Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C016 Finals Page 312 23-9-2008 #3 312 Handbook of Algorithms for Physical Design Automation Algorithm simulated_annealing(void) 1 T ← T 0 /* Initial Temperature */ 2do 3do 4 j = generate ( i )/*Move Strategies */ 5if accept ( C,T ) then /* Metropolis function */ 6 i = j 7 until cost is in equilibrium /* Temperature equilibrium */ 8 update(T) /* Temperature decrement */ 9 until cost cannot be reduced any further /* Stopping criteria */ FIGURE 16.1 Basic simulated annealing algorithm. Unfortunately, the placement problem described here does not exhibit optimal substructure. If we apply the greedy algorithm search strategy, we will usually get stuck in a local minimum. This means that c(i) ≥ c(j min ), ∀ j ∈ S(j min ) (16.1) where j min is th e local minimum state, and S(j min ) is the set of states reachable from the state j min . In many cases, there is a large disparity between the local minimum and the global minimum cost. We n eed a search strategy that avoids local minima and finds the global minimum. Simulated annealing is such a search strategy. At the heart of the simulated annealing algorithm is the Metropolis Monte Carlo procedure that was introduced to provide an efficient simulation of a collection of atoms in equilibrium at a given temperature [2]. The Metropolis procedure is the inner loop of the simulated annealing algorithm as shown in Figure16.1. Although the greedy algorithm forbids changes of state that increase the cost function, the Metropolis procedureallows moves to states that increase the cost function. Kirkpatrick et al. suggested that the Metropolis MonteCarlomethodcan beusedtosimulatethephysicalannealing process and to solve combinatorial optimization problems [1]. They suggested adding an outer loop that lowers the temperature fro m a high melting temperature in slow stages until the system freezes, and no further changes occur. At each temperature, the simulation must proceed long enough for the system to reach a steady state. The sequence of temperatures and the method to reach equilibrium at each temperature is known as annealing schedule. They showed that this same technique can be applied to combinatorial optimization problems if a cost function is used in place of energy, and the temperature is used as a control parameter. 16.2 ANNEALING SCHEDULES It has been shown that the simulated annealing algorithm, when started in an arb itrary state and given an appropriate annealing schedule, will eventually convergeto a global optimum [3]. Although these results required an infinite amount of computation time for the convergence guarantee, in practice, simulated annealing has been extremelysuccessful when appliedto circuit partitioningand placement problems. It has outperformed all other known algorithms if given sufficient time resources. The essential elements of the simulated annealing algorithm are summarized below in Figure16.1. The algorithm consists of two loops. Each execution of the inner loop generates new configurations to be evaluated at constant temperature. The acceptance of a new configuration j depends on the current temperature T and the change in cost between the current configuration i and the proposed configuration j as presented in Figure 16.2. All configuration changes that do not increase the cost are accepted as in any-iterative improvement algorithm, but moves with C > 0 Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C016 Finals Page 313 23-9-2008 #4 Placement Using Simulated Annealing 313 Algorithm accept ( C,T ) 1ifC ≤ 0 then /*new cost is less than or equal to the old cost */ 2 return(ACCEPT) /*accept the new configuration */ 3 else 4 randomly generate a number r between 0 and 1 5if r < e − C/T then return(ACCEPT) 6 else return (REJECT) FIGURE 16.2 Acceptance function for the simulated annealing algorithm. are accepted depending on the value of C and the value of T. The Boltzmann distribution e −C T that governs physical annealing is used as the criteria for determining acceptance of states with increased cost. In this simple formulation of simulated annealing, we designate that the inner loop is repeated until the average value o f the cost appears to have converged.As T is lowered from a high value, large uphill moves are mostly rejected. As T is lowered further, moves with yet lower values of C > 0 become largely rejected. In some sense, critical decisionsare made for those values of C > 0which are on the order of the value of T. Hence, simulated annealing operates in apseudohierarchical fashion with respect to C > 0 values as T is decreased. 