Find Count of all Overlapping Intervals

Question

I have startTime and endTime for all records like this:

{
 startTime : 21345678
 endTime   : 31345678
}

I am trying to find number of all the c

Answer 1

As you correctly mention, there are different approaches with varying complexity inherent to their execution. This basically covers how they are done and which one you implement actually depends on which your data and use case is best suited to.

Current Range Match

MongoDB 3.6 $lookup

The most simple approach can be employed using the new syntax of the $lookup operator with MongoDB 3.6 that allows a pipeline to be given as the expression to "self join" to the same collection. This can basically query the collection again for any items where the starttime "or" endtime of the current document falls between the same values of any other document, not including the original of course:

db.getCollection('collection').aggregate([
  { "$lookup": {
    "from": "collection",
    "let": {
      "_id": "$_id",
      "starttime": "$starttime",
      "endtime": "$endtime"
    },
    "pipeline": [
      { "$match": {
        "$expr": {
          "$and": [
            { "$ne": [ "$$_id", "$_id" },
            { "$or": [
              { "$and": [
                { "$gte": [ "$$starttime", "$starttime" ] },
                { "$lte": [ "$$starttime", "$endtime" ] }
              ]},
              { "$and": [
                { "$gte": [ "$$endtime", "$starttime" ] },
                { "$lte": [ "$$endtime", "$endtime" ] }
              ]}
            ]},
          ]
        },
        "as": "overlaps"
      }},
      { "$count": "count" },
    ]
  }},
  { "$match": { "overlaps.0": { "$exists": true }  } }
])

The single $lookup performs the "join" on the same collection allowing you to keep the "current document" values for the "_id", "starttime" and "endtime" values respectively via the "let" option of the pipeline stage. These will be available as "local variables" using the $$ prefix in subsequent "pipeline" of the expression.

Within this "sub-pipeline" you use the $match pipeline stage and the $expr query operator, which allows you to evaluate aggregation framework logical expressions as part of the query condition. This allows the comparison between values as it selects new documents matching the conditions.

The conditions simply look for the "processed documents" where the "_id" field is not equal to the "current document", $and where either the "starttime" $or "endtime" values of the "current document" falls between the same properties of the "processed document". Noting here that these as well as the respective $gte and $lte operators are the "aggregation comparison operators" and not the "query operator" form, as the returned result evaluated by $expr must be boolean in context. This is what the aggregation comparison operators actually do, and it's also the only way to pass in values for comparison.

Since we only want the "count" of the matches, the $count pipeline stage is used to do this. The result of the overall $lookup will be a "single element" array where there was a count, or an "empty array" where there was no match to the conditions.

An alternate case would be to "omit" the $count stage and simply allow the matching documents to return. This allows easy identification, but as an "array embedded within the document" you do need to be mindful of the number of "overlaps" that will be returned as whole documents and that this does not cause a breach of the BSON limit of 16MB. In most cases this should be fine, but for cases where you expect a large number of overlaps for a given document this can be a real case. So it's really something more to be aware of.

The $lookup pipeline stage in this context will "always" return an array in result, even if empty. The name of the output property "merging" into the existing document will be "overlaps" as specified in the "as" property to the $lookup stage.

Following the $lookup, we can then do a simple $match with a regular query expression employing the $exists test for the 0 index value of output array. Where there actually is some content in the array and therefore "overlaps" the condition will be true and the document returned, showing either the count or the documents "overlapping" as per your selection.

Other versions - Queries to "join"

The alternate case where your MongoDB lacks this support is to "join" manually by issuing the same query conditions outlined above for each document examined:

db.getCollection('collection').find().map( d => {
  var overlaps = db.getCollection('collection').find({
    "_id": { "$ne": d._id },
    "$or": [
      { "starttime": { "$gte": d.starttime, "$lte": d.endtime } },
      { "endtime": { "$gte": d.starttime, "$lte": d.endtime } }
    ]
  }).toArray();

  return ( overlaps.length !== 0 ) 
    ? Object.assign(
        d,
        {
          "overlaps": {
            "count": overlaps.length,
            "documents": overlaps
          }
        }
      )
    : null;
}).filter(e => e != null);

This is essentially the same logic except we actually need to go "back to the database" in order to issue the query to match the overlapping documents. This time it's the "query operators" used to find where the current document values fall between those of the processed document.

