1. FPGA Technology Mapping Dr. Philip Brisk Department of Computer Science and Engineering University of California, Riverside CS 223

2. Example

3. Theoretical Results Minimize Number of Logic Stages – Polynomial-time Minimize Total Number of LUTs (Area) – NP-Complete Minimize Power Consumption – NP-Complete

4. DAG Representation Since LUTs are reconfigurable, we don’t need to worry about the logic function of each gate during mapping

5. K-Feasible Cuts and LUT Mapping

6. Example: Node Duplication

7. FlowMap: An Optimal Technology Mapping Algorithm for Delay Optimization in Lookup-Table Based FPGA Designs Jason Cong and Yuzheng Ding IEEE Trans. CAD 13(1): 1-12, Jan. 1994

8. Cuts in a Directed Acyclic Graph (DAG) A cut is K-feasible if: This cut is 3-feasible

9. Edge Cut Size Each edge has non-negative capacity The edge cut size is the sum of the capacities of the forward edges that cross the cut All edge capacities are assumed to be 1

10. Example: Edge-cut size is 10

11. Volume and Height The volume of a cut is the number of vertices in X: Given an assignment of labels to vertices, the height of a cut is the largest label in X

12. Example: Volume=9, Height=2

13. FlowMap Algorithm (Overview) Labeling Phase – Computes a label for each node reflecting the level of the K-LUT that implements that node in a depth-optimal mapping solution Mapping Phase – Generates the K-LUT mapping solution based on node labels computed in the first phase

14. Subnetwork of a Node For node t, let N t denote the subnetwork consisting of every vertex s, such that there is a path from s to t

15. Conversion to a Network N t For node t, let N t denote the subnetwork consisting of every vertex s, such that there is a path from s to t We can ignore the logic function of each gate

16. Intuition Let LUT(t) represent a K-LUT that produces an output at node t Define a K-feasible cut where – denotes the set of nodes in LUT(t) – denotes the remaining nodes in – K-feasibility is ensured since LUT(t) has < K inputs If u has the maximum label in, then in the optimal mapping,

17. Minimizing the Level of LUT(t) There may be many K-feasible cuts in Lemma 1. Find the one that minimizes height! Note: This definition enumerates all K-feasible cuts at t – Key contribution: This can be done in O(Km) time, where m is the number of edges in

18. Example You get the existence of the 3-feasible cut in part (c) for free. Figuring out how to compute it is the hard part!

19. Lemma 2 Proof Strategy 1. Prove 2. Prove Consult the paper for details

20. Algorithmic Strategy Check if there is a K-feasible cut of height in – If so, pack along with the nodes in in the second phase of the algorithm. Otherwise, the minimum height among all K- feasible cuts in is is,, and is one such cut. – If so, use a new K-LUT for in the next phase.

21. How to efficiently test if has a K-feasible cut of height p – 1? Let p be the maximum label among all nodes of input(t) Equivalently, p is the maximum label of all nodes that belong to Collapse all nodes in with label > p along with t into a single sink t’; call the new network

22. More Theory Construct another network from – Details to follow… has a cut whose edge cut-size is no more than K if the max. flow in is at most K

23. Example … …

24. Algorithmic Strategy (Recap) Check if there is a K-feasible cut of height in – If so, pack along with the nodes in in the second phase of the algorithm. Otherwise, the minimum height among all K- feasible cuts in is is,, and is one such cut. – If so, use a new K-LUT for in the next phase.

25. Labeling Algorithm for K-LUTs For each node t in the DAG, taken in topological order – Let p be the max. label among all nodes of – Build networks,, and – Compute the maximum flow in – If the maximum flow is less than K, then: – Otherwise

26. Summary of Theoretical Results

27. FlowMap Algorithm

28. Post-processing for Area Reduction

31. WireMap: FPGA Technology Mapping for Improved Routability and Enhanced LUT Merging S. Jang, B. Chan, K. Chung, and A. Mishchenko ACM TRETS 2(2): article #14, June, 2009

32. And-Inverter Graph (AIG) https://en.wikipedia.org/wiki/And-inverter_graph AND INV

33. Generic FPGA Technology Mapping

34. Cut Enumeration The set of K-feasible cuts for an AND node n with predecessor nodes n 1 and n 2 Let A and B be two sets of cuts

35. Cut Enumeration Process vertices in topological order to ensure that cut sets for n 1 and n 2 are known before computing the cut set for n – The CUT set of an AND node is computed by merging the CUT sets of its predecessors and adding the trivial cut (containing just n) while keeping only the K-feasible cuts – Remove dominated cuts – Each AIG node is a 2-input AND

36. Depth-Oriented Mapping Keep the node at each level that minimizes depth (e.g., FlowMap)

37. Area Recovery Depth minimization may cause area duplication – Multiple cuts cover an AIG node – Increases LUT count // Area Flow Global View Selects cuts with more shared logic // Exact Local Area Local View Minimizes area exactly at each node

38. Area Flow Estimates sharing between cuts without the need to (re-)traverse them Area(n) is the area cost of the LUT that maps node n Leaf i is the ith leaf of the cut at n NumFanout(n) is 1 if n is not used in the current mapping for area flow computation n Leaf

39. Local View The exact local area of the current node is the area added to the mapping by using the current node Recursively compute the number of LUTs in the max. fanout free cone (MFFC) of the current node – Use a fast local DFS traversal Recursive Calls n

40. Producing a Mapped Network Assume one K-feasible representative cut is computed for each node

41. WireMap Objective – Reduce the number of LUT-to-LUT connections in addition to area reduction Rationale – Fewer nets will help the placer to generate a solution with reduced wirelength

42. Global View Heuristic Area Flow (from previous slide) Edge Flow (new idea) Area(n) is the area cost of the LUT that maps node n Edge(n) is the number of fanin edges to the LUT that maps node n

43. Global Edge/Area Recovery Alg. Find all cuts with min. area Use edge flow as tiebreaker No recursion; use the saved edgeflow computed at each predecessor node n Leaf

44. Local View The exact local area (edge count) of the current node is the area (edge count) added to the mapping by using the current node Recursively compute the number (edge count) of LUTs in the max. fanout free cone (MFFC) of the current node – Use a fast local DFS traversal Recursive Calls n

45. Local View Algorithm Find all cuts that minimize the exact area; use the exact edge count as a tiebreaker Edge count of a cut depends if the cut is representative of the node in the mapping If so, reference the node and the leaves of its representative cut Pointer manipulation in function calls (not shown)

46. WireMap Algorithm Xilinx Virtex-5 Dual Output LUT