% Boyd, Kim, Patil, and Horowitz, "Digital circuit optimization
% via geometric programming"
% Written for CVX by Almir Mutapcic 02/08/06
%
% Solves the problem of choosing gate scale factors x_i to give
% minimum ckt delay, subject to limits on the total area and power.
% Uses max gate arrival time T formulation that avoids evaluation
% of the delay over all paths in the circuit.
%
%   minimize   T_bar
%       s.t.   T_j <= T_bar      for j an output gate
%              T_j + d_i <= T_i  for j in FI(i)
%              P <= Pmax, A <= Amax
%              x >= 1
%
% where variables are x and T.
%
% We use the circuit topology presented in figure 1 (page 902),
% where we take gates 1, 3 and 6 to be inverters (INV),
% gates 2 and 7 to be three input NANDs (NAND3),
% and gates 4 and 5 to be two input NORs (NOR2).

%********************************************************************
% user specified data (specify problem constant and ckt topology)
%********************************************************************
m = 7;        % number of gates
Vdd = 5;      % supply voltage
Amax = 250;   % maximum area spec

% gate specs
INV   = struct('Cin',3, 'Cint',3, 'Rdrv',0.48, 'A',3,  'Ileak',0.006);
NAND3 = struct('Cin',4, 'Cint',6, 'Rdrv',0.48, 'A',8,  'Ileak',0.007);
NOR2  = struct('Cin',5, 'Cint',6, 'Rdrv',0.48, 'A',10, 'Ileak',0.009);

clear gates;
gates([1 3 6]) = INV;
gates([2 7])   = NAND3;
gates([4 5])   = NOR2;

% primary inputs and primary outputs labels (start with m+1)
primary_inputs = [8 9 10];
primary_outputs = [11 12];
M = m + length( primary_inputs ) + length( primary_outputs );

% fan-in cell array
FI = cell(M,1);
FI{1} = [8];
FI{2} = [8 9 10];
FI{3} = [10];
FI{4} = [1 2];
FI{5} = [2 3];
FI{6} = [4];
FI{7} = [3 4 5];
FI{8} = [];
FI{9} = [];
FI{10} = [];
FI{11} = [6];
FI{12} = [7];

% primary output has Cin capacitance (but has no Cload)
Cin_po = sparse(M,1);
Cin_po(primary_outputs) = [10 10];

% primary input has Cload capacitance (but has no Cin)
Cload_pi = sparse(M,1);
Cload_pi(primary_inputs) = [10 10 10];

% activity frequency of gates and primary inputs
f_gates = 0.001*ones(m,1);
f_pi = sparse(M,1);
f_pi(primary_inputs) = 0.001*[10 10 10];

%********************************************************************
% derived problem data (computed from user inputs)
%********************************************************************
% fan-out cell array (compute it from the fan-in cell array)
FO = cell(M,1);
for gate = [1:m primary_outputs]
  preds = FI{gate};
  for k = 1:length(preds)
    FO{preds(k)}(end+1) = gate;
  end
end

% input and internal capacitance of gates, and driving resistance
Cin_norm  = [gates.Cin]';
Cint_norm = [gates.Cint]';
Rdrv_norm = [gates.Rdrv]';

% area specification for each gate with unit scaling
A_norm = [gates.A]';

% leakage current of gate i with unit scaling
Ileak_norm = [gates.Ileak]';

%********************************************************************
% optimization (with tradeoff curve generation)
%********************************************************************
% objective is the upper bound on the overall delay
% and that is the max of arrival times for output gates
output_gates = [FI{primary_outputs}];

% varying parameters for the tradeoff curve
N = 25;
Pmax = linspace(10,20,N);
min_delay = zeros(N,1);
disp('Generating the optimal tradeoff curve...')
for n = 1:N
  fprintf('Pmax = %6.2f: ',Pmax(n));
  cvx_begin gp quiet
    % optimization variables
    variable x(m)                 % scale factor
    variable T(m)                 % arrival times

    % input capacitance is an affine function of sizes
    Cin  = Cin_norm.*x;
    Cint = Cint_norm.*x;

    % driving resistance is inversily proportional to sizes
    R = Rdrv_norm./x;

    % gate delay is the product of its driving resistance and load cap.
    Cload = cvx( zeros(m,1) );
    for gate = 1:m
      if ~ismember( FO{gate}, primary_outputs )
        Cload(gate) = sum( Cin(FO{gate}) );
      else
        Cload(gate) = Cin_po( FO{gate} );
      end
    end

    % delay
    D = 0.69*ones(m,1).*R.*( Cint + Cload );

    % total area
    area = A_norm'*x;

    % total power calculation
    Pdyn = Vdd^2*sum( f_pi(primary_inputs).*Cload_pi(primary_inputs) ) + ...
           Vdd^2*(f_gates'*(Cint + Cload));
    Pstat = Vdd*Ileak_norm'*x;
    power = Pdyn + Pstat;

    minimize( max( T(output_gates) ) )
    subject to
      % constraints
      x >= 1;
      area <= Amax;
      power <= Pmax(n);

      % create timing constraints
      for gate = 1:m
        if ~ismember( FI{gate}, primary_inputs )
          for j = FI{gate}
            % enforce T_j + D_j <= T_i over all gates j that drive i
            D(gate) + T(j) <= T(gate);
          end
        else
          % enforce D_i <= T_i for gates i connected to primary inputs
          D(gate) <= T(gate);
        end
      end
  cvx_end
  fprintf( 'delay = %3.2f\n', cvx_optval );
  min_delay(n) = cvx_optval;
end

% plot the tradeoff curve
figure, clf
plot(Pmax,min_delay);
xlabel('Pmax'); ylabel('Dmin');
title(['Tradeoff curve for Amax = ' num2str(Amax)])
disp('Optimal tradeoff curve plotted.')

Generating the optimal tradeoff curve...
Pmax =  10.00: delay = 14.20
Pmax =  10.42: delay = 12.33
Pmax =  10.83: delay = 11.51
Pmax =  11.25: delay = 11.02
Pmax =  11.67: delay = 10.72
Pmax =  12.08: delay = 10.48
Pmax =  12.50: delay = 10.27
Pmax =  12.92: delay = 10.08
Pmax =  13.33: delay = 9.92
Pmax =  13.75: delay = 9.78
Pmax =  14.17: delay = 9.66
Pmax =  14.58: delay = 9.54
Pmax =  15.00: delay = 9.44
Pmax =  15.42: delay = 9.35
Pmax =  15.83: delay = 9.26
Pmax =  16.25: delay = 9.18
Pmax =  16.67: delay = 9.15
Pmax =  17.08: delay = 9.15
Pmax =  17.50: delay = 9.15
Pmax =  17.92: delay = 9.15
Pmax =  18.33: delay = 9.15
Pmax =  18.75: delay = 9.15
Pmax =  19.17: delay = 9.15
Pmax =  19.58: delay = 9.15
Pmax =  20.00: delay = 9.15
Optimal tradeoff curve plotted.