1. Ramakrishna sramki@gmail.com
Lecture#2
CAD for VLSI
Ramakrishna

2. Combinatorial Optimization E.g Combinatorial Optimization Reference
Lecture# Outline
Combinatorial Optimization
E.g Combinatorial Optimization
Reference
G DeMicheli “Synthesis and Optimization of Digital Circuits” – Ch – 2.4
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3. Recap Need for CAD? Handling Complexity Graph Terminology
Combinatorial Optimization
Decision problems
Solution with certain characterisitics.
Optimization Problems
What is best/optimal/exact solution.
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4. Different Algorithms Branch-Bound Algorithm: Greedy Algorithm:
Part of the Decision tree is evaluated.
Sub-trees considered based on Current_Cost
Greedy Algorithm:
Based on series of local decisions and try to solve the sub problem.
Short/long path problems
Bellman -- si = min (si, sk+wk,i) i = 1,2 ..n
Bellman-ford -- extension to Bellman algorithm which is meant for graphs with cycles. Si j+1 = min[sij, (skj + wk,i)]
Liao_Wong Algorithm for long paths.
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5. A bin-packing Problem A collection of items T = {t1, t2, t3, …. tn}.
Every item tn has an integer size sk.
Set of bins B, each with fixed integer size b.
Can hold items if the sum of their sizes is b or less
Goal: Pack all items, using a minimum number of bins
Analogy:
1D packing problem, packing std cells in rows.
2D packing problem – Floor plan
3D packing problem – Multi layer routing, Resource allocation, Stacking.
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6. Packing Problem Bins with maximum size of 6 units.
We have items with sizes: { 1, 4, 2, 1, 2, 3, 5}
Cost function is simply the number of bins – B
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7. Greedy Approach Each step, go for local minimum
Simple and fast heuristic, does not produce global optimum
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8. Greedy Algorithm Contd…
Iteratively the final solution will be:
{1,4} { 2,1,2} {3} {5}
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9. Greedy Algorithm Contd…
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10. Greedy Algorithm Contd…
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11. Hierarchical approach
Partition the problem into smaller sub-problems
Initial partitioning {1, 4, 2, 1} {2, 3, 5} => {1,4} {2,1} {2, 3} {5}
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12. Simulated Annealing
Exploits the metal cooling and freeze into a crystalline structure (minimum energy structure, the annealing process).
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M. N., Teller, A.H. and Teller, E., Equations of State Calculations by Fast Computing Machines, J. Chem. Phys. 21, , 1958.
This paper/algorithm proposes a means of finding the equilibrium configuration of a collection of atoms at a given temperature.
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13. Simulated Annealing Contd…
Pincus figured the connection of this algorithm to the mathematical minimization and Kirkpatrick, S., Gerlatt, C. D. Jr., and Vecchi, M.P., Optimization by Simulated Annealing, Science 220, , 1983 showed it forms the basis of an optimization technique for combinatorial problems.
SA has the ability to avoid becoming trapped at local minima.
It uses random search and accepts changes that decrease the objective function f, but also accepts that increase it.
Changes that impact the objective function and increases it, SA incorporates them with a probability.
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14. Simulated Annealing Contd…
Step 1: Initialize – Start with a random assumption, Initialize a very high “temperature”
Step 2: Move – Perturb the system through a defined move
Step 3: Calculate score – calculate the change, due to the above move
Step 4: Choose -- Accept or reject the current move. The probability of acceptance depending on current temperature.
Step 5: Update and repeat – Update the temperature and move back to Step 2.
Continue the above process until Freezing Point is reached.
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15. Simulated Annealing Contd…
Pseudo Code:
Start with the system in known configuration,
T = temperature = hot; frozen = false;
While (! Frozen) {
Repeat {
Perturb system slightly
Compute ΔE, change in energy due to above move
If (ΔE < 0)
Accept it with probability = e – ΔE/T
} until (the system in thermal equilibrium at this T)
If (E still decreasing over the last few temperatures)
Then T = 0.x T
Else frozen = true }
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16. Simulated Annealing Start with any feasible solution.
Try to merge bins (improve)
At the beginning, allow splitting of bins (may make the cost worse)
Moves between solutions are probabilistic
Entire process is random
Merge: Choose two random bins and try to merge it
Split: Choose a random bin and split it up
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17. Simulated Annealing Contd…
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18. Simulated Annealing Contd…
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19. Simulated Annealing Contd….
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20. Simulated Annealing Contd….
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21. Simulated Annealing Contd….
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22. Class Work Derive the same using Exhaustive search (5)
Provide the detailed tree.
Apply the simulated annealing algorithm for a TSP with the given “N cities and distances among them, find a shortest tour of N cities by visiting each city exactly once and returning to starting city”. (5)
Cost of traveling is represented as a matrix:
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