33628ad6bad3df87b59448d7900f15c2.ppt

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Multi-Layer Channel Routing Complexity and Algorithm - Rajat K. Pal Section 5. 3: NP-completeness of Multi-Layer No-dogleg Routing Presented by Md. Jawaherul Alam #040805062 P

Channel E D A Channel C B F VLSI Layout

Channel nets: set of terminals to be connected A D Ch a terminals n n e l

Channel Routing Problem Channel Area minimizatio n requires number of track minimizatio The channel routing problem is the problem of computing an feasible route for the nets so that the number of tracks required is minimized

Channel Routing 1 2 3 M 1 0 a 3 n h a t 3 2

Channel Routing 1 2 3 V 1 0 3 3 H 2 r o u

Channel Routing 1 2 3 3 0 1 3 2

1 2 3 3 0 Channel Routing 2 1 0 3 2 No-dogleg routing 1 2 2 3 3 Dogleg routing 0 1 0 3 2

Parameters in No-dogleg Routing 1 4 0 6 0 3 3 9 7 0 3 2 0 0 5 0 0 0 5 I 7 I 1 I 6 I 4 I 5 I 3 I 9 I 2 I 8 4 0 6 0 4 1 8 2 0 9 0 0 8 3 0 0 7 0 5

Parameters in No-dogleg Routing 1 4 0 6 0 3 3 9 7 0 3 2 0 0 5 0 0 0 5 I 7 I 1 I 6 I 4 I 3 More horizontal I 9 layers: HVH routing I 2 I 8 4 0 6 0 4 1 8 2 0 9 0 0 8 3 0 0 7 0 5 Column density =3 Column density =5 dmax = maximum column density Lower bound on # tracks

Parameters in No-dogleg Routing 1 2 3 3 1 3 2 2 3 3 0 0 1 1 3 2

Parameters in No-dogleg Routing 1 2 3 3 vmax = longest path length + 1 Lower bound on # tracks 3 0 1 More vertical layers: VHV routing 3 2 2 1 VCG

Parameters in No-dogleg Routing 2 2 1 3 Not possible in no -dogleg VH routing Possible in nodogleg VHV routing 3 0 1 3 2 VCG 1 2

VHVH Routing V 2 2 2 1 3 H 1 H 2 0 1 V 1 3 2

VHVH Routing V 2 2 2 1 3 H 1 H 2 Tracks on H 1 layer has VHV routing 0 1 3 2 Tracks on H 2 layer has VH routing V 1

NPcompleteness of Multi-Layer No-dogleg Channel Routing

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

Polynomial-time Reducibility from X’ to X Any instance I’ of X’ Polynomial-time om yn l Po Size of I is in polynomial of I’ e im l-t ia Solution of X An instance I of X A solution of I Polynomial-time A solution of I’

3 -SAT problem a U= { a, b, c, d } : a set of literals Is there a truth b e + a )( aassignment of F = ( b + c + d )( d et b + +b+c) a, b, c, d that c : Logical AND of q number of 3 -element clauses, pl comelement in U makes F=1 ? P-each N d Is there a truth assignment of U that satisfies F ? F

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

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

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

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

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

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

MNVi. Hi ( MNVi. Hik ) problem Channel specification of multi-terminal net Is there a four layer Vi. Hi routing solution for the given instance using dmax/i ( k ) tracks?

MNVi. Hi+1 ( MNVi. Hi+1 k ) problem Channel specification of multi-terminal net Is there a four layer Vi. Hi+1 routing solution for the given instance using dmax/(i+1) ( k ) tracks?

3 -SAT IS 2 IS 3 ISi MNVHVHVH MNVi. Hi+1 MNVHVHk MNVHVHVHk MNVi. Hi+1 k

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

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

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

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

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

3 -SAT IS 2 IS 3 ISi MNVHVHVH MNVi. Hi+1 MNVHVHk MNVHVHVHk MNVi. Hi+1 k

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

3 -SAT IS 2 IS 3 ISi MNVHVHVH MNVi. Hi+1 MNVHVHk MNVHVHVHk MNVi. Hi+1 k

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

3 -SAT IS 2 IS 3 ISi MNVHVHVH MNVi. Hi+1 MNVHVHk MNVHVHVHk MNVi. Hi+1 k

MNVHVH is NP-complete • 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 0 0 1 2 3 1 1 4 3 4 2 4 1 2 3 4 1 3 2 4 G 1 2 3 1 3 2 4 L VCG G has 1 an I I 2 independent set I 3 of size n/2 if I and only 4 if the net has a VHVH 4 2 1 1 3 4 1 4 3 4 routing. M with n/2 tracks dmax = n 2 0 0 R

MNVHVH is NP-complete 0 0 1 2 3 1 1 4 3 4 2 4 1 2 3 4 1 3 2 4 G 1 2 3 1 3 2 4 L VCG G has 1 an I I 2 independent set I 3 of size n/2 if I and only 4 if the net has a VHVH 4 2 1 1 3 4 1 4 3 4 routing. M with n/2 tracks dmax = n 2 0 0 R

MNVHVH is NP-complete 0 0 1 2 3 1 1 4 3 4 2 3 1 2 3 4 1 3 2 I 1 I 2 4 I 3 I 4 G 1 2 3 4 2 1 1 3 4 1 4 3 4 2 3 2 0 0 1 3 2 4 L VCG M dmax = n R

MNVHVH is NP-complete 0 0 1 2 3 1 1 4 3 4 2 3 1 2 3 4 1 3 2 I 1 I 2 4 I 3 I 4 G 1 2 3 4 2 1 1 3 4 1 4 3 4 2 3 2 0 0 1 3 2 4 L VCG M dmax = n R

MNVHVH is NP-complete 0 0 1 2 3 1 1 4 3 4 2 3 1 2 3 4 1 3 2 I 3 4 I 2 G 1 2 3 4 2 1 1 3 4 1 4 3 4 2 3 2 0 0 1 3 2 4 L VCG M dmax = n R

3 -SAT IS 2 IS 3 ISi MNVHVHVH MNVi. Hi+1 MNVHVHk MNVHVHVHk MNVi. Hi+1 k

MNVHVHk is NP-complete Trivial MNVHVH is a special case of MNVHVHk where k = dmax/2

3 -SAT IS 2 IS 3 ISi MNVHVHVH MNVi. Hi+1 MNVHVHk MNVHVHVHk MNVi. Hi+1 k

Thank You

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

3 -SAT problem U= { a, b, c, d } : a set of literals F = ( b + c + d )( d et b + a )( a + b + c ) +e pl of 3 -element clauses, : Logical AND of q number comelement in U P-each N Is there a truth assignment of U that satisfies F ?