EE4271 VLSI Design

1. EE4271 VLSI Design Logic Synthesis EE 4271 VLSI Design

2. Logic Synthesis • Starts from RTL description in HDL or Boolean expressions • Outputs a standard cell netlist EE 4271 VLSI Design

3. Boolean Function • f: Bm Yn • B = {‘0’, ‘1’}, Y = {‘0’, ‘1’, ‘-’} • The function is incompletely specified, don’t care ‘-’ • For each output, space of Bm can be partitioned into • on-set: all inputs leading to output ‘1’ • off-set: all inputs leading to output ‘0’ • dc-set: all inputs leading to output ‘-’ EE 4271 VLSI Design

4. Points in Input Space • Assigning ‘1’ or ‘0’ to each of the m Boolean variables x1, x2, …, xm specifies a point in input space Bm • Example • (‘1’, ‘0’, ‘1’) in B3 • x1=‘1’ Λ x2=‘0’ Λ x3=‘1’ • x1•x2•x3 EE 4271 VLSI Design

5. Terminology • Literal: a Boolean variable or its complement • Minterm: a product of all input variables or their complements – a point in Bm • Cube: a product of input variables or their complements • The fewer number of variables, the bigger space covered EE 4271 VLSI Design

6. Boolean Function Representation • A Boolean function can be specified by a sum of minterms • The expression has a minterm for each point in the on-set EE 4271 VLSI Design

7. Implicant and Cover • An implicant is a cube whose points are either in the on-set or the dc-set • A prime implicant is an implicant that is not included in any other implicant • A set of prime implicants that together cover all points in the on-set (and some or all points of the dc-set) is called a prime cover EE 4271 VLSI Design

8. Irredundant Cover • An prime cover is irredundant when none of its prime implicants can be removed from the cover • An irredundant prime cover is minimal when the cover has the minimal number of prime implicants EE 4271 VLSI Design

9. Goal of Logic Synthesis • Find a minimum irredundant prime cover • An irredundant prime cover is not necessarily a minimum EE 4271 VLSI Design

10. Quine-McCluskey Algorithm • Calculate all prime implicants of the union of the on-set and dc-set, omitting prime implicants that only cover points of dc-set • Finds the minimum cost cover of all minterms in the on-set by the obtained prime implicants EE 4271 VLSI Design

11. Set Covering Problem • Given a universe U={e1, …, en}, a collection of subsets {S1, …, Sm} where each subset contains some elements in universe • cost wi is associated with each subset Si • find a subcollection C (cover) such that C covers the entire universe • Famous NP-complete problem • On-set U, a prime implicant an S EE 4271 VLSI Design

12. Example of Set Cover • U={1,2,3,4} • S1={1,2}, w=2 • S2={1,3,4}, w=3 • S3={3}, w=1 • S4={2,4}, w=2 • S5={2,3}, w=2 • S6={1,2,4}, w=3 • C={S3,S6} is the optimal solution • S3 is redundant in C={S1,S2,S3} , and C={S1,S2} is not optimal EE 4271 VLSI Design

13. Greedy Algorithm • C = empty [the cover] • E= empty [record elements which have been covered] • While there is uncovered element • Find the subset Si which is most cost effective, that is, the Si with smallest w(Si)/|Si-E|. [weight of subset over the elements in the subset but not covered so far] • For each e in Si-E, set price(e)=w(Si)/|Si-E| • Put all e in Si to E • Put Si in the current partial cover C EE 4271 VLSI Design

14. Running Example (Iter 1) • U={1,2,3,4} • S1={1,2}, w=10 • S2={1,3,4}, w=9 • S3={3}, w=7 • S4={2,4}, w=4 • S5={2,3}, w=2 • Iteration 1, E = empty • w(S1)/|S1-E|=10/2=5 • w(S2)/|S2-E|=9/3=3 • w(S3)/|S3-E|=7/1=7 • w(S4)/|S4-E|=4/2=2 • w(S5)/|S5-E|=2/2=1 • Pick S5 • E = {2,3} • C = {S5} EE 4271 VLSI Design

15. Running Example (Iter 2) • U={1,2,3,4} • S1={1,2}, w=10 • S2={1,3,4}, w=9 • S3={3}, w=7 • S4={2,4}, w=4 • S5={2,3}, w=2 • Iteration 2, E={2,3}, C = {S5} • w(S1)/|S1-E|=10/1=10 • w(S2)/|S2-E|=9/2=4.5 • w(S3)/|S3-E|=7/0=+infty • w(S4)/|S4-E|=4/1=4 • Pick S4 • E = {2,3,4} • C = {S5,S4} EE 4271 VLSI Design

