming driver circuitry) is minimized.The major focus in placement is on mini-mizing the length of connections, sincethis translates into the time required forpropagation of signals, and thus into thespeed of the finished system. However,the placements with the shortest impliedwire lengths may not be wirable, becauseof the presence of regions in which thewiring is too congested for the packag-ing technology. Congestion, therefore,should also be anticipated and minimizedduring the placement process. In wiring,it is desirable to maintain the minimumpossible wire lengths while minimizingsources of noise, such as cross talk be-tween adjacent wires. We show in thisand the next two sections how theseconflicting goals can be combined andmade the basis of an automatic optimiza-tion procedure.The tight schedules involved present

major obstacles to automation and opti-mization of large system design, evenwhen computers are employed to speedup the mechanical tasks and reduce thechance of error. Possibilities of feed-back, in which early stages of a designare redone to solve problems that be-came apparent only at later stages, aregreatly reduced as the scale of the over-all system being designed increases. Op-

673

._ea

60co

0Sum of pins on the two chips

Fig. I. Distribution of total number of pinsrequired in two-way partition of a micro-processor at various temperatures. Arrow in-dicates best solution obtained by rapidquenching as opposed to annealing.

timization procedures that can incorpo-rate, even approximately, informationabout the chance of success of laterstages of such complex designs will beincreasingly valuable in the limit of verylarge scale.System performance is almost always

achieved at the expense of design conve-nience. The partitioning problem pro-vides a clean example of this. ConsiderN circuits that are to be partitioned be-tween two chips. Propagating a signalacross a chip boundary is always slow,so the number of signals required tocross between the two must be mini-mized. Putting all the circuits on onechip eliminates signal crossings, but usu-ally there is no room. Instead, for laterconvenience, it is desirable to divide thecircuits about equally.

If we have connectivity information ina matrix whose elements {a,j} are thenumber of signals passing between cir-cuits i and j, and we indicate which chipcircuit i is placed on by a two-valuedvariable ,i = ± 1, then Nc, the numberof signals that must cross a chip bound-ary is given by X>jia,a14)(>xi - Rj)2. Cal-culating IiRi gives the difference be-tween the numbers of circuits on the twochips. Squaring this imbalance and intro-ducing a coefficient, X, to express therelative costs of imbalance and boundarycrossings, we obtain an objective func-tion, f, for the partition problem:

f'

a

2j I'Aj (7)

Reasonable values of X should satisfyX < z/2, where z is the average numberof circuits connected to a typical circuit(fan-in plus fan-out). Choosing X z/2implies giving equal weight to changes inthe balance and crossing scores.

674

The objective function f has preciselythe form of a Hamiltonian, or energyfunction, studied in the theory of randommagnets, when the common simplifyingassumption is made that the spins, pLi,have only two allowed orientations (upor down), as in the linear chain exampleof the previous section. It combines lo-cal, random, attractive ("ferromagnet-ic") interactions, resulting from the aij's,with a long-range repulsive ("antiferro-magnetic") interaction due to X. No con-figuration of the {,u,} can simultaneouslysatisfy all the interactions, so the systemis "frustrated," in the sense formalizedby Toulouse (9).

If the aij are completely uncorrelated,it can be shown (13) that this Hamilto-nian has a spin glass phase at low tem-peratures. This implies for the associatedmagnetic problem that there are manydegenerate "ground states" of nearlyequal energy and no obvious symmetry.The magnetic state of a spin glass is verystable at low temperatures (14), so theground states have energies well belowthe energies of the random high-tempera-ture states, and transforming one groundstate into another will usually requireconsiderable rearrangement. Thus thisanalogy has several implications for opti-mization of partition:

1) Even in the presence offrustration,significant improvements over a randomstarting partition are possible.

2) There will be many good near-opti-mal solutions, so a stochastic searchprocedure such as simulated annealingshould find some.

3) No one of the ground states is sig-nificantly better than the others, so it isnot very fruitful to search for the abso-lute optimum.

In developing Eq. 7 we made severalsevere simplifications, considering onlytwo-way partitioning and ignoring thefact that most signals connect more thantwo circuits. Objective functions analo-gous tof that include both complicationsare easily constructed. They no longerhave the simple quadratic form of Eq. 7,but the qualitative feature, frustration,remains dominant. The form of the Ham-iltonian makes no difference in the Me-tropolis Monte Carlo algorithm. Evalua-tion of the change in function when acircuit is shifted to a new chip remainsrapid as the definition of f becomesmore complicated.

