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Advanced Interconnect Optimizations. Buffers Improve Slack. RAT = 300 Delay = 350 Slack = -50. slack min = -50. RAT = 700 Delay = 600 Slack = 100. RAT = Required Arrival Time Slack = RAT - Delay. RAT = 300 Delay = 250 Slack = 50. Decouple capacitive load from critical path.

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## Advanced Interconnect Optimizations

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**Buffers Improve Slack**RAT = 300 Delay = 350 Slack = -50 slackmin = -50 RAT = 700 Delay = 600 Slack = 100 RAT = Required Arrival Time Slack = RAT - Delay RAT = 300 Delay = 250 Slack = 50 Decouple capacitive load from critical path slackmin = 50 RAT = 700 Delay = 400 Slack = 300**Timing Driven Buffering Problem Formulation**• Given • A Steiner tree • RAT at each sink • A buffer type • RC parameters • Candidate buffer locations • Find buffer insertion solution such that the slack at the driver is maximized**Candidate Solution Characteristics**• Each candidate solution is associated with • vi: a node • ci: downstream capacitance • qi: RAT vi is a sink ciis sink capacitance vis an internal node**Van Ginneken’s Algorithm**Candidate solutions are propagated toward the source Dynamic Programming**Solution Propagation: Add Wire**• c2 = c1 + cx • q2 = q1 – rcx2/2 – rxc1 • r: wire resistance per unit length • c: wire capacitance per unit length x (v1, c1, q1) (v2, c2, q2)**Solution Propagation: Insert Buffer**(v1, c1, q1) (v1, c1b, q1b) • c1b = Cb • q1b = q1 – Rbc1– tb • Cb: buffer input capacitance • Rb: buffer output resistance • tb: buffer intrinsic delay**Solution Propagation: Merge**• cmerge = cl + cr • qmerge = min(ql , qr) (v, cl , ql) (v, cr , qr)**Solution Propagation: Add Driver**(v0, c0, q0) (v0, c0d, q0d) • q0d = q0 – Rdc0 = slackmin • Rd: driver resistance • Pick solution with max slackmin**Example of Solution Propagation**• r = 1, c = 1 • Rb = 1, Cb = 1, tb = 1 • Rd = 1 2 2 (v1, 1, 20) Add wire (v2, 3, 16) (v2, 1, 12) v1 v1 Insert buffer Add wire Add wire (v3, 5, 8) (v3, 3, 8) v1 v1 slack = 3 slack = 5 Add driver Add driver**Example of Merging**Left candidates Right candidates Merged candidates**Solution Pruning**• Two candidate solutions • (v, c1, q1) • (v, c2, q2) • Solution 1 is inferior if • c1 > c2 : larger load • and q1 < q2 : tighter timing**Pruning When Insert Buffer**They have the same load cap Cb, only the one with max q is kept**(1)**(2) (3) Generating Candidates From Dr. Charles Alpert**(3)**(b) (a) Both (a) and (b) “look” the same to the source. Throw out the one with the worst slack (4) Pruning Candidates**(4)**(5) Candidate Example Continued**(5)**At driver, compute which candidate maximizes slack. Result is optimal. Candidate Example Continued After pruning**Left**Candidates Right Candidates Merging Branches**Critical**With pruning Pruning Merged Branches**Van Ginneken Example**(20,400) Buffer C=5, d=30 Wire C=10,d=150 (30,250) (5, 220) (20,400) Buffer C=5, d=50 C=5, d=30 Wire C=15,d=200 C=15,d=120 (30,250) (5, 220) (45, 50) (5, 0) (20,100) (5, 70) (20,400)**Van Ginneken Example Cont’d**(30,250) (5, 220) (45, 50) (5, 0) (20,100) (5, 70) (20,400) (5,0) is inferior to (5,70). (45,50) is inferior to (20,100) Wire C=10 (30,250) (5, 220) (20,100) (5, 70) (30,10) (15, -10) (20,400) Pick solution with largest slack, follow arrows to get solution**Basic Data Structure**Worse load cap (c1, q1) (c2, q2) (c3, q3) Better timing • Sorted list such that • c1 < c2 < c3 • If there is no inferior candidates q1 < q2 < q3**Prune Solution List**Increasing c (c1, q1) (c2, q2) (c3, q3) (c4, q4) N N q1 < q2? q1 < q3? q1 < q4? Prune 2 Prune 3 Y Y N q2 < q4? Prune 3 q2 < q3? Y N Prune 4 q3 < q4? N Prune 4 q3 < q4?**Pruning In Merging**Left candidates Right candidates ql1 < ql2 < qr1 < ql3 < qr2 (cl1, ql1) (cl2, ql2) (cl3, ql3) (cr1, qr1) (cr2, qr2) (cl1, ql1) (cl2, ql2) (cl3, ql3) (cr1, qr1) (cr2, qr2) Merged candidates (cl1+cr1, ql1) (cl2+cr1, ql2) (cl3+cr1, qr1) (cl3+cr2, ql3) (cl1, ql1) (cl2, ql2) (cl3, ql3) (cr1, qr1) (cr2, qr2) (cl1, ql1) (cl2, ql2) (cl3, ql3) (cr1, qr1) (cr2, qr2)**Van Ginneken Complexity**• Generate candidates from sinks to source • Quadratic runtime • Adding a wire does not change #candidates • Adding a buffer adds only one new candidate • Merging branches additive, not multiplicative • Linear time solution list pruning • Optimal for Elmore delay model**Multiple Buffer Types**• r = 1, c = 1 • Rb1 = 1, Cb1 = 1, tb1 = 1 • Rb2 = 0.5, Cb2 = 2, tb2 = 0.5 • Rd = 1 2 2 (v1, 1, 20) (v2, 3, 16) v1 (v2, 2, 14) (v2, 1, 12) v1 v1

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