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list manipulation - Identifying and isolating sections of overlap in a set of integer intervals



I have a very large array like the following:


{{1,10},{8,9},{9,20},{34,40},{41,42},{43,50},{47,53},...};

Now, imagine that every element in the array corresponds to an integer interval. I'd like to delete elements in the array that fall within a larger interval, and whenever there is an interval-interval overlap, I'd like to shrink the intersecting intervals to eliminate the overlap and then add the former section where the overlap occurred as a new element to a separate array (here I'm calling this overlapArray).


Perhaps this is best explained by example. Looking at the smaller case:


{{1,10},{8,9},{9,20},{34,40}...}

We notice that there is an overlap between the first three elements of the array, and that the element {8,9} falls entirely in this overlap section. So we this array becomes the following after our procedure:


{{1,7},{11,20},{34,40}...}


overlapArray = {{8,10}};

Now for the full example, this:


{{1,10},{8,9},{9,20},{34,40},{41,42},{43,50},{47,53},...};

Becomes:


{{1,7},{11,20},{34,40},{41,42},{43,46},{51,53}

overlapArray = {{8,10},{47,50}};


Is there an elegant way to do this with list operations in Mathematica v9?



Answer



I like this one :P


list = {{1, 10}, {8, 9}, {9, 20}, {34, 40}, {41, 42}, {43, 50}, {47, 53}}

Composition[
 Split[#, #2 - #1 == 1 &][[ All , {1, -1}]] &,
 Sort,
 DeleteCases[#, {_, 1}][[ All , 1]] &,
 Tally,

 Flatten,
 Range @@@ # &
  ][list]


 {{8, 10}, {47, 50}}

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