It is likely that the aij are somewhatcorrelated, since any design has consid-erable logical structure. Efforts to under-stand the nature of this structure byanalyzing the surface-to-volume ratio ofcomponents of electronic systems [as in"Rent's rule" (15)] conclude that the

-IFI F~ILILL

zo

Z0

Boundary

Fig. 2. Construction of a horizontal net-cross-ing histogram.

circuits in a typical system could beconnected with short-range interactionsif they were embedded in a space withdimension between two and three. Un-correlated connections, by contrast, canbe thought of as infinite-dimensional,since they are never short-range.The identification of Eq. 7 as a spin

glass Hamiltonian is not affected by thereduction to a two- or three-dimensionalproblem, as long as XN z/2. The de-gree of ground state degeneracy in-creases with decreasing dimensionality.For the uncorrelated model, there aretypically of order N"/2 nearly degenerateground states (14), while in two and threedimensions, 2 N, for some small value,a, are expected (16). This implies thatfinding a near-optimum solution shouldbecome easier, the lower the effectivedimensionality of the problem. The en-tropy, measurable as shown in Eq. 6,provides a measure of the degeneracy ofsolutions. S(1) is the logarithm of thenumber of solutions equal to or betterthan the average result encountered attemperature T.As an example of the partitioning

problem, we have taken the logic designfor a single-chip IBM "370 microproces-sor" (17) and considered partitioning itinto two chips. The original design hasapproximately 5000 primitive logic gatesand 200 external signals (the chip has 200logic pins). The results of this study areplotted in Fig. 1. If one randomly assignsgates to the two chips, one finds thedistribution marked T = co for the num-ber of pins required. Each of the twochips (with about 2500 circuits) wouldneed 3000 pins. The other distributionsin Fig. 1 show the results of simulatedannealing.Monte Carlo annealing is simple to

implement in this case. Each proposedconfiguration change simply flips a ran-domly chosen circuit from one chip tothe other. The new number of externalconnections, C, to the two chips is calcu-lated (an external connection is a netwith circuits on both chips, or a circuit

SCIENCE, VOL. 220

T-0.1

T-1.05

T=1.3 T-1.6 T-2.4

1000 20 34I I1000i 2000 3000 4000 5000 60

connected to one of the pins of theoriginal single-chip design), as is the newbalance score, B, calculated as in deriv-ing Eq. 7. The objective function analo-gous to Eq. 7 is

f = C +XB (8)

where C is the sum of the number ofexternal connections on the two chipsand B is the balance score. For thisexample, A = 0.01.For the annealing schedule we chose

to start at a high "temperature,"To = 10, where essentially all proposedcircuit flips are accepted, then cool ex-ponentially, T, = (T1/To)'To, with the ra-tio T11To = 0.9. At each temperatureenough flips are attempted that eitherthere are ten accepted flips per circuit onthe average (for this case, 50,000 accept-ed flips at each temperature), or thenumber of attempts exceeds 100 timesthe number of circuits before ten flipsper circuit have been accepted. If thedesired number of acceptances is notachieved at three successive tempera-

Initial position

El Cl 3[ 5 6 7

262 11 12 E II 15 16 111

S2 2~~23 2g25 22721

3

i') 333 336 is824

82651 52 53 55 N

*,:X: .62. t*r: 64 6 7

662 71 72 73 %fIJ W 4 4556

:&1. 8 Z 83 84 87

91 92 93 94 95 97

tures, the system is considered "frozen"and annealing stops.The finite temperature curves in Fig. 1

show the distribution of pins per chip forthe configurations sampled at T 2.5,1.0, and 0.1. As one would expect fromthe statistical mechanical analog, the dis-tribution shifts to fewer pins and shar-pens as the temperature is decreased.The sharpening is one consequence ofthe decrease in the number of configura-tions that contribute to the equilibriumensemble at the lower temnperature. Inthe language of statistical mechanics, theentropy of the system decreases. For,this sample run in the low-temnperaturelimit, the two chips required 353 and 321pins, respectively. There are 237 netsconnecting the two chips (requiring a pinon each chip) in addition to the 200inputs and outputs of the original chip.The final partition in this example hasthe circuits exactly evenly distributedbetween the two partitions. Using amore complicated balance score, whichdid not penalize imbalance of less than

8 9 10

18 19

28 E [

3*1i 39 E49 [

4mm1 RE EM4 69 E

78 79 80

4 89 90

98 99

a 306 428 555 759 879 945 898 672 210

T i 1250

99 98 E3 92 83 72:2 s *47t :8.

93 95 55 84 52 25 :5 # *Z. .:760~~~~~~~~~X644 97 69 71 4 6 :': :z:

80990 53 87 9 4 743

89 27 67 78 E2 23 12 10 11

29 41 15 16 1 91 21 3717

785 51 1i Ifj 30 49 I1fI28 18 79 5 1 36 jj J

9 ElB 19 B " Ell 116 8 7

C 477 700 892 782 820 932 909 736 489

100 tircuits, we found partitions result-ing in hips with 271 and 183 pins.

If, in ead of slowly cooling, one wereto start frqm a random partition andaccept only Sips that reduce the objec-tive function (equivalent to setting T = 0in the Metropolis rule), the result is chipswith approximately 700 pins (severalsuch runs led to results with 677 to730 pins). Rapid cooling results in asystem frozen into a metastable statefar fromn the optimnal configuration. Thebest result obtained after several rapidquenches is indicated by the arrow inFig. 1.