16. Running Example (Iter 3) • U={1,2,3,4} • S1={1,2}, w=10 • S2={1,3,4}, w=9 • S3={3}, w=7 • S4={2,4}, w=4 • S5={2,3}, w=2 • Iteration 3, E={2,3,4}, C={S5,S4} • w(S1)/|S1-E|=10/1=10 • w(S2)/|S2-E|=9/1=9 • w(S3)/|S3-E|=7/0=+infty • Pick S2 • E = {2,3,4,1} • C = {S5,S4,S2} EE 4271 VLSI Design

17. Approximation Ratio • Denote by OPT the weight of the optimal solution • Denote by ALG the weight of the solution of our algorithm • Our algorithm is an r-approximation to the optimal solution if for all instances, ALG<= rOPT • r=1 means that our algorithm computes the optimal solution • We would like to minimize r EE 4271 VLSI Design

18. Proof of Approximation - I • Order all elements {e1, …, en} in which they are covered by the algorithm, e.g., E = {2,3,4,1} • Denote by OPT the optimal solution • At any iteration, the uncovered (remaining) elements in the universe (i.e., U-E in the beginning of the iteration) can be covered by a cover of weight at most OPT (since OPT covers all elements) • There exists one subset Si with w(Si)/|Si-E|<= OPT/|U-E| (average weight per element). Otherwise, on average covering any remaining element needs weight>OPT/|U-E|, thus we need a cover with weight>OPT to cover all remaining elements EE 4271 VLSI Design

19. Proof of Approximation - II • Our algorithm picks the Si with smallest w(Si)/|Si-E|, so its w(Si)/|Si-E|<= OPT/|U-E|. • When an element ek is covered, |U-E|<=n-(k-1)=n-k+1 • price(ek)<=OPT/(n-k+1) • price(ek) gives the weight of our solution • price(ek)=OPT(1/n+1/(n-1)+…+1)=lnn OPT EE 4271 VLSI Design

20. Tight Bound • n elements and a collection of m=n+1 subsets • U={1,2,…n} • S1={1}, w(S1)=1/n • S2={2}, w(S2)=1/(n-1) • S3={3}, w(S3)=1/(n-2) • … • Sn={n}, w(Sn)=1 • Sn+1={1,2,…,n}, w(Sn+1)=1.0001 • Our solution {S1,S2,…,Sn}, weight = ln n • Optimal solution {Sn+1}, weight = 1.0001 EE 4271 VLSI Design

21. Better solution? • There is no (1-)lnn-approximation for Weighted Set Cover problem unless NPDTIME(nloglogn) • Uriel Feige, A threshold of ln n for approximating set cover, Journal of the ACM, Vol. 45, No. 4, pp. 634 - 652, 1998 • There is no better approximation algorithm running in polynomial time for the general weighted set cover problem • It is possible to design better (constant) approximation for specific cases • Bounded size EE 4271 VLSI Design

22. Unweighted & Size Bounded - I When w(S)=|S| and # elements in any S <= a constant D, the lnn-approximation algorithm works However, we can have better approximation ratio Recall that in each iteration, we pick the subset with minimum |Si|/|Si-E|.There exists one subset Si with |Si|/|Si-E|<= OPT/|U-E|. We pick the subset with |Si-E| (which are uncovered elements so far) at least |Si||U-E|/OPT >= |U-E|/OPT We actually pick the subset which covers at least |U-E|/OPT uncovered elements, where |U-E| is # uncovered elements so far (in the beginning of the current iteration) EE 4271 VLSI Design 22

23. Unweighted & Size Bounded - II Initially we have n uncovered elements After iteration 1, # uncovered elements <= n – n/OPT After iteration 2, # uncovered elements <= (n – n/OPT) – (n – n/OPT)/OPT After iteration k, # uncovered elements <= n(1-1/OPT)k Solve n(1-1/OPT)k<=x to find the smallest k, i.e., # iterations such that # uncovered elements <= x Since (1-1/OPT)OPT • k/OPT < (1/e)k/OPT, we have n/(ek/OPT)<=x  k>=OPT ln(n/x) Each iteration, one subset is picked and the last x elements need at most x subsets EE 4271 VLSI Design 23

24. Unweighted & Size Bounded - III Our solution has at most OPT ln(n/x) + x subsets Since # elements for any subset is bounded by a constant D. OPT>=n/D, i.e., n/OPT<=D. Set x=n/D, so x<=OPT. OPT ln(n/x) + x<=OPT ln(D)+OPT Our algorithm is a ln(D)+1 approximation which is a constant approximation EE 4271 VLSI Design 24

25. Summary Primary implicant Minimum cost irredundant prime cover Set Cover Greedy Algorithm Approximation Ratio EE 4271 VLSI Design 25