function [p,q,D] = dp_dtw2(M,t_a,t_b)
% [p,q] = dp(M) 
%    Use dynamic programming to find a min-cost path through matrix M.
%    Return state sequence in p,q
% 2003-03-15 dpwe@ee.columbia.edu

% Copyright (c) 2003 Dan Ellis <dpwe@ee.columbia.edu>
% released under GPL - see file COPYRIGHT

[r,c] = size(M);

% costs
D = zeros(r+1, c+1);
D(1,:) = NaN;
D(:,1) = NaN;
D(1,1) = 0;
D(2:(r+1), 2:(c+1)) = M;

% traceback
phi = zeros(r,c);

for i = 1:r; 
  for j = 1:c;
    [dmax, tb] = min([D(i, j)+M(i,j)*(t_a(i)+t_b(j)), D(i, j+1)+M(i,j)*t_a(i), D(i+1, j)+M(i,j)*t_b(j)]);
    D(i+1,j+1) = D(i+1,j+1)+dmax;
    phi(i,j) = tb;
  end
end

% Traceback from top left
i = r; 
j = c;
p = i;
q = j;
while i > 1 & j > 1
  tb = phi(i,j);
  if (tb == 1)
    i = i-1;
    j = j-1;
  elseif (tb == 2)
    i = i-1;
  elseif (tb == 3)
    j = j-1;
  else    
    error;
  end
  p = [i,p];
  q = [j,q];
end
p=[1,p];
q=[1,q];

% Strip off the edges of the D matrix before returning
D = D(2:(r+1),2:(c+1));