1. Multi-Layer Channel Routing Complexity and Algorithm - Rajat K. Pal Md. Jawaherul Alam #040805062P Presented by Section 5.3: NP-completeness of Multi-Layer No-dogleg Routing

2. Channel C E A D B F VLSI Layout Channel

3. AD nets: set of terminals to be connected terminals Channel routing

4. Channel Area minimization requires number of track minimization The channel routing problem is the problem of computing a feasible route for the nets so that the number of tracks required is minimized Channel Routing Problem

5. Channel Routing 233 320 1 1 Manhattan routing

6. 233 320 1 1 Channel Routing VH routing

7. 233 320 1 1 Channel Routing

8. 233 320 1 1 2 0 233 320 1 1 2 0 Dogleg routing No-dogleg routing

9. 1406033970320050005 4060418209008300705 I4I4 I8I8 I9I9 I2I2 I1I1 I3I3 I7I7 I5I5 I6I6 Parameters in No-dogleg Routing

10. 1406033970320050005 4060418209008300705 I4I4 I8I8 I9I9 I2I2 I1I1 I3I3 I7I7 I5I5 I6I6 Column density =3 Column density =5 More horizontal layers: HVH routing d max = maximum column density Lower bound on # tracks

11. 233 320 1 1 Parameters in No-dogleg Routing 233 320 1 1

12. 233 320 1 1 2 3 1 VCG v max = longest path length + 1 Lower bound on # tracks More vertical layers: VHV routing

13. Parameters in No-dogleg Routing 2 3 1 VCG 213 320 2 1 Not possible in no- dogleg VH routing Possible in no- dogleg VHV routing

14. VHVH Routing 213 320 2 1 V1V1 V2V2 H2H2 H1H1

15. 213 320 2 1 V1V1 V2V2 H2H2 H1H1 Tracks on H 1 layer has VHV routing Tracks on H 2 layer has VH routing

16. NP-completeness of Multi-Layer No-dogleg Channel Routing

17. NP completeness A decision problem X is NP-complete if X  NP, i.e. for any yes instance I of X, there is a polynomial (in  I  ) sized certificate, which can be verified in polynomial ( in  I  ) time. A polynomial-time solution of X implies a polynomial-time solution of any problem X’  NP. Polynomial-time reducibility

18. Polynomial-time Reducibility from X’ to X Any instance I’ of X’ An instance I of X Size of I is in polynomial of  I’  Polynomial-time A solution of I Polynomial-time A solution of I’ Solution of X Any instance I’ of X’ Polynomial-time A solution of I’

19. 3-SAT problem U= { a, b, c, d } : a set of literals F = ( b +  c + d )(  d +  b + a )(  a + b + c ) : Logical AND of q number of 3-element clauses, each element in U Is there a truth assignment of U that satisfies F ? a b c d F Is there a truth assignment of a,b,c,d that makes F=1 ? NP-complete

20. IS 3 problem A undirected graph G = ( V, E ) with n vertices Is there an independent set of size  n/3  ?

21. IS 2 problem A undirected graph G = ( V, E ) with n vertices Is there an independent set of size  n/2  ?

22. IS i problem; i ≥ 4 A undirected graph G = ( V, E ) with n vertices Is there an independent set of size  n/i  ?

23. MNVHVH problem Channel specification of multi-terminal net Is there a four layer VHVH routing solution for the given instance using  d max /2  tracks? Multi-terminal no-dogleg VHVH channel routing

24. MNVHVHk problem Channel specification of multi-terminal net Is there a four layer VHVH routing solution for the given instance using k tracks?

25. MNVHVHVH ( MNVHVHVHk ) problem Channel specification of multi-terminal net Is there a four layer VHVHVH routing solution for the given instance using  d max /3  ( k ) tracks?

26. MNV i H i ( MNV i H i k ) problem Channel specification of multi-terminal net Is there a four layer V i H i routing solution for the given instance using  d max /i  ( k ) tracks?

27. MNV i H i+1 ( MNV i H i+1 k ) problem Channel specification of multi-terminal net Is there a four layer V i H i+1 routing solution for the given instance using  d max /(i+1)  ( k ) tracks?

