generating all unique pairs from a list of numbers, n choose 2

Question

i have a list of elements (let's say integers), and i need to make all possible 2-pair comparisons. my approach is O(n^2), and i am wondering if there is a faster way. here is my implementation in java.

public class Pair {
 public int x, y;
 public Pair(int x, int y) {
  this.x = x;
  this.y = y;
 }
}

public List<Pair> getAllPairs(List<Integer> numbers) {
 List<Pair> pairs = new ArrayList<Pair>();
 int total = numbers.size();
 for(int i=0; i < total; i++) {
  int num1 = numbers.get(i).intValue();
  for(int j=i+1; j < total; j++) {
   int num2 = numbers.get(j).intValue();
   pairs.add(new Pair(num1,num2));
  }
 }
 return pairs;
}

please note that i don't allow self-pairing, so there should be ((n(n+1))/2) - n possible pairs. what i have currently works, but as n increases, it is taking me an unbearable long amount of time to get the pairs. is there any way to turn the O(n^2) algorithm above to something sub-quadratic? any help is appreciated.

by the way, i also tried the algorithm below, but when i benchmark, empirically, it performs worst than what i had above. i had thought that by avoiding an inner loop this would speed things up. shouldn't this algorithm below be faster? i would think that it's O(n)? if not, please explain and let me know. thanks.

public List<Pair> getAllPairs(List<Integer> numbers) {
 int n = list.size();
 int i = 0;
 int j = i + 1;
 while(true) {
  int num1 = list.get(i);
  int num2 = list.get(j);
  pairs.add(new Pair(num1,num2));

  j++;

  if(j >= n) {
   i++;
   j = i + 1;
  }

  if(i >= n - 1) {
   break;
  }
 }
}

Answer 1

Well, you can't, right?

The result is a list with n*(n-1)/2 elements, no matter what those elements are, just to populate this list (say with zeros) takes O(n^2) time, since n*(n-1)/2 = O(n^2) ...

Answer 2

You cannot make it sub-quadric, because as you said - the output is itself quadric - and to create it, you need at least #elements_in_output ops.

However, you could do some "cheating" create your list on the fly :
You can create a class CombinationsGetter that implements Iterable<Pair> , and implement its Iterator<Pair> . This way, you will be able to iterate on the elements on the fly, without creating the list first, which might decrease latency for your application.

Note : It will still be quadric! The time to generate the list on the fly will just be distributed between more operations.

---

EDIT: Another solution, which is faster then the naive approach - is multithreading .
Create a few threads, each will get his "slice" of the data - and generate relevant pairs, and create its own partial list.
Later - you can use ArrayList.addAll() to convert those different lists into one.
Note: though complexity is stiil O(n^2) , it is likely to be much faster - since the creation of pairs is done in parallel, and ArrayList.addAll() is implemented much more effieciently then the trivial insert one by one elements.

EDIT2: Your second code is still O(n^2) , even though it is a "single loop" - the loop itself will repeat O(n^2) times. Have a look at your variable i . It increases only when j==n , and it decreases j back to i+1 when it does it. So, it will result in n + (n-1) + ... + 1 iterations, and this is sum of arithmetic progression , and gets us back to O(n^2) as expected.

We cannot get better then O(n^2), because we are trying to create O(n^2) distinct Pair objects.