Combining two lists with minimum distance between elements - algorithm

Question

I have to lists like these:
a = ["1a","2a","3a","4a","5a","6a","7a","8a","9a","10a","11a","12a","13a","14a"]
b = ["1b","2b","3b","4b","5b","6b","7b","8b","9b","10b","11b","12b","13b","14b"]
And what I want is to combine them, so that there is at least a difference of n elements between an element from a and it's corresponding element in b.
As an example, if my n is 10, and "3a" is in position 3 and "3b" is in position 5, that isn't a solution since there's only a distance of 2 between these corresponding elements.
I have already solved this for the purpose I want through a brute force method: shuffle the union of the two arrays and see if the constraint is met; if not, shuffle again and so on... Needless to say, that for 14 elements array, sometimes there is 5 to 10 second computation to yield a solution with a minimum distance of 10. Even though that's kind of ok for what I am looking for, I am curious about how I could solve this in a more optimized way.
I am currently using Python, but code in any language (or pseudo-code) is more than welcomed.
EDIT: The context of this problem is something like a questionnarie, in which around 100 participants are expected to join in. Therefore, I am not necessarily interested in all the solutions, but rather something like the first 100.
Thanks.

Answer 1

For your specific scenario, you could use a randomized approach -- though not as random as what you've already tried. Something like this:
start with a random permutation of the items in both lists
create a new permutation by creating a copy of the other and randomly swapping two items
measure the quality of the permutations, e.g., the sum of the distances of each pair of related items, or the minimum of such distances
if the quality of the new permutation is better than that of the original permutation, keep the new one, otherwise discard the new one and continue with the original permutation
repeat from 2. until each distance is at least 10 or until quality does not improve over a number of iterations
The difference to shuffling the whole list in each iteration (as in your approach) is that in each iteration the permutation can only get better, until a satisfying solution is found.
Each time you run this algorithm, the result will be slightly different, so you can run it 100 times for 100 different solutions. Of course, this algorithm does not guarantee to find a solution (much less all such solutions), but it should be fast enough so that you could just restart it in case it fails.
In Python, this could look somewhat like this (slightly simplified, but still working):
def shuffle(A, B):
# original positions, i.e. types of questions
kind = dict([(item, i) for i, item in list(enumerate(A)) + list(enumerate(B))])
# get positions of elements of kinds, and return sum of their distances
def quality(perm):
pos = dict([(kind[item], i) for i, item in enumerate(perm)])
return sum(abs(pos[kind[item]] - i) for i, item in enumerate(perm))
# initial permutation and quality
current = A + B
random.shuffle(current)
best = quality(current)
# improve upon initial permutation by randomly swapping items
for g in range(1000):
i = random.randint(0, len(current)-1)
j = random.randint(0, len(current)-1)
copy = current[:]
copy[i], copy[j] = copy[j], copy[i]
q = quality(copy)
if q > best:
current, best = copy, q
return current
Example output for print shuffle(a, b):
['14b', '2a', '13b', '3a', '9b', '4a', '6a', '1a', '8a', '5b', '12b', '11a', '10b', '7b', '4b', '11b', '5a', '7a', '8b', '12a', '13a', '14a', '1b', '2b', '3b', '6b', '10a', '9a']

Answer 2

As I understand from your question, it is possible to perform all the ordering by relying exclusively on the indices of the arrays (i.e., on pure integers) and thus the problem can be reduced to create (valid) ranges instead of analysing each element.
for each a <= total_items-n , valid b = if(a + n == total_items){total_items} else{[a + n, total_items]}
For example:
n = 10;
total_items = 15;
for a = 1 -> valid b = [11, 15]
for a = 2 -> valid b = [12, 15]
, etc.
This would be perfomed 4 times: forwards and backwards for a respect to b and the same for b respect to a.
In this way you would reduce the number of iterations to its minimum expression and would get, as an output, a set of "solutions" for each position, rather than a one-to-one binding (that is what you have right now, isn't it?).

Answer 3

If there are equivalents in Python to .NET's Lists and LINQ, then you may be able to directly convert the following code. It generates up to 100 lists really quickly: I press "debug" to run it and up pops a windows with the results in much less than a second.
' VS2012
Option Infer On
Module Module1
Dim minDistance As Integer = 10
Dim rand As New Random ' a random number generator
Function OkToAppend(current As List(Of Integer), x As Integer) As Boolean
' see if the previous minDistance values contain the number x
Return Not (current.Skip(current.Count - minDistance).Take(minDistance).Contains(x))
End Function
Function GenerateList() As List(Of String)
' We don't need to start with strings: integers will make it faster.
' The "a" and "b" suffixes can be sprinkled on at random once the
' list is created.
Dim numbersToUse() As Integer = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}
Dim pool As New List(Of Integer)
' we need all the numbers twice
pool.AddRange(numbersToUse)
pool.AddRange(numbersToUse)
Dim newList As New List(Of Integer)
Dim pos As Integer
For i = 0 To pool.Count - 1
' limit the effort it puts in
Dim sanity As Integer = pool.Count * 10
Do
pos = rand.Next(0, pool.Count)
sanity -= 1
Loop Until OkToAppend(newList, pool(pos)) OrElse sanity = 0
If sanity > 0 Then ' it worked
' append the value to the list
newList.Add(pool(pos))
' remove the value which has been used
pool.RemoveAt(pos)
Else ' give up on this arrangement
Return Nothing
End If
Next
' Create the final list with "a" and "b" stuck on each value.
Dim stringList As New List(Of String)
Dim usedA(numbersToUse.Length) As Boolean
Dim usedB(numbersToUse.Length) As Boolean
For i = 0 To newList.Count - 1
Dim z = newList(i)
Dim suffix As String = ""
If usedA(z) Then
suffix = "b"
ElseIf usedB(z) Then
suffix = "a"
End If
' rand.Next(2) generates an integer in the range [0,2)
If suffix.Length = 0 Then suffix = If(rand.Next(2) = 1, "a", "b")
If suffix = "a" Then
usedA(z) = True
Else
usedB(z) = True
End If
stringList.Add(z.ToString & suffix)
Next
Return stringList
End Function
Sub Main()
Dim arrangements As New List(Of List(Of String))
For i = 1 To 100
Dim thisArrangement = GenerateList()
If thisArrangement IsNot Nothing Then
arrangements.Add(thisArrangement)
End If
Next
'TODO: remove duplicate entries and generate more to make it up to
' the required quantity.
For Each a In arrangements
' outputs the elements of a with ", " as a separator
Console.WriteLine(String.Join(", ", a))
Next
' wait for user to press enter
Console.ReadLine()
End Sub
End Module