Random Unique Pairs - algorithm

Question

I have a list of 100 items. I'd like to randomly pair these items with each other. These pairs must be unique, so there are 4950 possibilities (100 choose 2) total.
Of all 4950 pairs, I'd like to have 1000 pairs randomly selected. But they key is, I'd like each item (of the 100 items) to overall appear the same amount of times (here, 20 times).
I tried to implement this with code a couple of times. And it worked fine when I tried with a lower amount of pairs chosen, but each time I try with the full 1000 pairs, I get stuck in a loop.
Does anyone have an idea for an approach? And what if I change the number of pairs I wish to select (e.g., 1500 rather than 1000 random pairs)?
My attempt (written in VBA):
Dim City1(4951) As Integer
Dim City2(4951) As Integer
Dim CityCounter(101) As Integer
Dim PairCounter(4951) As Integer
Dim i As Integer
Dim j As Integer
Dim k As Integer
i = 1
While i < 101
CityCounter(i) = 0
i = i + 1
Wend
i = 1
While i < 4951
PairCounter(i) = 0
i = i + 1
Wend
i = 1
j = 1
While j < 101
k = j + 1
While k < 101
City1(i) = j
City2(i) = k
k = k + 1
i = i + 1
Wend
j = j + 1
Wend
Dim temp As Integer
i = 1
While i < 1001
temp = Random(1,4950)
While ((PairCounter(temp) = 1) Or (CityCounter( (City1(temp)) ) = 20) Or (CityCounter( (City2(temp)) ) = 20))
temp = Random(1,4950)
Wend
PairCounter(temp) = 1
CityCounter( (City1(temp)) ) = (CityCounter( (City1(temp)) ) + 1)
CityCounter( (City2(temp)) ) = (CityCounter( (City2(temp)) ) + 1)
i = i + 1
Wend

Answer 1

Take a list, scramble it, and mark every two elements off as a pair. Add these pairs to a list of pairs. Ensure that list of pairs is sorted.
Scramble the list of pairs, and add each pair to a "staged" pair list. Check if it's in the list of pairs. If it's in the list of pairs, scramble and start over. If you get the entire list without any duplicates, add the staged pair list to the pair list and start this paragraph over.
Since this involves a nondeterministic step at the end I'm not sure how slow it will be, but it should work.

Answer 2

This is old thread, but I was looking for something similar, and finaly did it myself.
The algorithm is not 100% random (after being a bit "tired" with unsuccessfull random trials starts systematic screening of the table :) - anyway for me - "random enough") but works reasonably fast, and returns required table (unfortunalety not always, but...) usually every second or third use (look in A1 if there is your reqired number of pairs for each item).
Here is VBA code to be run in Excel environment.
Output is directed to current sheet starting from A1 cell.
Option Explicit
Public generalmax%, oldgeneralmax%, generalmin%, alloweddiff%, i&
Public outtable() As Integer
Const maxpair = 100, upperlimit = 20
Sub generate_random_unique_pairs()
'by Kaper 2015.02 for stackoverflow.com/questions/14884975
Dim x%, y%, counter%
Randomize
ReDim outtable(1 To maxpair + 1, 1 To maxpair + 1)
Range("A1").Resize(maxpair + 1, maxpair + 1).ClearContents
alloweddiff = 1
Do
i = i + 1
If counter > (0.5 * upperlimit) Then 'try some systematic approach
For x = 1 To maxpair - 1 ' top-left or:' To 1 Step -1 ' bottom-right
For y = x + 1 To maxpair
Call test_and_fill(x, y, counter)
Next y
Next x
If counter > 0 Then
alloweddiff = alloweddiff + 1
counter = 0
End If
End If
' mostly used - random mode
x = WorksheetFunction.RandBetween(1, maxpair - 1)
y = WorksheetFunction.RandBetween(x + 1, maxpair)
counter = counter + 1
Call test_and_fill(x, y, counter)
If counter = 0 Then alloweddiff = WorksheetFunction.Max(alloweddiff, 1)
If i > (2.5 * upperlimit) Then Exit Do
Loop Until generalmin = upperlimit
Range("A1").Resize(maxpair + 1, maxpair + 1).Value = outtable
Range("A1").Value = generalmin
Application.StatusBar = ""
End Sub
Sub test_and_fill(x%, y%, ByRef counter%)
Dim temprowx%, temprowy%, tempcolx%, tempcoly%, tempmax%, j%
tempcolx = outtable(1, x + 1)
tempcoly = outtable(1, y + 1)
temprowx = outtable(x + 1, 1)
temprowy = outtable(y + 1, 1)
tempmax = 1+ WorksheetFunction.Max(tempcolx, tempcoly, temprowx, temprowy)
If tempmax <= (generalmin + alloweddiff) And tempmax <= upperlimit And outtable(y + 1, x + 1) = 0 Then
counter = 0
outtable(y + 1, x + 1) = 1
outtable(x + 1, y + 1) = 1
outtable(x + 1, 1) = 1 + outtable(x + 1, 1)
outtable(y + 1, 1) = 1 + outtable(y + 1, 1)
outtable(1, x + 1) = 1 + outtable(1, x + 1)
outtable(1, y + 1) = 1 + outtable(1, y + 1)
generalmax = WorksheetFunction.Max(generalmax, outtable(x + 1, 1), outtable(y + 1, 1), outtable(1, x + 1), outtable(1, y + 1))
generalmin = outtable(x + 1, 1)
For j = 1 To maxpair
If outtable(j + 1, 1) < generalmin Then generalmin = outtable(j + 1, 1)
If outtable(1, j + 1) < generalmin Then generalmin = outtable(1, j + 1)
Next j
If generalmax > oldgeneralmax Then
oldgeneralmax = generalmax
Application.StatusBar = "Working on pairs " & generalmax & "Total progress (non-linear): " & Format(1# * generalmax / upperlimit, "0%")
End If
alloweddiff = alloweddiff - 1
i = 0
End If
End Sub

Answer 3

Have an array appeared[] which keeps track of how many times each item already appeared in answer. Let's say each element has to appear k times. Iterate over the array, and while current element has its appeared value less than k, choose a random pair for it from that element who also have appeared less than k times. Add that pair to answer and increase appearance count for both.

Answer 4

create a 2-dimensional 100*100 matrix of booleans, all False
of these 10K booleans, set 1K of them to true, with the following constraints:
the diagonal should stay empty
no row or column should have more than 20 true values
at the end, every row and column should have 20 True values.
Now, there is the X=Y diagonal symmetry. Just add the following constraints:
the triangle at one side of the diagonal should stay empty
in the above constraints, the restrictions for rows&columns should be combined/added