Placement

Placement is. a further refinement ofthe logic partitioning process, in whichthe circuits are given physical positions(11, 12, 18, 19). In principle, the twostages could be combined, although thisis not often possible in practice. Theobjectives in placement are to minimize

T- 10000

5 53 52 1 9 :2:2tE

78 23 64 95 41 [i :dI E 36

25 Xj9 3 7016 91 8 3149 4-

25 33 15 83 IN I i f1378 ___

aimAM 84 79 4 1 5 871 388 illI

6 90 97 94 PINI 58 89 27

.1.37- 98 55 s6:3: 10 67 72

1170 7 3 30 29 69 E 49838

838 II2 11M!4. Ct-51 18 4 143646.. ___ 93 12 28 99 92 71 E

b 513 920 1091 1266 1349 1327 1147 858 503

T-0

405

563

559

560

591

595

562

558

395

d

71 92 93 94 877 3 : *9Z :4.

83 84 97 73 72 p .3:1: k..95 99 98 4 52 12 23

5 15 90 4 4 E2

16 25 78 55 69 10 a E 11 1

79 89 27 67 53 64 Ii I

8 19 18 29 49 = IIlI# I|I16 f3 t 61 511g7 9 iJ 41 E 4

30 28 i 91 36 Plol - 33

373 471 569 613 580 625 632 590 334

Fig. 3. Ninety-eight chips on a ceramic module from the IBM 3081. Chips are identified by number (I to 100, with 20 and 100 absent) and function.The dark squares comprise an adder, the three types of squares with ruled lines are chips that control and supply data to the adder, the lightly dot-ted chips perform logical arithmetic (bitwise AND, OR, and so on), and the open squares denote general-purpose registers, which serve botharithmetic units. The numbers at the left and lower edges of the module image are the vertical and horizontal net-crossing histograms,respectively. (a) Original chip placement; (b) a configuration at T = 10,000; (c) T = 1250; (d) a zero-temperature result.

13 MAY 1983 675

signal propagation times or distanceswhile satisfying prescribed ejectricalconstraints, without creating rdions socongested that there will not be roomlater to connect the circuits with-actualwire.

Physical design of computers includesseveral distinct categories of placementproblems, depending on the packagesinvolved (20). The larger objects to beplaced include chips that must reside in ahigher level package, such as a printedcircuit card or fired ceramic "module"(21). These chip carriers must in turn beplaced on a backplane or "board,"which is simply a very large printedcircuit card. The chips seen today con-tain from tens to tens of thousands oflogic circuits, and each chip carrier orboard will provide from one to ten thou-sand interconnections. The partition andplacement problems decouple poorly inthis situation, since the choice of whichchip should carry a given piece of logicwill be influenced by the position of thatchip.The simplest placement problems

arise in designing chips with structuredlayout rules. These are called "gate ar-ray" or "master slice" chips. In thesechips, standard logic circuits, such asthree- or four-input NOR's, are pre-placed in a regular grid arrangement, andthe designer specifies only the signalwiring, which occupies the final, highest,layers of the chip. The circuits may all beidentical, or they may be described interms of a few standard groupings of twoor more adjacent cells.As an example of a placement problem

with realistic complexity without toomany complications arising from pack-age idiosyncrasies, we consider 98 chipspackaged on one multilayer ceramicmodule of the IBM 3081 processor (21).Each chip can be placed on any of 100sites, in a 10 x 10 grid on the top surfaceof the module. Information about theconnections to be made through the sig-nal-carrying planes of the module is con-tained in a "netlist," which groups setsof pins that see the same signal.The state of the system can be briefly

represented by a list of the 98 chips withtheir x and y coordinates, or a list of thecontents of each of the 100 legal loca-tions. A sufficient set of moves to use forannealing is interchanges of the contentsof two locations. This results in eitherthe interchange of two chips or the inter-change of a chip and a vacancy. Formore efficient search at low tempera-tures, it is helpful to allow restrictions onthe distance across which an interchangemay occur.To measure congestion at the same

676

time as wire length,intermediate analysnet-crossing histogrnis summarized in Fipackage surface b!boundaries. In this eboundaries betweercolumns of chip sithen contains the nting each boundary.wire must be routedary crossed, the sumhistogram of Fig. 2horizontal extentsbounding each net, Eto the horizontal wConstructing a vertitogram and summinsimilar estimate oflength.The peak of the h

lower bound to themust be provided ineach net requires

100

80

as

.2

DL

Temg

Fig. 4. Specific heat asture for the design of X

*------a

S1b

Fig. 5. Examples of (ashaped wire rearrangen

we use a convenient channel somewhere on the boundary. Tois of the layout, a combine this information into a singleam. Its construction objective function, we introduce aig. 2. We divide the threshold level for each histogram-any a set of natural amount of wire that will nearly exhaust,xample, we use the the available wire capacity-and theni adjacent rows or sum for all histogram elements that ex-ites. The histogram ceed the threshold the square of theimber of nets cross- excess over threshold. Adding this quan-Since at least one tity to the estimated length gives the