Because the results are already returned from the server, there is no BSON limit restriction on adding content to the output. You might have memory restrictions, but that's another issue. Simply put we return the array rather than cursor via .toArray() so we have the matching documents and can simply access the array length to obtain a count. If you don't actually need the documents, then using .count() instead of .find() is far more efficient since there is not the document fetching overhead.

The output is then simply merged with the existing document, where the other important distinction is that since theses are "multiple queries" there is no way of providing the condition that they must "match" something. So this leaves us with considering there will be results where the count ( or array length ) is 0 and all we can do at this time is return a null value which we can later .filter() from the result array. Other methods of iterating the cursor employ the same basic principle of "discarding" results where we do not want them. But nothing stops the query being run on the server and this filtering is "post processing" in some form or the other.

Reducing Complexity

So the above approaches work with the structure as described, but of course the overall complexity requires that for each document you must essentially examine every other document in the collection in order to look for overlaps. Therefore whilst using $lookup allows for some "efficiency" in reduction of transport and response overhead, it still suffers the same problem that you are still essentially comparing each document to everything.

A better solution "where you can make it fit" is to instead store a "hard value"* representative of the interval on each document. For instance we could "presume" that there are solid "booking" periods of one hour within a day for a total of 24 booking periods. This "could" be represented something like:

{ "_id": "A", "booking": [ 10, 11, 12 ] }
{ "_id": "B", "booking": [ 12, 13, 14 ] }
{ "_id": "C", "booking": [ 7, 8 ] }
{ "_id": "D", "booking": [ 9, 10, 11 ] }

With data organized like that where there was a set indicator for the interval the complexity is greatly reduced since it's really just a matter of "grouping" on the interval value from the array within the "booking" property:

db.booking.aggregate([
  { "$unwind": "$booking" },
  { "$group": { "_id": "$booking", "docs": { "$push": "$_id" } } },
  { "$match": { "docs.1": { "$exists": true } } }
])

And the output:

{ "_id" : 10, "docs" : [ "A", "D" ] }
{ "_id" : 11, "docs" : [ "A", "D" ] }
{ "_id" : 12, "docs" : [ "A", "B" ] }

That correctly identifies that for the 10 and 11 intervals both "A" and "D" contain the overlap, whilst "B" and "A" overlap on 12. Other intervals and documents matching are excluded via the same $exists test except this time on the 1 index ( or second array element being present ) in order to see that there was "more than one" document in the grouping, hence indicating an overlap.

This simply employs the $unwind aggregation pipeline stage to "deconstruct/denormalize" the array content so we can access the inner values for grouping. This is exactly what happens in the $group stage where the "key" provided is the booking interval id and the $push operator is used to "collect" data about the current document which was found in that group. The $match is as explained earlier.

This can even be expanded for alternate presentation:

db.booking.aggregate([
  { "$unwind": "$booking" },
  { "$group": { "_id": "$booking", "docs": { "$push": "$_id" } } },
  { "$match": { "docs.1": { "$exists": true } } },
  { "$unwind": "$docs" },
  { "$group": {
    "_id": "$docs",
    "intervals": { "$push": "$_id" }  
  }}
])

With output:

{ "_id" : "B", "intervals" : [ 12 ] }
{ "_id" : "D", "intervals" : [ 10, 11 ] }
{ "_id" : "A", "intervals" : [ 10, 11, 12 ] }

It's a simplified demonstration, but where the data you have would allow it for the sort of analysis required then this is the far more efficient approach. So if you can keep the "granularity" to be fixed to "set" intervals which can be commonly recorded on each document, then the analysis and reporting can use the latter approach to quickly and efficiently identify such overlaps.

Essentially, this is how you would implement what you basically mentioned as a "better" approach anyway, and the first being a "slight" improvement over what you originally theorized. See which one actually suits your situation, but this should explain the implementation and the differences.