# Digital circuit sizing (vectorized)

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    % Boyd, Kim, Vandenberghe, and Hassibi, "A Tutorial on Geometric Programming"
    % Written for CVX by Almir Mutapcic 02/08/06
    % (a figure is generated)
    %
    % Solves the problem of choosing gate scale factors x_i to give
    % minimum ckt delay, subject to limits on the total area and power.
    %
    %   minimize   D
    %       s.t.   P <= Pmax, A <= Amax
    %              x >= 1
    %
    % where variables are scale factors x.
    %
    % This code uses matrices in order to evaluate signal paths
    % through the circuit (thus, it uses vectorize Matlab features).
    % It is specific to the digital circuit shown in figure 4 (page 28)
    % of GP tutorial paper.

    % digital circuit shown in figure 4 (page 28) of GP tutorial paper
    m = 7;  % number of cells
    n = 8;  % number of edges
    A = sparse(m,n);

    % A is standard cell-edge incidence matrix of the circuit
    % A_ij = 1 if edge j comes out of cell i, -1 if it comes in, 0 otherwise
      A(1,1) =     1;
      A(2,2) =     1;
      A(2,3) =     1;
      A(3,4) =     1;
      A(3,8) =     1;
      A(4,1) =    -1;
      A(4,2) =    -1;
      A(4,5) =     1;
      A(4,6) =     1;
      A(5,3) =    -1;
      A(5,4) =    -1;
      A(5,7) =     1;
      A(6,5) =    -1;
      A(7,6) =    -1;
      A(7,7) =    -1;
      A(7,8) =    -1;

    % decompose A into edge outgoing and edge-incoming part
    Aout = double(A > 0);
    Ain = double(A < 0);

    % problem constants
    f = [1 0.8 1 0.7 0.7 0.5 0.5]';
    e = [1 2 1 1.5 1.5 1 2]';
    Cout6 = 10;
    Cout7 = 10;

    a     = ones(m,1);
    alpha = ones(m,1);
    beta  = ones(m,1);
    gamma = ones(m,1);

    % varying parameters for an optimal trade-off curve
    N = 20;
    Pmax = linspace(10,100,N);
    Amax = [25 50 100];
    min_delay = zeros(length(Amax),N);

    disp('Generating the optimal tradeoff curve...')

    for k = 1:length(Amax)
        fprintf( 'Amax = %d:\n', Amax(k) );
        for n = 1:N
            fprintf( '    Pmax = %6.2f: ', Pmax(n) );
            cvx_begin gp quiet
              % optimization variables
              variable x(m)                 % scale factors
              variable t(m)                 % arrival times

              % objective is the upper bound on the overall delay
              % and that is the max of arrival times for output gates 6 and 7
              minimize( max( t(6),t(7) ) )
              subject to
                % input capacitance is an affine function of sizes
                cin = alpha + beta.*x;

                % load capacitance is the input capacitance times the fan-out matrix
                % given by Fout = Aout*Ain'
                cload = (Aout*Ain')*cin;
                cload(6) = Cout6;          % load capacitance of the output gate 6
                cload(7) = Cout7;          % load capacitance of othe utput gate 7

                % delay is the product of its driving resistance R = gamma./x and cload
                d = cload.*gamma./x;

                % power and area definitions
                power = (f.*e)'*x;
                area = a'*x;

                % scale size, power, and area constraints
                x >= 1;
                power <= Pmax(n);
                area <= Amax(k);

                % create timing constraints
                % these constraints enforce t_j + d_j <= t_i over all gates j that drive gate i
                Aout'*t + Ain'*d <= Ain'*t;

                % for gates with inputs not connected to other gates we enforce d_i <= t_i
                d(1:3) <= t(1:3);
            cvx_end
            fprintf( 'delay = %3.2f\n', cvx_optval );
            min_delay(k,n) = cvx_optval;
        end
    end

    % plot the tradeoff curve
    plot(Pmax,min_delay(1,:), Pmax,min_delay(2,:), Pmax,min_delay(3,:));
    xlabel('Pmax'); ylabel('Dmin');
    disp('Optimal tradeoff curve plotted.')

    Generating the optimal tradeoff curve...
    Amax = 25:
        Pmax =  10.00: delay = 12.21
        Pmax =  14.74: delay = 9.41
        Pmax =  19.47: delay = 8.01
        Pmax =  24.21: delay = 7.11
        Pmax =  28.95: delay = 6.80
        Pmax =  33.68: delay = 6.80
        Pmax =  38.42: delay = 6.80
        Pmax =  43.16: delay = 6.80
        Pmax =  47.89: delay = 6.80
        Pmax =  52.63: delay = 6.80
        Pmax =  57.37: delay = 6.80
        Pmax =  62.11: delay = 6.80
        Pmax =  66.84: delay = 6.80
        Pmax =  71.58: delay = 6.80
        Pmax =  76.32: delay = 6.80
        Pmax =  81.05: delay = 6.80
        Pmax =  85.79: delay = 6.80
        Pmax =  90.53: delay = 6.80
        Pmax =  95.26: delay = 6.80
        Pmax = 100.00: delay = 6.80
    Amax = 50:
        Pmax =  10.00: delay = 12.21
        Pmax =  14.74: delay = 9.41
        Pmax =  19.47: delay = 8.01
        Pmax =  24.21: delay = 7.11
        Pmax =  28.95: delay = 6.46
        Pmax =  33.68: delay = 5.97
        Pmax =  38.42: delay = 5.59
        Pmax =  43.16: delay = 5.27
        Pmax =  47.89: delay = 5.01
        Pmax =  52.63: delay = 4.79
        Pmax =  57.37: delay = 4.71
        Pmax =  62.11: delay = 4.71
        Pmax =  66.84: delay = 4.71
        Pmax =  71.58: delay = 4.71
        Pmax =  76.32: delay = 4.71
        Pmax =  81.05: delay = 4.71
        Pmax =  85.79: delay = 4.71
        Pmax =  90.53: delay = 4.71
        Pmax =  95.26: delay = 4.71
        Pmax = 100.00: delay = 4.71
    Amax = 100:
        Pmax =  10.00: delay = 12.21
        Pmax =  14.74: delay = 9.41
        Pmax =  19.47: delay = 8.01
        Pmax =  24.21: delay = 7.11
        Pmax =  28.95: delay = 6.46
        Pmax =  33.68: delay = 5.97
        Pmax =  38.42: delay = 5.59
        Pmax =  43.16: delay = 5.27
        Pmax =  47.89: delay = 5.01
        Pmax =  52.63: delay = 4.79
        Pmax =  57.37: delay = 4.60
        Pmax =  62.11: delay = 4.43
        Pmax =  66.84: delay = 4.28
        Pmax =  71.58: delay = 4.15
        Pmax =  76.32: delay = 4.03
        Pmax =  81.05: delay = 3.92
        Pmax =  85.79: delay = 3.82
        Pmax =  90.53: delay = 3.73
        Pmax =  95.26: delay = 3.65
        Pmax = 100.00: delay = 3.57
    Optimal tradeoff curve plotted.