graphs and networks - Binary decision diagrams (BDD)

I would like to use Mathematica to help me simplify a given Binary Decision Diagram (i.e. pick the optimal variable ordering). I don't seem to find anything at all on the topic, although it is something everyone with a little CS background knows very well.

Given MMA's ability of dealing with graphs and boolean functions the task seems feasible, but I wouldn't want to waste a long time to solve a problem that is already solved. Is anybody aware of MMA resources on the matter?

Answer

This answer is a modification of my answer given in the discussion Creating Identification/Classification trees.

With this solution I am trying to achieve the simplification by using the impurity function applied to the data (the truth table in this case).

Make the truth table from the linked Wikipedia article (Binary Decision Diagram):

truthTable = {{0, 0, 0, 1}, {0, 0, 1, 0}, {0, 1, 0, 0}, {0, 1, 1, 
    1}, {1, 0, 0, 0}, {1, 0, 1, 0}, {1, 1, 0, 1}, {1, 1, 1, 1}};
truthTable = Map[ToString, truthTable, {-1}];
colNames = {"x1", "x2", "x3", "f"};
TableForm[truthTable, TableHeadings -> {None, colNames}]

Import the package:

Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/AVCDecisionTreeForest.m"]

Build a decision tree:

dtree = BuildDecisionTree[truthTable, "ImpurityFunction" -> "Gini"]

(* {{0.125, "0", 2, Symbol, 
  8}, {{0.125, "0", 1, Symbol, 

   4}, {{0.5, "0", 3, Symbol, 
    2}, {{{1, "1"}}}, {{{1, "0"}}}}, {{{2, "0"}}}}, {{0.125, "0", 1, Symbol, 
   4}, {{0.5, "0", 3, Symbol, 2}, {{{1, "0"}}}, {{{1, "1"}}}}, {{{2, "1"}}}}} *)

The impurity function "Gini" is used by default. Another possible value is "Entropy". Note that to make the decision tree I converted the 0-1 values in the truth table into strings.

Visualize the tree:

trules = dtree // DecisionTreeToRules;
trules = trules /. ({m_, v_, cInd_Integer, s_, n_, id_} :> {m, v, colNames[[cInd]], s, n, id});
LayeredGraphPlot[trules, VertexLabeling -> True]

The last commands produce the plot:

The second row of a leaf shows the corresponding number of records and label.

The tree rules can be manipulated to produce a graph closer to the one in the Wikipedia article, if that is desired.

See these blog posts for application examples, parameters explanations, and tweaking discussions:

Decision trees posts at MathematicaForPrediction at WordPress.

Comments

plotting - How to draw lines between specified dots on ListPlot?

I would like to create a plot where I have unconnected dots and some connected. So far, I have figured out how to draw the dots. My code is the following: ListPlot[{{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4,13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full] I have thought using ListLinePlot command, but I don't know how to specify to the command to draw only selected lines between the dots. Do have any suggestions/hints on how to do that? Thank you. Answer One possibility would be to use Epilog with Line : ListPlot[ {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4, 13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full, Epilog -> { Line[ ...

dynamic - How can I make a clickable ArrayPlot that returns input?

I would like to create a dynamic ArrayPlot so that the rectangles, when clicked, provide the input. Can I use ArrayPlot for this? Or is there something else I should have to use? Answer ArrayPlot is much more than just a simple array like Grid : it represents a ranged 2D dataset, and its visualization can be finetuned by options like DataReversed and DataRange . These features make it quite complicated to reproduce the same layout and order with Grid . Here I offer AnnotatedArrayPlot which comes in handy when your dataset is more than just a flat 2D array. The dynamic interface allows highlighting individual cells and possibly interacting with them. AnnotatedArrayPlot works the same way as ArrayPlot and accepts the same options plus Enabled , HighlightCoordinates , HighlightStyle and HighlightElementFunction . data = {{Missing["HasSomeMoreData"], GrayLevel[ 1], {RGBColor[0, 1, 1], RGBColor[0, 0, 1], GrayLevel[1]}, RGBColor[0, 1, 0]}, {GrayLevel[0], GrayLevel...

Is there a way to do conditional matrix loop using 'continue'

I have the following: n = 3; m = 5; ww = RandomReal[{0, 0.1}, {n, n}]; uu = RandomReal[{0, 1}, {m, n}]; pp = RandomReal[{0, 1}, {n, n}]; ss = RandomInteger[{0, 5}, {m, n}]; Grid[{{"ww", "uu", "pp", "ss"}, {ww // TableForm, uu // TableForm, pp // TableForm, ss // TableForm}}, Spacings -> {5, 2}, Dividers -> All] where I would like to look at every element of matrix ss and produce a matrix tt , with zeroes at the locations in ss which have zeroes, and in all other positions do the following: tt = (-1/Subscript[ww, m]) Log[(1 - uu)/(Subscript[pp, m - 1])], where Subscript[ww, m] is the value at index of ww matrix and where Subscript[pp, m - 1] is the value at index-1 of pp matrix. So for example if the first value ever read from matrix ss happens to be 2, then value taken from matrix ww would be from the row 2, but from pp would be from row 1. Also how to tell difference between a 0 as a valid value from within the matrix elemen...

Blog

Search This Blog