DAOmap: A Depth-optimal Area Optimization Mapping Algorithm for FPGA Designs Deming Chen, Jason Cong ， Computer Science Department ， UCLA Presented by Shikang Xu 1

Outline Introduction Related Works Definitions and Problem Fomulation Algorithm Description Discuss of Techniques 2

Introduction The LUT-based FPGA architecture dominates the existing programmable chip industry FPGA technology mapping converts a given Boolean circuit into a functionally equivalent network comprised only of LUTs 3

Related Works Area Minimization – Chortle-crf, [Francis, et al, DAC’91] – MIS-pga, [Murgai, et al, ICCAD’91] – Praetor, [Cong, et al, FPGA’99] – Anti-fuse FPGA Mapper, [Kang, et al, ASPDAC’04] Delay Minimization – DAG-Map, [Chen, et al, DTC’92] – FlowMap, [Cong, et al, ICCAD’92] – Edge-map, [Yang, et al, ICCAD’94] Power Minimization – PowerMinMap, [Li, et al, ASPDAC’03] – Emap, [Lamoureux, et al, ICCAD’03] – DVmap, [Chen, et al, FPGA’04] Simultaneous Delay and Area Minimization(Area Minimization under Timing Constraints ) – FlowMap-r, [Cong, et al, TVLSI’94] – CutMap, [Cong, et al, FPGA’95] – BoolMap-D, [Legl, et al, DAC’96] Adopted from Deming Chen, Jason Cong, Computer Science Department, UCLA 4

Definitions Cone (Ov):- A subnetwork of the original network, consisting of v and some of its predecessors, such that for any node w in Ov, there is a path from w to v in Ov. Fanin cone (Fv):- The maximum cone of v, consisting of all PI predecessors of v Input(Ov):- Denotes the set of distinct nodes outside Ov which supply inputs to the gates in Ov. Cut:- It is a partitioning (X,X’) of a cone Ov such that X’ is a cone of v. Cut-set:- It is represented as V(X,X’), and consists of input(X’) 5

Definitons Cutsize: It is the cardinality of the cut-set. A cut is said to be K-feasible if the cutsize is <=K Level: The level of a node v is the length of the longest path from any PI to the node v. Depth : The depth of a network is the largest node level in the network. Mapping Depth: The largest optimal delay of the mapped circuit. Picture adopted from Deming Chen, Jason Cong, Computer Science Department, UCLA 6 a b c d e v FvFv 3-feasible cone C v PIs Delay of 2

Problem Formulation Area Minimization under Timing Constraint: Given: a Boolean network; Unity delay model (1 LUT contributes unit delay) Goal: cover the network with K-feasible cones (K-LUTs), such that Optimal mapping depth is guaranteed Area (number of LUTs) is minimized 7

Algorithm Description A Cut-enumeration-based method consisting of cut generation and cut selection Cut generation traverses the network from the PI to the PO, and combines subcuts on the fanin nodes of a target node to generate all the cuts on the target node After generating the cuts, the network is traversed from the PO to the PI, and the cuts are selected to produce the LUT mapping result. 8

Cut Enumeration Cut enumeration means generating all K-feasible cuts of a cone for a given node effectively f(K, v) represents all the K-feasible cuts rooted at node v, operator + is Boolean OR, K is Boolean AND on its operands, but filtering out all the resulting p-terms with more than K variables. 9

Cut Enumeration: Example 10 All the cuts rooted on node s can be generated by combining the cuts rooted on its fanin nodes q and r. The cuts on the fanin nodes are called subcuts. Combining C1 with C2 will form a new cut Cs = {m, n, o, p} rooted on s. If the input of the new cut exceeds K, the cut is discarded.

Cut Enumeration: Time propagation The arrival time propagates through each of the cut, and each cut represents a LUT and hence a unit delay. The minimum arrival time at a node v is where C represents every cut generated for v through cut enumeration. Arr i is the minimum arrival time on input signal i of C. There can be several cuts with Arr i, form a set Xv 11

Cut enumeration: Area Propagation Similar to the arrival time, the area can also be propagated. The area is calculated as Where Uc is the area contributed by the cut C, A i is the estimated area of the cone rooted on signal i and f(i) is the fanout number of signal i. That means that the area on i is shared and distributed into other fanout nodes of i. 12

Delay and Area Propagation a c d yx z b w e f g Delay 1, Area 1 Optimal Delay = 1 Area = 1 Optimal Delay = 2 Area = 2 Delay 1, Area 1 Delay 2, Area 3 Delay 2, Area 2 Optimal Delay = 1 Area = 1 Optimal Delay = 1 Area = 1 Propagation process visits cuts and nodes iteratively The longest best delay on the POs is the optimal mapping delay Adopted from Deming Chen, Jason Cong, Computer Science Department, UCLA 13

Area propagation under Timing constraints To guarantee optimal mapping depth, we need to propagate the estimated area together with the minimum arrival time A v represents the best achievable area under the constraint that it also generates the optimal mapping delay upto the point of v With these formulae, the areas of cuts and nodes are iteratively calculated until the enumeration process reaches the POs. During the cut selection process when we know that v is not on a critical path, a cut C not belonging to Xv can be chosen as long as it does not violate the timing constraint. 14

Cost function of a cut Some Key parameters I C : cutsize of C N C : number of nodes covered by C f(v): fanout number of the root node v Rc: number of reconvergent path 15

Example of Cost function In the example above C1 and C2 have the same cutsize, but C2 is better C2 covers two sets of reconvergent paths Having a cut rooted at node 5 will reduce potential duplications 16

Global Duplication Cost Adjustment Consider potential node duplications Check the sub-cuts for multiple fanouts Propagate adjusted cost globally 17

Cut Selection From POs to PIs Critical paths: optimal delay + best area available Non-critical paths: relaxed delay + better area 18

Cut Selection Greedily pick cuts with smallest costs will forfeit some optimization factors in term of reducing duplication locally. Use heuristics to guide the selection procedure – Iterative Cut Selection Procedure – Local Cost Adjustment Input Sharing Slack Distribution Cut Probing 19

Efficiency With DAOmap, the researchers report a better area values with a lower runtime, when compared to CutMap. The impact of the various techniques used, on the final area values is shown here. 20

Efficiency: Impact of Techniques Input sharing proves to be the most important technique to reduce area because it reduces the number of edges and node duplications The mincost propagation is trying to evaluate how accurate our cost estimation model is. Global duplication cost adjustment offers the next largest gain, which shows that duplication of nodes adds to the area cost 21

Summary A cut enumeration based cut selection and generation process for LUT Novel techniques make DAOmap gained significant amount of area and runtime reduction over a state-of-the-art algorithm CutMap 22