28. 3-SAT MNV i H i+1 MNV i H i+1 k MNV i H i MNV i H i k MNVHVH MNVHVHk MNVHVHVH MNVHVHVHk IS i IS 2 IS 3

29. IS 3 is NP-complete IS3  NP : trivial Given a guess of  n/3  vertices, check whether they are independent A undirected graph G with n vertices Is there an independent set of size  n/3  ? IS3 is NP-complete Reduction from 3-SAT problem

30. U= { a, b, c, d } F = ( b +  c + d )(  d +  b + a )(  a + b + c ) IS 3 is NP-complete b d !c !d !b a c b !a G(F) F is satisfiable if and only if G (F) has an independent size of size q q clauses 3q vertices

31. U= { a, b, c, d } F = ( b +  c + d )(  d +  b + a )(  a + b + c ) IS 3 is NP-complete b d !c !d !b a c b !a G(F) q clauses 3q vertices F is satisfiable if and only if G (F) has an independent size of size q a=0, b=1, c=0, d=0 F= 1

32. U= { a, b, c, d } F = ( b +  c + d )(  d +  b + a )(  a + b + c ) IS 3 is NP-complete b d !c !d !b a c b !a G(F) q clauses 3q vertices a=0, b=1, c=0, d=0 F= 1 b, !d, !a

33. U= { a, b, c, d } F = ( b +  c + d )(  d +  b + a )(  a + b + c ) IS 3 is NP-complete b d !c !d !b a c b !a G(F) q clauses 3q vertices a=1, c=1, d=1 U= { a, b, c, d } F = ( b +  c + d )(  d +  b + a )(  a + b + c )

34. 3-SAT MNV i H i+1 MNV i H i+1 k MNV i H i MNV i H i k MNVHVH MNVHVHk MNVHVHVH MNVHVHVHk IS i IS 2 IS 3

35. U= { a, b, c, d } F = ( b +  c + d )(  d +  b + a )(  a + b + c ) IS 2 is NP-complete b d !c !d !b a c b !a G(F) F is satisfiable if and only if G (F) has an independent size of size 2 q q clauses 3q vertices q vertices 4q vertices

36. 3-SAT MNV i H i+1 MNV i H i+1 k MNV i H i MNV i H i k MNVHVH MNVHVHk MNVHVHVH MNVHVHVHk IS i IS 2 IS 3

37. U= { a, b, c, d } F = ( b +  c + d )(  d +  b + a )(  a + b + c ) IS i is NP-complete b d !c !d !b a c b !a G(F) q clauses 3q vertices K (i-3)q+i iq+i vertices F is satisfiable if and only if G (F) has an independent size of size q+1

38. MNV i H i+1 MNV i H i+1 k MNV i H i MNV i H i k MNVHVH MNVHVHk MNVHVHVH MNVHVHVHk IS i IS 2 IS 3 3-SAT

39. MNVHVH  NP Given a guess of a feasible routing solution of an instance of MNVHVH, verify whether the guess is a valid solution MNVHVH is NP-complete Reduction from IS 2 problem MNVHVH is NP-complete

40. 3 4 2 1 G 000012311434241234 123421134143420000 I2I2 I1I1 I3I3 I4I4 LRM d max = n VCG 3 4 2 1 G has an independent set of size  n /2  if and only if the net has a VHVH routing with  n /2  tracks

41. 3 4 2 1 G MNVHVH is NP-complete 000012311434241234 123421134143420000 I2I2 I1I1 I3I3 I4I4 LRM d max = n VCG 4 1 G has an independent set of size  n /2  if and only if the net has a VHVH routing with  n /2  tracks 3 2

42. 3 4 2 1 G MNVHVH is NP-complete 000012311434241234 123421134143420000 I2I2 I1I1 I3I3 I4I4 LRM d max = n VCG 4 1 3 2 23 32

43. 3 4 2 1 G MNVHVH is NP-complete 000012311434241234 123421134143420000 I2I2 I1I1 I3I3 I4I4 LRM d max = n VCG 4 1 3 2 23 32

44. 3 4 2 1 G MNVHVH is NP-complete 000012311434241234 123421134143420000 I2I2 I3I3 LRM d max = n VCG 4 1 3 2 23 32

45. MNV i H i+1 MNV i H i+1 k MNV i H i MNV i H i k MNVHVH MNVHVHk MNVHVHVH MNVHVHVHk IS i IS 2 IS 3 3-SAT

46. MNVHVHk is NP-complete Trivial MNVHVH is a special case of MNVHVHk where k =  d max /2 

47. MNV i H i+1 MNV i H i+1 k MNV i H i MNV i H i k MNVHVH MNVHVHk MNVHVHVH MNVHVHVHk IS i IS 2 IS 3 3-SAT

48. ThankYou

49. ab c d F Is there a truth assignment of a,b,c,d that makes F=1 ?

50. 3-SAT problem U= { a, b, c, d } : a set of literals F = ( b +  c + d )(  d +  b + a )(  a + b + c ) : Logical AND of q number of 3-element clauses, each element in U Is there a truth assignment of U that satisfies F ? NP-complete