16.3 SIMULATED ANNEALING AND PLACEMENT The critical ingredients for a implementing a successful placer based on simulated annealing are the simulated annealing cooling schedule, the cost fu nction to be evaluated, and the generation of new state configurations or move strategies. Although simulated annealing placers are quite straightfor- ward to implement, the best results in terms of quality and execution time have been obtained with careful attention to these details. We discuss each of these aspects of simulated annealing in turn. 16.4 SIMULATED ANNEALING COOLING SCHEDULES A simulated annealin g cooling schedule is differentiated by the implementation of four lines of the basic annealing schedule presented in Figure16.1: initial temperature selection, temperature equilibrium criteria, temperature update, and stopping criteria. A common implementation is easily coded as shown in Figure 16.3. Here, iterations is a variable which counts the number of Metropolis cycles or inner loop executions, numberOfMovesis a variable which counts the number of generated new configurations in an iteration, I max is the predetermined maximum number of iterations, N max is the predetermined maximum number of moves generated per iteration, and ∝ is the temperature multiplier. Although the previous implementation is simple, effective, and easily programmed, it has a major drawback: at low temperatures, the running time is very long because many candidates for movesare rejected before each moveto a different configuration.To remedy this inefficiency, various approaches have been proposed to speed the algorithm including parallel implementations [4–6] as well as rejectionless hill climbing [7]. Lam studied the problem and proposed a statistical annealing schedule [8]. Lam’s schedule is based on the observation that annealing is successful if the system is kept close to thermal equilibrium as the temperature is lowered. However, to keep the system in equilibrium at all times requires that the temperature decrements b e infinitesimal; a long time would have passed before the system is frozen, and annealing is stopped. From a practical standpoint, a good annealing schedule must, therefore, achieve a compromise between the quality of the final solution and the computation time. To determine when the system is in equilibrium so that the Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C016 Finals Page 314 23-9-2008 #5 314 Handbook of Algorithms for Physical Design Automation 1 T ← largeNumber /* Initial Temperature */ 1b iterations ← 0 2do 2b numberOfMoves ← 0 3do 4 j = generate ( i )/*Move Strategies */ 4b increment numberOfMoves 5if accept ( C,T ) then /* Metropolis function */ 6 i = j 7 while numberOfMoves ≤ N max /* Temperature equilibrium */ 8 T ← α T, 0.8 ≤ α ≤ 0.99 /* Temperature decrement */ 8b increment iterations 9 while iterations ≤ I max /* Stopping criteria */ FIGURE 16.3 Simple simulated annealing algorithm. temperature c ould be lowered, we need an equilibrium cr iterion [ 9]. A system is close to equilibrium at temperature T if the following condition is satisfied µ(s) −λσ(s) ≤¯c ≤ µ(s) +λσ(s) (16.2) where ¯c is the average cost of the system s = 1/T is th e inverse temperature µ(s) and σ(s) are the mean and standard deviation of the cost if the system were in thermal equilibrium at temperature T The parameter λ, which can be made as small as desired to ensure a good approximation of equilibrium, realizes the compromise between the quality of the final solution and the computa- tion time: the smaller the λ, the better is the quality of the final solution and the longer is the computation time. Simulated annealing has been applied to the placementproblem in the TimberWolfsystem. Com- plete accounts of the implementations of simulated annealing for earlier versions of the TimberWolf placement programs have been published [10–18]. The inclusion of the resu lts of a theoretically derived statistical annealing schedule have been responsible for the very significant reduction in the CPU time required by TimberWolf. We now presentthe adapationof Lam’s statisticalannealing schedule[8]found in TimberWo lf. In his work, Lam showed theoretically that theoptimum acceptance rate ofproposed newconfigurations is approximately 44 percent. In Lam’s algo rithm, a range limiter w indow (first described in Ref. [10]) is used to keep the acceptance rate (denoted as ρ) as close as possible to 44 percent. (The range limiter window bounds the magnitude of the perturbation (or move distance) from the current state. The range limiter window size is designed to increase the acceptance rate at a given temperature. Changes in cost are on the order of the move distance. Therefore, reducing the move distance yields smaller values and hence an elevated acceptance rate.) In the beginning of the execution of this algorithm, the temperature T is set to a very high value ( effectively infinity). Even with the range limiter