Iacross each bound- objective function that was used.i of the entries in the Figure 3 shows the stages of a simulat-

is the sum of the ed annealing run on the 98-chip module.of the rectangles Figure 3a shows the chip locations from

ind is a lower bound the original design, with vertical and,ire length required. horizontal net-crossing histograms indi-ical net-crossing his- cated. The different shading patterns dis-ig its entries gives a tinguish the groups of chips that carryf the vertical wire out different functions. Each such group

was designed and placed together, usual-iistogram provides a ly by a single designer. The net-crossingamount of wire that histograms show that the center of thethe worst case, since layout is much more congested than theat least one wiring edges, most likely because the chips

known to have the most critical timingconstraints were placed in the center ofthe module to allow the greatest number

Clustering of other chips to be close to them.Heating the original design until the

/I\ chips diffuse about freely quickly pro-duces a random-looking arrangement,Fig. 3b. Cooling very slowly until thechips move sluggishly and the objectivefunction ceases to decrease rapidly withchange of temperature produced the re-sult in Fig. 3c. The net-crossing histo-grams have peaks comparable to thepeak heights in the original placement,

__03_ 104but are much flatter. At this "freezing

103 104 point," we find that the functionally re-perature lated groups of chips have reorganizeda function of tempera- from the melt, but now are spatiallyFig. 3, a to d.

separated in an overall arrangementquite different from the original place-ment. In the final result, Fig. 3d, thehistogram peaks are about 30 percentless than in the original placement. Inte-grating them, we find that total wirelength, estimated in this way, is de-creased by about 10 percent. The com-puting requirements for this examplewere modest: 250,000 interchanges wereattempted, requiring 12 minutes of com-

* putation on an IBM 3033.Between the temperature at which

clusters form and freezing starts (Fig. 3c)and the final result (Fig. 3d) there aremany further local rearrangements. Thefunctional groups have remained in thesame regions, but their shapes and rela-tive alignments continue to changethroughout the low-temperature part ofthe annealing process. This illustrates

* that the introduction of temperature to) L-shaped and (b) z- the optimization process permits a con-nents. trolled, adaptive division of the problem

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through the evolution of natural clustersat the freezing temperature. Early pre-scription of natural clusters is also acentral feature of several sophisticatedplacement programs used in master slicechip placement (22, 23).A quantity corresponding to the ther-

modynamic specific heat is defined forthis problem by taking the derivativewith respect to temperature of the aver-age value of the objective function ob-served at a given temperature. This isplotted in Fig. 4. Just as a maximum inthe specific heat of a fluid indicates theonset of freezing or the formation ofclusters, we find specific heat maxima attwo temperatures, each indicating a dif-ferent type of ordering in the problem.The higher temperature peak corre-sponds to the aggregation of clusters offunctionally related objects, driven apartby the congestion term in the scoring.The lower temperature peak indicatesthe further decrease in wire length ob-tained by local rearrangements. This sortof measurement can be useful in practiceas a means of determining the tempera-ture ranges in which the important rear-rangements in the design are occurring,where slower cooling with be helpful.

Wiring

After placement, specific legal rout-ings must be found for the wires neededto connect the circuits. The techniquestypically applied to generate such rout-ings are sequential in nature, treating onewire at a time with incomplete informa-tion about the positions and effects of theother wires (11, 24). Annealing is inher-ently free of this sequence dependence.In this section we describe a simulatedannealing approach to wiring, using theceramic module of the last section as anexample.Nets with many pins must first be

broken into connections-pairs of pinsjoined by a single continuous wire. This"ordering" of each net is highly depen-dent on the nature of the circuits beingconnected and the package technology.Orderings permitting more than two pinsto be connected are sometimes allowed,but will not be discussed here.The usual procedure, given an order-

ing, is first to construct a coarse-scalerouting for each connection from whichthe ultimate detailed wiring can be com-pleted. Package technologies and struc-tured image chips have prearranged ar-eas of fixed capacity for the wires. Forthe rough routing to be successful, itmust not call for wire densities that ex-ceed this capacity.13 MAY 1983

RandomGrid size 10

- u 1

-e 111lhm.- fl E

M.C. Z-pathsGrid sizd 10

e.I Ir_ - _

_-

Fig. 6 (left). Wire density in the 98-chip module with the connections randomly assigned toperimeter routes. Chips are in the original placement. Fig. 7 (right). Wire density aftersimulated annealing of the wire routing, using Z-shaped moves.

