# Computing the equivalence classes of the symmetric transitive closure of a relation

## 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 = [email protected];

(*the adjacency matrix of atomic elements in pairs:*)
SparseArray[pairs ~Append~ {i_, i_} -> 1, {m, m}];

% + %\[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}}

pairs //. x_ :> Union @@@ Gather[x, # ⋂ #2 =!= {} &]

{{1, 6, 10, 13, 36},
{12, 14, 16, 23},
{11, 21, 22, 35},
{3, 5,  9},
{17, 20, 24, 25, 28, 32},
{4, 7, 19, 26, 30, 33, 34},
{2, 8, 15, 18, 27, 29, 31}}


Here's code for version 7:

Needs["Combinatorica"]

gr = FromUnorderedPairs @ pairs;

ConnectedComponents @ gr

{{1, 6, 10, 13, 36},
{2, 8, 15, 18, 27, 29, 31},
{3, 5, 9},
{4, 7, 19, 26, 30, 33, 34},
{11, 21, 22, 35},
{12, 14, 16, 23},
{17, 20, 24, 25, 28, 32}}

GraphPlot[gr, VertexLabeling -> True]
`