dimensions encompassing the entire chip, the acceptance rate ρ approaches 100 percent. Because a further increase in range limiter dimensions cannot decrease ρ, there clearly must be a region of operation for the algorithm in which ρ is above th e ideal value of 44 percent. Also, as T gets sufficiently low, the range limiter dimensions reduce to their minimum values. Then, as ρ drops below 44 percent, there is no way for it to return to a higher level. It is therefore apparent Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C016 Finals Page 315 23-9-2008 #6 Placement Using Simulated Annealing 315 Acceptance rate r 1.0 0.44 Region 1 Region 2 Region 3 Generated new configurations FIGURE 16.4 Anticipated plot of the acceptance rate versus generated new configurations. that there is a region of operation in which ρ falls from 44 percent toward zero as T approaches zero. The anticipated three regions of operation (ρ above 0.44, ρ equals 0.44, and ρ below 0.44) are illustrated in Figure 16.4. One disadvantage of the schedule developed by Lam is its inability to accurately predict when the execution of the algorithm will end from the beginning of the run. That is, it is not known how many new configurations will be generated during the course of the execution of the algorithm. In an effort to gain a different perspective on Lam’s theory, the authors of TimberWolf measured ρ versus generated new configurations for executions on several industrial circuits. One objective was to determine the percentage of the run (i.e., the percentage of the total new configurations generated) devoted to each of the three regions of operation. These percentages were remarkably similar for the very wide range of circuit sizes which were tested. A typical plot is shown in Figure 16.5. Acceptance rate r 15 percent 50 percent 35 percent Generated new configurations 1.0 0.44 FIGURE 16.5 Typical measured acceptance rate versus generated new configurations as obtained from exper- iments conducted on s everal industrial circuits, showing the percentage of the run spent in each region of operation. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C016 Finals Page 316 23-9-2008 #7 316 Handbook of Algorithms for Physical Design Automation They discovered that for region 1 (which encompasses approximately 15 percent of the run), ρ versus generated new configurations could be modeled by an exponential function. This function has a peak value of 1.0, and passes through the point where ρ first reduces to 0.44. Furthermore, they found that region 3 could also be modeled by an exponential function with peak value 0.44 and minimum value0.0. In region 2, the acceptance r ate is flat, but they d iscovered that the decrease in the range limiter window dimensions as a function of generated new configurations can also be modeled by exponential functional form. That all three regions can be modeled by exponential functions is not surprising in light of the use of the (exponential) Boltzmann-like factor used to govern acceptance or rejection of new configurations. Here they define an iteration (represented by I where 1 ≤ I ≤ I max ) to correspond to an interval along the horizontal axis in Figure16.6. That is, N max new configurations are generated during iteration I. An iteration defines a set of N max moves during which the range limiter window dimensions remain con stant. In simulated annealing, the more new configurations generated during the course of a run, the higher the probability of achieving a better solution. However, extensive experimentation suggested the existence of a diminishing return on the number of new configurations generated. Therefore, a default number of moves can be determined for which the best results can be obtained with high probability. The default total number of moves during a run is set to total moves = 1500N 4/3 c (16.3) where N c is the numberof cells. In TimberWolf implementations, they set I max equal to 150 iterations. Therefore: N max = 10N 4/3 c (16.4) Note that the range limiter dimensions ar e actually changed 50 per cent of 150 times, or 75 times during the course of a run (i.e., its dimensions only change during region 2 of the operation of the annealing algorithm). Becausewe knowthat theacceptance r ate behavior described inFigure16.5along withthe default values of I max and N max yield close to the best possible results for simulated annealing, the algorithm Acceptance rate r 0.15I max 0.65I max 1.0I max Generated new configurations 1.0 0.44 FIGURE 16.6 Target acceptance rate versus iteration. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C016 Finals Page 317 23-9-2008 #8 Placement Using Simulated Annealing 317 is forced to strictly obey that acceptance rate behavior through the use of a feedback mechanism. That is, for each iteration I (I varies from 1 to I max ), one can compute the target acceptance rate (ρ T I ) as shown in Figure16.6. To ensure that significant further reductions in the cost are not possible, the target acceptance rate is set to be below 1 percent at the last iteration (I max ). One can force the actual acceptance rate to track the target acceptance rate by using negative feedback control on the temperature T: T = 1 − ρ I − ρ T I K T (16.5) where K is a damp ing constant used to stabilize the control of the value of T (in TimberWolf implementations,a very suitable value of K is 40). T is updated every update_limitmoves(as defined in the description of our simulated annealing algorithm in Figure 16.7). Note that T can increase as well as decrease as the executionof the algorithm proceeds,and the range limiter windowdimensions decrease exponentially as a function o f the number of iterations. In Lam’s schedule by contrast, T decreases monotonically but the range limiter window dimensions fluctuate up o r down. Clearly these two parameters are closely related. It is sufficient to dictate the functional form for either one, and let the other parame ter adapt to monitore d cond itions. Algorithm simulated_annealing ( X 0 ) 1 X ← X 0 /* set current configuration equal to initial configuration */ 2 T ← set_initial_T () /* sufficiently sample configuration space to ascertain value of T yielding an initial acceptance rate slightly below 100 percent */ 3 I ← 1 4 while I ≤ I max do 5 N ← 0 /* N is the number of moves attempted so far during iteration I */ 6 set_range_limiter_size(I) /* sets range limiter window dimensions */ 7 up ← 0 /* update counter */ 8 while N ≤ N max do 9 N ← N + 1 10 up ← up + 1 11 if up = update_limit then /* we need to update the temperature T */ 12 up ← 0 /* reset the counter */ 13 if ρ I <ρ T I then 14 raise_temp (T) 15 else if ρ I >ρ T I then 16 lower_temp (T) 17 Y = generate(X) /* propose a new configuration */ 18 C = C(Y) − C(X) /* compute the cost change */ 19 if accept ( C,T )then 20 X ← Y /* accept the new configuration to be the current config. */ 21 I ← I + 1 FIGURE 16.7 Advanced simulated annealing algorithm. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C016 Finals Page 318 23-9-2008 #9 318 Handbook of Algorithms for Physical Design Automation The heuristic adaptation of Lam’s schedule shown in Figure 16.7 did not show a difference in placement quality for a given execution time as compared to Lam’s original version and was adopted as the annealing schedule in TimberWolf. The TimberWolf approach generates a fixed number of moves for a circuit of a given size, and therefore, the number of iterations is known a priori. 16.5 COST FUNCTIONS One of the advantages of the simulated annealing algorithm is its ability to accommodate any cost function. In fact, there are no constraints on the form of the cost function. However, recent research has shown that the best results are linear or logarithmically related terms or variables. Siarryet al. have “noticed improved convergence toward the correct results when using no rmalized variables instead of unnormalized real variables range exploration with the same simulated annealing algorithm” [19]. Traditionally, a common cost function for simulated annealing row-based placers is the weighted summation of total half-perimeter wirelength, timing penalty, overlap penalty, row leng th control penalty, and congestion penalty: C = β w W +β t P t + β o P o + β r P r + β c P c (16.6) where W = N N n=1 max v i ,v j ∈n |x i − x j |+max v i ,v j ∈n |y i − y j | (16.7) P t = N P p=1 D p (16.8) D p = f (R, C, l, t g ) (16.9) P o = k=l O x (k, l) 2 (16.10) P r = N R r=1 |L(r) −L d (r)| (16.11) P c = N x m=1 N y n=1 C g (m, n) (16.12) C g (m, n) = 0, (d mn ≤ s mn ) d mn − s mn , (d mn > s mn ) (16.13) The wirelength term W is the summation overall nets where each net consists of a set of terminals v i ,and(x i , y i ) is the coordinate of v i . The constant N N represents the total number of nets present in the design. The timing penalty P t is the summation of all N P path delays in the circuit. The generalized delay function D p is shown as a complex function of resistance R, capacitance C, wirelength l of path, and propagation delay t g through the circuit. The timing model may utilize lookup tables, Elmore delay calculations, or simple lumped capacitance calculations. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C016 Finals Page 319 23-9-2008 #10 Placement Using Simulated Annealing 319 The overlap penalty function O x (k, l) returns the amount of overlap of cells k and l in the x direction of the row (as we assume horizontal rows). The overlap term is used to insure a legal placement at the end of annealing, that is, no two cells overlap in a row or area. The row length penalty function is present to ensure that each row in a standard cell placement is filled to a desire length. The function L(r) returns the length of row r and the function L d (r) returns the desired length of row r. The congestion cost P C is calcu lated by overlaying a two-dimensional global bin structure over the design. Global routing is performed on each net by mapping each terminal vertex v i to its corresponding bin (m, n), collapsing the terminals within a bin, and interconnecting the terminals spanning the bins. Each time a net crosses a bin, the demand for an bin edge is incremented. The total demand for a bin is the sum of all bin edges. The geometry of the design determines the routing supply s mn available for the bin. An overflow occurs if the demand of a bin d mn exceeds its supply s dm . The congestion is the sum overflows over all global routing bins. Each of the terms of the cost function are multiplied by a scaling factor β i to balance the relative importance of the term. To achieve good results over many different circuits and conditions, a feedback mechanism was proposed to control the individual β i [12]: β i I+1 = max 0, β i I + P i − P T i P T i (16.14) where the scaling factor at the next iteration β i I+1 is calculated from the current scaling factor β i I (at the Ith iteration) and an error term representing the deviation of penalty P i from the ideal target penalty P T i . While this does help improve the final result and drive the penalty terms to zero, this method does not adequately determine the initial scaling factor β i 0 and m ay require a damping factor similar to Equation 16.5 to prevent numerical large oscillations of the scaling factor. Furthermore, to achieve satisfactory results, this method requires a significant tuning effort. Nevertheless, many simulated annealing placers used cost functions of this general form. In fact, the early versions of TimberWolfSC, the row-based simulated annealing placer used the following cost function [12]: C = W +β o P o + β r P r (16.15) For floorplanning or macrocell placement problems, the overlap penalty becomes two dimensional and an additional term is sometimes added to minimize wasted area between cells known as white space: P S = A C (s) A T (16.16) where A c (s) is the total area of the chip including white space and A T is the sum of all of the cell areas. In this case, the scaling factor has been defined as [4]: β s = K 0 P s < 1 K 1 e r P s ≥ 1 (16.17) where K 0 and K 1 are two constants such that K 0 K 1 e r to ensure feasibility. However, these straightforward functions suffer in that they fail dimensional analysis as the individual terms are not unit compatible. This makes the cost function unfit for general use and susceptible to problems tuning the weight factors. While one can attempt to optimize the weight factors using a set of benchmark circuits, constant weighting factors precludes optimal solutions over a sufficiently large dynamic range. In addition, the feedback control of these weighting factors becomes more unstable as the dynamic range increases; it will become increasingly difficult to Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C016 Finals Page 320 23-9-2008 #11 320 Handbook of Algorithms for Physical Design Automation balance the linear terms again quadratic terms. Clearly, as problems change in size and topology, the relative attention paid to individual terms of the cost function will vary enormously. Yet most published works on placement h ave cost functions of the mixed form of Equation16.6. This is due to an over reliance on b enchmarks as a performancemeasure. Benchmarks offered at given technology node are similar in terms of scale and mask the mixed unit cost function problem. One can avoid the problematic mixed unit cost function by rewriting the cost function in terms of a single dimension, length, ma king all terms unit compatible. The timing penalty can be rewritten in terms of a path length penalty where the bounds are given or derived from timing analysis [17]: P p = ⎧ ⎨ ⎩ length(p) −upperBound(p) length(p)>upperBound(p) lowerBound(p) −length(p) length(p)<lowerBound(p) 0otherwise (16.18) length(p) = ∀n∈p W n (16.19) The cell overlap penalty may be completely eliminated through the use of cell shifting and the row length control penalty may be eliminated by careful attention to row bounds during n ew state generation [15]. The floorplanning area term can be rewritten as the squ are root of its area and this was utilized in TimberWolfMC [18]. The congestion penalty is more challenging but it can be rewritten in terms of detour length o r the additional length needed by a net to avoid a congested area. Kahng and Xu have shown how to effectively calculate the