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1 function order = best_first_elim_order(G, node_sizes, stage)
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2 % BEST_FIRST_ELIM_ORDER Greedily search for an optimal elimination order.
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3 % order = best_first_elim_order(moral_graph, node_sizes)
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4 %
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5 % Find an order in which to eliminate nodes from the graph in such a way as to try and minimize the
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6 % weight of the resulting triangulated graph. The weight of a graph is the sum of the weights of each
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7 % of its cliques; the weight of a clique is the product of the weights of each of its members; the
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8 % weight of a node is the number of values it can take on.
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9 %
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10 % Since this is an NP-hard problem, we use the following greedy heuristic:
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11 % at each step, eliminate that node which will result in the addition of the least
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12 % number of fill-in edges, breaking ties by choosing the node that induces the lighest clique.
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13 % For details, see
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14 % - Kjaerulff, "Triangulation of graphs -- algorithms giving small total state space",
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15 % Univ. Aalborg tech report, 1990 (www.cs.auc.dk/~uk)
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16 % - C. Huang and A. Darwiche, "Inference in Belief Networks: A procedural guide",
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17 % Intl. J. Approx. Reasoning, 11, 1994
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18 %
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19
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20 % Warning: This code is pretty old and could probably be made faster.
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21
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22 n = length(G);
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23 if nargin < 3, stage = { 1:n }; end % no constraints
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24
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25 % For long DBNs, it may be useful to eliminate all the nodes in slice t before slice t+1.
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26 % This will ensure that the jtree has a repeating structure (at least away from both edges).
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27 % This is why we have stages.
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28 % See the discussion of splicing jtrees on p68 of
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29 % Geoff Zweig's PhD thesis, Dept. Comp. Sci., UC Berkeley, 1998.
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30 % This constraint can increase the clique size significantly.
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31
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32 MG = G; % copy the original graph
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33 uneliminated = ones(1,n);
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34 order = zeros(1,n);
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35 t = 1; % Counts which time slice we are on
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36 for i=1:n
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37 U = find(uneliminated);
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38 valid = myintersect(U, stage{t});
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39 % Choose the best node from the set of valid candidates
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40 score1 = zeros(1,length(valid));
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41 score2 = zeros(1,length(valid));
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42 for j=1:length(valid)
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43 k = valid(j);
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44 ns = myintersect(neighbors(G, k), U);
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45 l = length(ns);
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46 M = MG(ns,ns);
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47 score1(j) = l^2 - sum(M(:)); % num. added edges
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48 score2(j) = prod(node_sizes([k ns])); % weight of clique
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49 end
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50 j1s = find(score1==min(score1));
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51 j = j1s(argmin(score2(j1s)));
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52 k = valid(j);
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53 uneliminated(k) = 0;
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54 order(i) = k;
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55 ns = myintersect(neighbors(G, k), U);
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56 if ~isempty(ns)
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57 G(ns,ns) = 1;
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58 G = setdiag(G,0);
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59 end
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60 if ~any(logical(uneliminated(stage{t}))) % are we allowed to the next slice?
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61 t = t + 1;
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62 end
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63 end
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64
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