Interconnect Layout Optimization by Simultaneous Steiner Tree Construction and Buffer Insertion

1. Interconnect Layout Optimization by Simultaneous Steiner Tree Construction and Buffer Insertion Takumi Okamoto , Jason Kong (ICCAD’96) Presented By Cesare Ferri

2. Buffer insertion and Interconnect Topology optimizations have an important role for Timing optimizations of VLSI circuits. Previous optimizations algorithms consider independently the 2 problems: the buffer insertion Steiner Tree construction (topology optimiz.) From the previous Lesson

3. The algorithm (BA-tree) addresses simultaneously the Steiner Tree construction problem and the Buffer insertion problem. It makes use of two others algorithms: Heuristic A-tree Algorithm Van Ginneken algorithm (Buffer insertion) Proposed Algorithm

4. Given: a source S0 and sinks S1..Sn with given positions and RAT associated with each Si Find: A Steiner tree Ts that spans S and has buffers inserted Objective : Maximized the RAT at the source Problem Formulation sink3 sink4 Source S0 sink2 sink1 RAT1

5. Steiner Tree: A tree connecting all terminals as well as other added virtual nodes (Steiner nodes). Rectilinear Steiner Tree: Steiner tree such that edges can only run horizontally and vertically. A-Tree: Shortest path rectilinear Steiner tree efficient algorithms can find excellent approximations of the optimal A-tree A-Tree Basic Concepts Steiner tree Rectilinear Steiner tree

6. The algorithm consists of 2 phases: Bottom up tree construction (A-tree alg.) Top down buffer insertion (Van Ginneken alg.) The first phase recursively calls the A-tree algorithm Overall Algorithm

7. Recursive A-tree creation Every pair of sub tree roots v and w are evaluated by computing the RAT at the root of of subtree Tr which results from merging of Tv and Tw First Phase – Recursive Merging

8. Top Down Buffer Insertion (Van Ginneken algorithm ) The option that gives the Maximum Required Arrival Time at root is chosen Traces back the computation of the first phase that led this option Second Phase -

9. Experimental Results Sequential A-tree, Buffer insertion Proposed alg. Table: RAT at source (ns) 75% bigger RAT than the sequential alg Net with # sinks

10. The BA-tree algorithm was presented, which derives buffered Steiner tree so that the RAT at the source is Maximized achieves Steiner tree construction and buffer insertion simultaneously Experimental Results show that the algorithm increases the timing slack by up 75% Future Work: Including the total capacitance minimization and their trade off with the RAT at the source Incorporating optimal wiresizing for further delay optimizzation Conclusions

11. optimal wire sizing and buffer insertion for low power nuno alves 7 / december / 2006

12. what’s the paper about? idea is simple: they want to improve delay while take power into account on VLSI circuits. how can we improve delay & routability ? by sizing wires by inserting buffers • sizing wires? yes! as we shrink down circuit size, wire becomes a contributor to to signal delay and time. by widening wires we reduce resistance, but we also increase capacitance • inserting buffers? yes! read slides from previous class

13. extension from van ginneken • this work is an extension from van ginneken work that takes into account: • signal slew • low power • On a circuit, we have the following: • length (l) , width (w), capacitance (c) and resistance (r) of a wire • capacitance and delay of a buffer • Model of buffer delay includes slew of the signal

14. firstly, applies van ginneken algorithm: Algorithm computes the optimal (input capacitance,required arrival time) pairs: For each achievable arrival time, it finds the smallest load achieving it Find optimal buffer configurations. secondly, applies a wire width algorithm from previous tree: Does a similar thing as van ginneken algorithm. It computes the optimal (input capacitance,required arrival time) with different wire widths How much we can scale the wire widths is user specified algorithm maximizing required arrival time

15. algorithm to include wire width length (L) Load = Ck + (l*w1)*L Ck RAT = Tk Load = Ck + (l*w2)*L Load = Ck + (l*w…)*L

16. Same thing as van ginneken algorithm But we include power as a capacitive value, in addition to (load, required time) pairs algorithm to include power consumption

17. experimental results

18. Minimum-Buffered Routing of Non-Critical Nets for Slew Rate and Reliability Control C. Alpert, A. Kahng, B. Liu, I. Mandoiu, A. Zelikovsky Presenter: Elif Alpaslan

19. Electrical correctness in large interconnects is an important requirement that arises before timing optimization of circuit Elimination of all electrical violations even for non-critical nets is a prerequisite to initiating a meaningful placement and timing optimizations Bounding load capacitance at gate output is a well-known VLSI design methodology to ensure electrical correctness of the nets Bounding the load capacitance at gate output : (+) improves coupling noise immunity reduces degradation of signal transition edges reduces delay uncertainty due to coupling noise improves reliability with respect to hot-carries oxide breakdown and AC self heating in interconnects guarantees bounded input rise/fall times at buffers and sinks Motivation

20. Given: Net N with source r and set of sinks S Binary routing tree T = (r, V, E) for N Input capacitance csfor each sink s S Buffer input capacitance Cb Unit-length wire capacitance Cw Capacitive load upper-bound CU Buffer-skew bound D Find: bufferingof the routing tree T such that The load cap of each buffer and of the source r is at most CU The buffer skew is at most D The number of inserted buffers is minimized Minimum-Buffered Routing Problem

21. T=(r, V, E) : routing tree for net N T= (r, V, E, B) : buffered routing tree, B is set of buffers located in edges of T For any b in B {r}, the subtree driven by b, is the maximal subtree of Db of T which is rooted at b and has no internal buffers. Cw = unit length wire segment capacitance Cb = input capacitance of buffer cv = input capacitance of sink or buffer v le = length of wire segment ce = capacitance of wire segment Cu = upper-bound on capacitive load on each buffer Load model: lumped capacitive load model Problem Formulation

22. Linear Time Greedy Algorithm with a single non-inverting buffer type Definitions used in the algorithm: Critical Vertex p: a vertex of a routing tree T is critical if p is a bottom-most point of T such that Tp can not be driven by a single buffer. Heaviest Child u of p: u is a heaviest child of p if it accumulates more capacitance than any other child of p. Algorithm 1: Routed Net Buffering

23. Algorithm 1: Routed Net Buffering Insert buffer at top of heaviest edge if CU > c(Tu)+c(u,p) Insert buffer on edge (u,p) if CU  c(Tu)+c(u,p)