We can model the rough routing prob-lem (and even simple cases of detailedembedding) by lumping all actual pinpositions into a regular grid of points,which are treated as the sources andsinks of all connections. The wires arethen to be routed along the links thatconnect adjacent grid points.The objectives in global routing are to

minimize wire length and, often, thenumber of bends in wires, while spread-ing the wire as evenly as possible tosimplify exact embedding and later revi-sion. Wires are to be routed aroundregions in which wire demand exceedscapacity if possible, so that they will not"'overflow," requiring drastic rearrange-ments of the other wires during exactembedding. Wire bends are costly inpackages that confine the north-southand east-west wires to different layers,since each bend requires a connectionbetween two layers. Two classes ofmoves that maintain the minimum wirelength are shown in Fig. 5. In the L-shaped move of Fig. 5a, only the essen-tial bends are permitted, while the Z-shaped move of Fig. Sb introduces oneextra bend. We will explore the optimi-zation possible with these two moves.For a simple objective function that

will reward the most balanced arrange-ment of wire, we calculate the square ofthe number of wires on each link of thenetwork, sum the squares for all links,and term the result F. If there are NLlinks and Nw wires, a global routingprogram that deals with a high density ofwires will attempt to route precisely theaverage number of wires, NwINL, alongeach link. In this limit F is boundedbelow by Nw2INL. One can use the sameobjective function for a low-density (orhigh-resolution) limit appropriate for de-

tailed wiring. In that case, all the linkshave either one or no wires, and linkswith two or more wires are illegal. Forthis limit the best possible value ofF willbe Nw/NL.For the L-shaped moves, F has a

relatively simple form. Let Ei = +1along the links that connection i has forone orientation, - I for the other orienta-tion, and 0 otherwise. Let ai, be I if theith connection can run through the vthlink in either of its two positions, and 0otherwise. Note that a is just Ei,2. Thenif t,j = ± 1 indicates which route the ithconnection has taken, we obtain for thenumber of wires along the vth link,

n = I aiv(eei,p + 1) + n,(0)i 2 (9)

where n,(0) is the contribution fromstraight wires, which cannot move with-out increasing their length, or blockages.Summing the n,7 gives

F = >Jj1xitj + LhiVpi + constants (10)iJi

which has the form of the Hamiltonianfor a random magnetic alloy or spinglass, like that discussed earlier. The"random field," hi, felt by each movableconnection reflects the difference, on theaverage, between the congestion associ-ated with the two possible paths:

hi = I Ei, [2nf(O) + I aj,lV J

(11)

The interaction between two wires isproportional to the number of links onwhich the two nets can overlap, its signdepending on their orientation conven-tions:

iii = IE ,

v 4(12)

677

Both Ju1 and hi vanish, on average, so it isthe fluctuations in the terms that makeup F which will control the nature of thelow-energy states. This is also true inspin glasses. We have not tried to exhibita functional form for the objective func-tion with Z-moves allowed, but simplycalculate it by first constructing the actu-al amounts of wire found along each link.To assess the value of annealing in

wiring this model, we studied an ensem-ble of randomly situated connections,under various statistical assumptions.Here we consider routing wires for the98 chips on a module considered earlier.First, we show in Fig. 6 the arrangement

200

c

c0

S

Exco

150

100

50

0

1.0

0.8

0.6

0.4

0.2

0.01.0

0.8

0.6

0.4

0.2

0

of wire that results from assigning eachwire to an L-shaped path, choosing ori-entations at random. The thickness ofthe links is proportional to the number ofwires on each link. The congested areathat gave rise to the peaks in the histo-grams discussed above is seen in thewiring just below and to the right of thecenter of the module. The maximumnumbers of wires along a single link inFig. 6 are 173 (x direction) and 143 (ydirection), so the design is also aniso-tropic. Various ways of rearranging thewiring paths were studied. Monte Carloannealing with Z-moves gave the bestsolution, shown in Fig. 7. In this exam-

Fig. 8. Histogram of the maxi-mum wire densities within agiven column of x-links, for

_- the various methods of rout-ing.

L-!

1~~~~~~~-----_-__-_-_I@~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

r--=-

I.

2 4 63 5

Channel position

8 109

0.0

0.0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

Fig. 9. Results at four temperatures for a clustered 400-city traveling salesman problem. Thepoints are uniformly distributed in nine regions. (a) T = 1.2, a = 2.0567; (b) T = 0.8,a = 1.515; (c) T = 0.4, a = 1.055; (d) T = 0.0, a = 0.7839.

ple, the largest numbers of wires on asingle link are 105 (x) and 96 (y).We compare the various methods of

improving the wire arrangement by plot-ting (Fig. 8) the highest wire densityfound in each column of x-links for eachof the methods. The unevenness of thedensity profiles was already seen whenwe considered net-crossing histogramsas input information to direct placement.The lines shown represent random as-signment of wires with L-moves; align-ing wires in the direction of least averagecongestion-that is, along h,-followedby cooling for one pass at zero T; simu-lated annealing with L-moves only; andannealing with Z-moves. Finally, thelight dashed line shows the optimumresult, in which the wires are distributedwith all links carrying as close to theaverage weight as possible. The opti-mum cannot be attained in this examplewithout stretching wires beyond theirminimum length, because the connec-tions are too unevenly arranged. Anymethod of optimization gives a signifi-cant improvement over the estimate ob-tained by assigning wire routings at ran-dom. All reduce the peak wire density ona link by more than 45 percent. Simulat-ed annealing with Z-moves improved therandom routing by 57 percent, averagingresults for both x and y links.

