function D = dijk(A,s,t) %DIJK Shortest paths from nodes 's' to nodes 't' using Dijkstra algorithm. % D = dijk(A,s,t) % A = n x n node-node weighted adjacency matrix of arc lengths % (Note: A(i,j) = 0 => Arc (i,j) does not exist; % A(i,j) = NaN => Arc (i,j) exists with 0 weight) % s = FROM node indices % = [] (default), paths from all nodes % t = TO node indices % = [] (default), paths to all nodes % D = |s| x |t| matrix of shortest path distances from 's' to 't' % = [D(i,j)], where D(i,j) = distance from node 'i' to node 'j' % % (If A is a triangular matrix, then computationally intensive node % selection step not needed since graph is acyclic (triangularity is a % sufficient, but not a necessary, condition for a graph to be acyclic) % and A can have non-negative elements) % % (If |s| >> |t|, then DIJK is faster if DIJK(A',t,s) used, where D is now % transposed and P now represents successor indices) % % (Based on Fig. 4.6 in Ahuja, Magnanti, and Orlin, Network Flows, % Prentice-Hall, 1993, p. 109.) % Copyright (c) 1998-2000 by Michael G. Kay % Matlog Version 1.3 29-Aug-2000 % % Modified by JBT, Dec 2000, to delete paths % Input Error Checking ****************************************************** error(nargchk(1,3,nargin)); [n,cA] = size(A); if nargin < 2 | isempty(s), s = (1:n)'; else s = s(:); end if nargin < 3 | isempty(t), t = (1:n)'; else t = t(:); end if ~any(any(tril(A) ~= 0)) % A is upper triangular isAcyclic = 1; elseif ~any(any(triu(A) ~= 0)) % A is lower triangular isAcyclic = 2; else % Graph may not be acyclic isAcyclic = 0; end if n ~= cA error('A must be a square matrix'); elseif ~isAcyclic & any(any(A < 0)) error('A must be non-negative'); elseif any(s < 1 | s > n) error(['''s'' must be an integer between 1 and ',num2str(n)]); elseif any(t < 1 | t > n) error(['''t'' must be an integer between 1 and ',num2str(n)]); end % End (Input Error Checking) ************************************************ A = A'; % Use transpose to speed-up FIND for sparse A D = zeros(length(s),length(t)); P = zeros(length(s),n); for i = 1:length(s) j = s(i); Di = Inf*ones(n,1); Di(j) = 0; isLab = logical(zeros(length(t),1)); if isAcyclic == 1 nLab = j - 1; elseif isAcyclic == 2 nLab = n - j; else nLab = 0; UnLab = 1:n; isUnLab = logical(ones(n,1)); end while nLab < n & ~all(isLab) if isAcyclic Dj = Di(j); else % Node selection [Dj,jj] = min(Di(isUnLab)); j = UnLab(jj); UnLab(jj) = []; isUnLab(j) = 0; end nLab = nLab + 1; if length(t) < n, isLab = isLab | (j == t); end [jA,kA,Aj] = find(A(:,j)); Aj(isnan(Aj)) = 0; if isempty(Aj), Dk = Inf; else Dk = Dj + Aj; end P(i,jA(Dk < Di(jA))) = j; Di(jA) = min(Di(jA),Dk); if isAcyclic == 1 % Increment node index for upper triangular A j = j + 1; elseif isAcyclic == 2 % Decrement node index for lower triangular A j = j - 1; end %disp( num2str( nLab )); end D(i,:) = Di(t)'; end