detour length from a congestion map [20]. Sun and Sechen [15] proposed just two terms in their cost function for Timberwolf version 7 whereas the co mmercial version of TimberWolf (aka InternetCAD itools) uses the following strict length-based cost function which utilizes half-perimeter, timing, and deto ur costs: C = W +β t P t + β d P d (16.20) 16.6 MOVE STRATEGIES Most simulated annealing placement algorithms predominately use two new configuration strategies or moves: a relocation of a single cell to a new position and a pairwise exchange of cells. Sechen and Lee [12] proposed a bin structure to automatically control the ratio of single cell relocations to pairwise exchanges. Each standard cell row is divided into bins. The center of each cell is assigned to a bin. A new move is proposed as follows: A cell a is randomly chosen. A new position is chosen which resides within the range limiter window and its corresponding bin is calculated. If the bin is empty, a single cell move to this position is performed. Otherwise, randomly pick cell b from the cells in the bin. Cells a and b are exchanged as shown in Figure 16.8 Although the primary new state strategy is the single and pairwise exchange of cells, other new state generators have been proposed and adopted. In row-based standard cell p lacement algorithms, cell o rientation and exchange of adjacent cells are common moves at low temperatures. Floor- planning or macrocell placers are augmented with aspect ratio modification, and p in optimization moves. Simulated annealing device placement algorithms are further enhanced with transistor fold- ing, diffusion merging, cell grouping, and symmetry operations. Hustin and Sangiovanni-Vincentelli proposed a dynamicand adaptive move strategy that optimizes the amountof work performedat each temperature. They compute a quality factor for each type of move m for a given temperature T [21]: Q T m = j∈A m |C j | G m (16.21) Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C016 Finals Page 321 23-9-2008 #12 Placement Using Simulated Annealing 321 Range limiter window Single cell move Cell c Pairwise exchange Cell b Cell a Now divided into bins FIGURE 16.8 Automatic move strategy. where G m is the number of generated moves A m is subset of accepted moves of type m,thatis,A m ⊆ G m |C j | is absolute value of the change in cost due to the accepted move m The probability of proposing the move m at a given temperature is then given by p T m = Q T m m Q T m (16.22) As you can see, the quality factor and hen ce probability of selecting a move m will be high when moves of this type are frequently accepted or when the average change in cost is large at the current temperature. This method discourages small delta cost moves at high temperatures where they would have little impact on the progress of explo ring the state space and discourages large delta cost moves at low temperatures where such moves would drastically perturb the current state and have little chance of acceptance. Sechen and Lee’s work describes many of the details of im plementing a simulated annealing placer. It is the basis for many of the advanced works in the field. It is available in source code in the SPEC CPU2000 benchmark set [22]. 16.7 MULTILEVEL METHODS To reduce the executiontime of simulated annealing placement, multilevel methods were introduced. Mallela and Grover were the first to introduce a two-step annealing p rocess to standard cell place- ment to reduce runtime [23]. The execution time is reduced by effectively reducing the size of the problem through clustering of the standard cells. First they form clusters of cells based on their inter- connections. Cells that are highly interconnected will be placed into the same cluster. The execution time of the clustering algorithm is only a small fraction of simulated annealing placement time. The clustered netlist is then placed using simulated annealing placement. Because the numb e r of total cells has been reduced, the execution time of the simulated annealing problem is reduced. Next, the clusters are broken up and the original netlist is restored. Then a final low temperature simulated . percentage of the run spent in each region of operation. Alpert /Handbook of Algorithms for Physical Design Automation AU7242_C016 Finals Page 316 23-9-2008 #7 316 Handbook of Algorithms for Physical Design. algorithm. Alpert /Handbook of Algorithms for Physical Design Automation AU7242_C016 Finals Page 318 23-9-2008 #9 318 Handbook of Algorithms for Physical Design Automation The heuristic adaptation of Lam’s. Alpert /Handbook of Algorithms for Physical Design Automation AU7242_C016 Finals Page 312 23-9-2008 #3 312 Handbook of Algorithms for Physical Design Automation Algorithm simulated_annealing(void) 1 T ← T 0 /*