1. FlowMap: An Optimal Technology Mapping Algorithm for Delay Optimisation in Lookup- Table Based FPGA Designs 04/06/2016 1 Presented by Qiwei Jin

2. Overview The paper and the authors. Some background information. The algorithm in detail. Results and Conclusion. Questions for discussion. 2

3. About the Paper Originally published in 1992, won IEEE Circuit & Systems Society best paper award in 1994. 238 citations in total, 33 self. The first algorithm to solve a conventionally NP-hard depth minimisation problem in polynomial time. The algorithm is a key component in most commercial FPGA compilers. FlowMap-r and other more sophisticated algorithms published by the authors at the same year or later for both depth and area minimisation. 3

4. Jason Cong Chairman of Computer Science Department, UCLA. Was Assistant Professor in 1994 when this paper was published. Got Promoted to Associate Professor in the same year. His company Aplus was acquired by Magma in 2004 for "$13 million in stock, cash and incentives“. 4 Picture borrowed from Jason Cong’s homepage

5. Yuzheng Ding Very low profile person, no picture, no home page, not even on FaceBook. RA in UCLA for PhD when this paper was published. May have left university for work (Mentor Graphics) after graduation. Still working actively with Jason Cong, latest paper published in year 2008. 5

6. Background FPGA (Field-Programmable Gate Array): Programmable hardware. 6 Xilinx Virtex 5 FPGA

7. Background Cont. 7 For more information, go to Wayne Luk’s Custom Computing Course

8. Background Cont. FPGAs are essentially a bunch of wires and LUTs (Look-Up Tables) that can be configured to emulate the behaviour of a digital circuit. FPGAs can be configured by Hardware Description Language (HDL, such as VHDL). Based on the HDL, a netlist can be generated automatically by some algorithm (FlowMap!). 8

9. Background Cont. = 9 Addr.Value 00000 00010... 11111 4-Input 1-Output LUT (16 entries in total)

10. Background Cont. Mappings from ASIC to FPGAs are not necessary one to one. The question is how to achieve the optimal condition? 10 =

11. Background Cont. Trade-offs: – Area (number of LUTs used) – Depth (delay of the circuit) FlowMap focuses on depth optimisation 11 depth

12. Depth Minimisation Example 12

13. The Key Idea of Depth Minimisation Try to pack as many gates in different levels into a LUT as possible. Number of LUT used (Area) is not the primary concern. The problem is equivalent to generating optimal code for expressions containing common subexpressions, hence NP-Hard.* Conventional method will decompose the Boolean network into a forest of trees before processing. FlowMap can find an optimal mapping directly from a Boolean network within polynomial time. Let’s see how. 13 * A. Aho, S. C. Johnson, “Optimal Code Generation for Expression Trees”, 23, 3, 488-501 (1976).

14. Preliminaries Input(T) Cut (X, X) Node Cut Size Edge Capacity Edge Cut Size Whether a cut is K-feasible Height of a Cut

15. The FlowMap Algorithm 2 Phases – Node Labeling: define the optimal depth of the LUT mapping solution for N t. – LUT Mapping: generate the LUT network based on the labeling in the first phase.

16. Phase 1: Node Labelling 16

17. Phase 1: Node Labelling Cont. 17

18. Phase 2: LUT Mapping 18

19. Phase 2: LUT Mapping Cont. 19

20. The FlowMap Pseudocode 20

21. Enhancements Maximising Cut Volume During Mapping Postprocessing (flow-pack) Operations to reduce number of K-LUTs 21

22. Results 22

23. Conclusion The paper presents the first algorithm to compute a NP-hard problem in polynomial time. Compared to other algorithms FlowMap is about to reduce up to 7% of the LUT network and reduce up to 50% of the number of LUTs. 23

24. Questions It is claimed that a minimum height K-feasible cut can be found in O(Km) time, where K is the number of input of LUT and m is the number of edges of in the network. But it is not clear to me how it is derived. 24

25. Questions Cont. It would be interesting to see the time taken to compute the mapping for the testing cases with FlowMap vs. Other Algorithms. The testing cases are generally small in size, it would be more convincing to see some large size examples. 25