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list manipulation - Computing the equivalence classes of the symmetric transitive closure of a relation


I have a list of pairs, for example:


pairs={{13, 10}, {12, 14}, {10, 36}, {35, 11}, {3, 5}, {1, 6},

{20, 24}, {21, 22}, {33, 7}, {31, 8}, {31, 27}, {32, 25}, 
{21, 35}, {34, 19}, {18, 15}, {14, 16}, {9, 5}, {4, 7}, 
{1, 13}, {15, 2}, {6, 36}, {4, 34}, {8, 2}, {9, 3}, {25, 20},
{19, 26}, {22, 11}, {23, 12}, {32, 28}, {30, 33}, {23, 16},
{24, 17}, {29, 27}, {26, 30}, {17, 28}, {18, 29}};

pairs can be seen as the definition of a relation $R$. $x$ and $y$ satisfy the relation if and only if {x,y} $\in$ pairs. I need to compute the equivalence classes of the symmetric transitive closure of $R$.


In other words, I need to compute a list eqvclss. The elements of eqvclss are lists themselves. For example, 13, 10, 36, 6, 1, ... should all be in the same list in eqvclss. (If you understand that, then I explained the question properly; if you don't, say so in the comments so I can try to improve).



Answer



ConnectedComponents



Using Daniel Lichtblau's answer to a related question


ConnectedComponents[pairs] //Sort /@ # & //Sort (* thanks: CarlWoll *)


{{3, 5, 9},
{11, 21, 22, 35},
{12, 14, 16, 23},
{1, 6, 10, 13, 36},
{17, 20, 24, 25, 28, 32},
{2, 8, 15, 18, 27, 29, 31},

{4, 7, 19, 26, 30, 33, 34}}



In versions prior to 10.3 use


 ConnectedComponents[Graph[UndirectedEdge @@@ pairs]] //Sort /@ # & //Sort

MatrixPower


Implementing transitive closure using MatrixPower:


m = Max@pairs;

(*the adjacency matrix of atomic elements in pairs:*)

SparseArray[pairs ~Append~ {i_, i_} -> 1, {m, m}];

(*symmetrize the adjacency matrix:*)
% + %\[Transpose] // Sign;

(*find the transitive closure:*)
Sign @ MatrixPower[N@%, m];

(*eliminate duplicate rows,and extract the atomic elements of pairs in each row:*)
Select[DeleteDuplicates @ Normal @ %, Tr@# > 1 &];

Join @@ Position[#, 1] & /@ %;

(*organize:*)
Sort[Sort /@ %]


{{3, 5, 9},
{11, 21, 22, 35},
{12, 14, 16, 23},
{1, 6, 10, 13, 36},

{17, 20, 24, 25, 28, 32},
{2, 8, 15, 18, 27, 29, 31},
{4, 7, 19, 26, 30, 33, 34}}



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