Traveling Salesmen

Quantitative analysis of the simulatedannealing algorithm or comparison be-tween it and other heuristics requiresproblems simpler than physical design ofcomputers. There is an extensive litera-ture on algorithms for the traveling sales-man problem (3, 4), so it provides anatural context for this discussion.

If the cost of travel between two citiesis proportional to the distance betweenthem, then each instance of a travelingsalesman problem is simply a list of thepositions of N cities. For example, anarrangement of N points positioned atrandom in a square generates one in-stance. The distance can be calculated ineither the Euclidean metric or a "Man-hattan" metric, in which the distancebetween two points is the sum of theirseparations along the two coordinateaxes. The latter is appropriate for physi-cal design applications, and easier tocompute, so we will adopt it.We let the side of the square have

length N"2, so that the average distancebetween each city and its nearest neigh-bor is independent of N. It can be shownthat this choice of length units leaves theoptimal tour length per step independentof N, when one averages over many

instances, keeping N fixed (25). Call thisaverage optimal step length a. To bounda from above, a numerical experimentwas performed with the following"greedy" heuristic algorithm. Fromeach city, go to the nearest city notalready on the tour. From the Nth city,return directly to the first. In the worstcase, the ratio of the length of such agreedy tour to the optimal tour is propor-tional to ln(N) (26), but on average, wefind that its step length is about 1. 12. Thevariance of the greedy step length de-creases as N- 1/2, so the situation envi-sioned in the worst case analysis is unob-servably rare for large N.To construct a simulated annealing

algorithm, we need a means of represent-ing the tour and a means of generatingrandom rearrangements of the tour.Each tour can be described by a permut-ed list of the numbers 1 to N, whichrepresents the cities. A powerful andgeneral set of moves was introduced byLin and Kernighan (27, 28). Each moveconsists of reversing the direction inwhich a section of the tour is traversed.More complicated moves have beenused to enhance the searching effective-ness of iterative improvement. We findwith the adaptive divide-and-conquer ef-fect of annealing at intermediate tem-peratures that the subsequence reversalmoves are sufficient (29).An annealing schedule was deter-

mined empirically. The temperature atwhich segments flow about freely will beof order N'2, since that is the averagebond length when the tour is highly ran-dom. Temperatures less than 1 should becold. We were able to anneal into locallyoptimal solutions with a c 0.95 for N upto 6000 sites. The largest traveling sales-man problem in the plane for which aproved exact solution has been obtainedand published (to our knowledge) has318 points (30).Real cities are not uniformly distribut-

ed, but are clumped, with dense andsparse regions. To introduce this featureinto an ensemble of traveling salesmanproblems, albeit in an exaggerated form,we confine the randomly distributed cit-ies to nine distinct regions with emptygaps between them. The temperaturegives the simulated annealing method ameans of separating out the problem ofthe coarse structure of the tour from thelocal details. At temperatures, such asT = 1.2 (Fig. 9a), where the small-scalestructure of the paths is completely dis-ordered, the longer steps across the gapsare already becoming infrequent andsteps joining regions more than one gapare eliminated. The configurations stud-ied below T = 0.8 (for instance, Fig. 9b)had the minimal number of long steps,

but the detailed arrangement of the longsteps continued to change down toT = 0.4 (Fig. 9c). Below T = 0.4, nofurther changes in the arrangement of thelong steps were seen, but the small-scalestructure within each region continued toevolve, with the result shown in Fig. 9d.

Summary and Conclusions

Implementing the appropriate Metrop-olis algorithm to simulate annealing of acombinatorial optimization problem isstraightforward, and easily extended tonew problems. Four ingredients areneeded: a concise description of a con-figuration of the system; a random gener-ator of "moves" or rearrangements ofthe elements in a configuration; a quanti-tative objective function containing thetrade-offs that have to be made; and anannealing schedule of the temperaturesand length of times for which the systemis to be evolved. The annealing schedulemay be developed by trial and error for agiven problem, or may consist of justwarming the system until it is obviouslymelted, then cooling in slow stages untildiffusion of the components ceases. In-venting the most effective sets of movesand deciding which factors to incorpo-rate into the objective function requireinsight into the problem being solved andmay not be obvious. However, existingmethods of iterative improvement canprovide natural elements on which tobase a simulated annealing algorithm.The connection with statistical me-

chanics offers some novel perspectiveson familiar optimization problems. Meanfield theory for the ordered state at lowtemperatures may be of use in estimatingthe average results to be obtained byoptimization. The comparison with mod-els of disordered interacting systemsgives insight into the ease or difficulty offinding heuristic solutions of the associ-ated optimization problems, and pro-vides a classification more discriminat-ing than the blanket "worst-case" as-signment of many optimization problemsto the NP-complete category. It appearsthat for the large optimization problemsthat arise in current engineering practicea "most probable" or average behavioranalysis will be more useful in assessingthe value of a heuristic than the tradition-al worst-case arguments. For such analy-sis to be useful and accurate, betterknowledge of the appropriate ensemblesis required.

