How to sort integer numbers in a huge text file?

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Problem Statement

I'm given a very large list of numbers one at a time, and I need to print the "median number".

To be more clear there can be "125,000,000" numbers and it is guaranteed that each number is less than "1.e+18".

It's for a contest, so there is memory limit (20 MB tops) and time limit (5 seconds tops).

"The median number" is the one that is in the middle of sorted numbers.
For instance if this is the list of numbers:

23 8 16 42 15 4 108

After sorting numbers:

1) 4 2) 8 3) 15 4) 16 5) 23 6) 42 7) 108

"The median number" would be 16;

So I searched for it in the Internet but I couldn't find any answer that pass these limits.

Approach

My approach was to get all the numbers, save them in a text file, sort them, then get "the median number".

Ideas

1. I know I can read all the numbers from the file and put them in a vector then easily sort them.
But this will exceed memory limit.
2. So I came up with this idea to sort numbers as I put them in text file.
It would be like there is a loop that after getting next number from the console reads the file (line by line) and when it reaches right place, inserts number there and doesn't touch other numbers.
But the problem is that I can't insert a line in the middle of text file because it will overwrite on other numbers.
3. So I created two files that one of them had numbers that were already inputted and the other one read the first file and copy it's number in itself until reach right place then insert last given number then continue copying remaining numbers.
But it spends too much time so it exceeded time limit.

Request

So I want to either optimize one of these ideas in order to pass limits or any new idea that pass these limits.

Preference

I prefer to use second idea because unlike other two, it passes limits but I can't do it because I don't know how to insert a line in the middle of a text file. So if I learn this, rest of the process will be so easy.

Attempted Solution

This is a function that receives a number and, by reading through a file, finds best place for it and puts it there.
As the matter of fact it represents my third idea.
So it works (I tested it with lots of inputs) but the problem as I mentioned before is with the time limit.

void insertNewCombinedNumber ( int combinedNumber ) {     char combinedNumberCharacterArray[ 20 ];     bool isInserted = false;      ofstream combinedNumbersOutputFile;     ifstream combinedNumbersInputFile;      // Operate on First File     if ( isFirstCombinedFileActive )     {         combinedNumbersOutputFile.open ( "Combined Numbers - File 01.txt" );         combinedNumbersInputFile.open ( "Combined Numbers - File 02.txt" );     }     // Operate on Second File     else     {         combinedNumbersOutputFile.open ( "Combined Numbers - File 02.txt" );         combinedNumbersInputFile.open ( "Combined Numbers - File 01.txt" );     }      if ( !combinedNumbersInputFile )     {         combinedNumbersInputFile.close ();          ofstream combinedNumbersInputCreateFile ( "Combined Numbers - File 02.txt" );         combinedNumbersInputCreateFile.close ();          combinedNumbersInputFile.open ( "Combined Numbers - File 02.txt" );     }      combinedNumbersInputFile.getline ( combinedNumberCharacterArray , 20 );      for ( int i = 0; !combinedNumbersInputFile.eof (); i++ )     {         if ( !isInserted && combinedNumber <= characterArrayToDecimal ( combinedNumberCharacterArray ) )         {             combinedNumbersOutputFile << combinedNumber << endl;             isInserted = true;         }          combinedNumbersOutputFile << combinedNumberCharacterArray << endl;          combinedNumbersInputFile.getline ( combinedNumberCharacterArray , 20 );     }      if ( !isInserted )     {         combinedNumbersOutputFile << combinedNumber << endl;         isInserted = true;     }      isFirstCombinedFileActive = !isFirstCombinedFileActive;      combinedNumbersOutputFile.close ();     combinedNumbersInputFile.close (); }

I will assume the list of numbers is already in binary form (because we will need multiple passes through the data, and each time converting text to binary would take extra processing time). That would be a file of 1GB (125M * 64bit).

It is also not clear if the OS disk caching of that file would count for the memory limit. I will assume it's not, because reading a 1GB file cold from disk multiple times would already take more than 5 seconds.

So let's start with a simple example of how this could be done (we will optimize and adjust this later):

• First create a histogram of ranges of numbers (say groups of 1 million as an example, but this won't work yet - see below)
• So create an uint32 array of size max value / 1 million (too big for now) where we will put the count of the buckets (0-999999, 1000000-1999999, and so on).
• Loop through the list of numbers and each time increment the n-th value of the array (the bucket where the number belongs).
• Now that we have an array with counts we can easily calculate in which bucket (or range) the median will be.
• Loop through the list again and now only store the numbers that fit in that range in an array.
• Sort the array and calculate which item is the median (also using the counts of all buckets).

Of course, we need to adjust the above a bit.

First, instead of using ranges of 1 million, it's better to use a power of two. That way we can simply use an and with a mask to get the position in the bucket/count list (instead of using a more expensive division).

Second, for using buckets with ranges of 1 million we would have to create an array that is way too big.

So the best option would be to do 3 passes: first with ranges of say 1e12, and then for the range the median is in, we loop again with ranges of 1e6 (but instead use powers of 2).

This way you would only have to sort the numbers belonging to one small bucket instead of the whole set of 125 million. Sorting takes O(n log n).

An example with the numbers given in the question:

23 8 16 42 15 4 108

Use buckets/ranges of 16 - first pass:

array_pos   count 0 (0-15)      3 1 (16-31)     2 2 (32-47)     1 3 (48-63)     0 4 (64-79)     0 5 (80-95)     0 6 (96-111)    1

We can now calculate that the median must be in the bucket at array_pos 1.

remember/store these values: Count before bucket 16-31: 3 Count  after bucket 16-31: 2

Second pass - read values for bucket (16-31) - (again, if the bucket sizes are a power of two we can use some bitmasking to quickly check if the number is in the range):

23 16

Sort this small array and calculate the position of the median using the 2 counts (before and after).

count 3     16 -> median     23 2