Freezing, at the temperatures wherelarge clusters form, sets a limit on theenergies reachable by a rapidly cooledspin glass. Further energy lowering ispossible only by slow annealing. We

expect similar freezing effects to limit theeffectiveness of the common device ofemploying iterative improvement repeat-edly from different random starting con-figurations.

Simulated annealing extends two ofthe most widely used heuristic tech-niques. The temperature distinguishesclasses of rearrangements, so that rear-rangements causing large changes in theobjective function occur at high tempera-tures, while the small changes are de-ferred until low temperatures. This is anadaptive form of the divide-and-conquerapproach. Like most iterative improve-ment schemes, the Metropolis algorithmproceeds in small steps from one config-uration to the next, but the temperaturekeeps the algorithm from getting stuckby permitting uphill moves. Our numeri-cal studies suggest that results of goodquality are obtained with annealingschedules in which the amount of com-putational effort scales as N or as a smallpower of N. The slow increase of effortwith increasing N and the generality ofthe method give promise that simulatedannealing will be a very widely applica-ble heuristic optimization technique.Dunham (5) has described iterative

improvement as the natural frameworkfor heuristic design, calling it "design bynatural selection." [See Lin (6) for afuller discussion.] In simulated anneal-ing, we appear to have found a richerframework for the construction of heu-ristic algorithms, since the extra controlprovided by introducing a temperatureallows us to separate out problems ondifferent scales.

Simulation of the process of arriving atan optimal design by annealing undercontrol of a schedule is an example of anevolutionary process modeled accurate-ly by purely stochastic means. In fact, itmay be a better model of selection pro-cesses in nature than is iterative im-provement. Also, it provides an intrigu-ing instance of "artificial intelligence,"in which the computer has arrived al-most uninstructed at a solution thatmight have been thought to require theintervention of human intelligence.

References and Notes

1. E. L. Lawlor, Combinatorial Optimization(Holt, Rinehart & Winston, New York, 1976).

2. A. V. Aho, J. E. Hopcroft, J. D. Ullman, TheDesign and Analysis of Computer Algorithms(Addison-Wesley, Reading, Mass., 1974).

3. M. R. Garey and D. S. Johnson, Computers andIntractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979).

4. R. Karp, Math. Oper. Res. 2, 209 (1977).5. B. Dunham, Synthese 15, 254 (1963).6. S. Lin, Networks 5, 33 (1975).7. For a concise and elegant presentation of the

basic ideas of statistical mechanics, see E. Shro-dinger, Statistical Thermodynamics (CambridgeUniv. Press, London, 1946).

8. N. Metropolis, A. Rosenbluth, M. Rosenbluth,A. Teller, E. Teller, J. Chem. Phys. 21, 1087(1953).

9. G. Toulouse, Commun. Phys. 2, 115 (1977).

10. For review articles, see C. Castellani, C. DiCas-tro, L. Peliti, Eds., Disordered .Systems andLocalization (Springer, New York, 1981).

11. J. Soukup, Proc. IEEE 69, 1281 (1981).12. M. A. Breuer, Ed., Design Automation of Digi-

tal Systems (Prentice-Hall, Engelwood Cliffs,N.J., 1972).

13. D. Sherrington and S. Kirkpatrick, Phys. Rev.Lett. 35, 1792 (1975); S. Kirkpatrick and D.Sherrington, Phys. Rev. B 17, 4384 (1978).

14. A. P. Young and S. Kirkpatrick, Phys. Rev. B25, 440 (1982).

15. B. Mandelbrot, Fractals: Form, Chance, andDimension (Freeman, San Francisco, 1979), pp.237-239.

16. S. Kirkpatrick, Phys. Rev. B 16, 4630 (1977).17. C. Davis, G. Maley, R. Simmons, H. Stoller, R.

Warren, T. Wohr, in Proceedings of the IEEEInternational Conference on Circuits and Com-puters, N. B. Guy Rabbat, Ed. (IEEE, NewYork, 1980), pp. 669-673.

18. M. A. Hanan, P. K. Wolff, B. J. Agule, J. Des.Autom. Fault-Tolerant Comput. 2, 145 (1978).

19. M. Breuer, ibid. 1, 343 (1977).20. P. W. Case, M. Correia, W. Gianopulos, W. R.

Heller, H. Ofek, T. C. Raymond, R. L. Simek,C. B. Steiglitz, IBM J. Res. Dev. 25, 631 (1981).

21. A. J. Blodgett and D. R. Barbout, ibid. 26, 30(1982); A. J. Blodgett, in Proceedings of theElectronics and Computers Conference (IEEE,New York, 1980), pp. 283-285.

22. K. A. Chen, M. Feuer, K. H. Khokhani, N.Nan, S. Schmidt, in Proceedings of the 14thIEEE Design Automation Conference (New Or-leans, La., 1977), pp. 298-302.

23. K. W. Lallier, J. B. Hickson, Jr., R. K. Jackson,paper presented at the European Conference onDesign Automation, September 1981.

24. D. Hightower, in Proceedings of the 6th IEEEDesign Automation Workshop (Miami Beach,Fla., June 1969), pp. 1-24.

25. J. Beardwood, J. H. Halton, J. M. Hammers-

ley, Proc. Cambridge Philos. Soc. 55, 299(1959).

26. D. J. Resenkrantz, R. E. Steams, P. M. Lewis,SIAM (Soc. Ind. Appl. Math.) J. Comput. 6, 563(1977).

27. S. Lin, Bell Syst. Tech. J. 44, 2245 (1965).28. __ and B. W. Kernighan, Oper. Res. 21, 498

(1973).29. V. teemy has described an approach to the

traveling salesman problem similar to ours in amanuscript received after this article was sub-mitted for publication.

30. H. Crowder and M. W. Padberg, Manage. Sci.26, 495 (1980).

31. The experience and collaborative efforts ofmany of our colleagues have been essential tothis work. In particular, we thank J. Cooper, W.Donath, B. Dunham, T. Enger, W. Heller, J.Hickson, G. Hsi, D. Jepsen, H. Koch, R.Linsker, C. Mehanian, S. Rothman, and U.Schultz.

Induced Bone Morphogenesis

Bone Cell Differentiationand Growth Factors

Marshall R. Urist, Robert J. DeLange, G. A. M. Finerman

Bone differs from other tissue not onlyin physiochemical structure but also inits extraordinary capacity for growth,continuous internal remodeling, and re-generation throughout postfetal life,even in long-lived higher vertebrates.How much of this capacity can be ac-

more than a century and is measured inreactions of periosteum and endosteumto injury, diet, vitamins, and hormones.Bone-derived growth factors (BDGF)stimulate osteoprogenitor cells to prolif-erate in serum-free tissue culture media(3, 4). The mechanisms of action of BMP

Summary. Bone morphogenetic protein and bone-derived growth factors arebiochemical tools for research on induced cell differentiation and local mechanismscontrolling cell proliferation. Bone morphogenetic protein irreversibly induces differen-tiation of perivascular mesenchymal-type cells into osteoprogenitor cells. Bone-derived growth factors are secreted by and for osteoprogenitor cells and stimulateDNA synthesis. Bone generation and regeneration are attributable to the co-efficiencyof bone morphogenetic protein and bone-derived growth factors.

counted for by proliferation of prediffer-entiated osteoprogenitor cells and howmuch can be attributed to induced differ-entiation of mesenchymal-type cellshave been challenging questions for along time. A basic assumption is thatregeneration occurs by a combination ofthe two processes. The process of in-duced cell differentiation has been ob-served from measurements of the quanti-ties of bone formed in response to im-plants of either bone matrix or purifiedbone morphogenetic protein (BMP) inextraskeletal (1) and intraskeletal (2)sites. The osteoprogenitor cell prolifera-tion process has been well known for

680

and BDGF are primarily local, but sec-ondary systemic immunologic reactionscould have either permissive or depres-sive effects.Recent progress in the field, surveyed

in this article, suggest that BMP andBDGF are coefficient; BMP initiates thecovert stage and BDGF stimulates theovert stage of bone development. Theeffects of BMP are observed on morpho-logically unspecialized mesenchymal-type cells either in systems in vitro or invivo. The action of BDGF is demonstra-ble only in tissue culture, ostensibly onmorphologically differentiated bonecells.

The theory of cell differentiation byinduction originates in observations ontransplants of embryonic tissues and is amain tenet of modern developmental bi-ology. Development begins with a mor-phogenetic phase and ends with a cyto-differentiation phase (5). The morpho-genetic phase consists of cell disaggre-gation, migration, reaggregation, andproliferation. Through interaction of in-tra- and extracellular influences, cytodif-ferentiation follows and a mature func-tional specialized tissue emerges. As cy-todifferentiation occurs, pattern forma-tion is established by the positionalvalues of cells in a three-dimensionalextracellular coordinate system (6). Pat-tern formation is a difficult concept toexplain because it is heritable, encom-passes morphogenesis, and is the culmi-nation of manifold physiochemical pro-cesses. Present evidence indicates thatchondro-osteogenetic gene activation isinduced at the onset of the morphogenet-ic phase of bone development and isregulated by a combination of extra- andintracellular factors.

Previous studies on extracellular ma-trix factors consisted chiefly of biochem-ical interpretations of descriptive mor-phology. The emphasis has been onuncertainties about when the morpholog-ically predifferentiated (protodifferen-tiated or covert) stage of developmentbegins, if and when an inductive agent istransferred from extracellular matrix toresponding cell surfaces, and how alter-ations in the genome occur. Since alter-ations ultimately occur at the level ofDNA, clear-cut distinctions between ex-

Dr. Urist is director of the Bone Research Labora-tory and a professor of surgery (orthopedics), at theUniversity of California Los Angeles (UCLA) Medi-cal School, Los Angeles 90024. Dr. DeLange is aprofessor of biological chemistry, at UCLA, and Dr.Finerman is a professor of orthopedic surgery atUCLA.

SCIENCE, VOL. 220