Big-O time complexity of recursive algorithm with nested for loops

Question

I have a recursive algorithm with two nested for loops. I'm trying to figure out what the Big-O time complexity will be.

public Set<Person> getDistinctCombinedPersons(Collection<Person> persons) {
  return permutatePersons(new ArrayList(persons), new HashSet<>(persons));
}

---

private Set<Person> permutatePersons(List<Person> personList, Set<Person> personSet) {
  if(personList.isEmpty() {
    return personSet;
  }

  Set<Person> deepCopyPersonSet = new HashSet<>(personSet);

  for(Person lPerson : personList) {
    for(Person sPerson : deepCopyPersonSet) {
      Person uniquePerson = CombinePeople.combine(lPerson, sPerson);
      personSet.add(uniquePerson);
    }
  }

  personList.remove(personList.size()-1);

  return permutatePersons(personList, personSet);
}

Answer 1

Assuming that you call permutatePersons with a list of length N the following recursion applies:

T(N) = T(N-1) + O(N^2)

That's because in every recursive step you call function with list of length N-1 (where N the current length) and also you do computations of total complexity O(N^2) (outer loop O(N) -just traversing list and inner loop traversing the hash map in O(N) -O(1) for each element and total N element, So the nested loops are overall O(N^2)).

You can easily see:

T(N) = T(N-1) + O(N^2) = T(N-2) + O(N^2) + O((N-1)^2) =...

= O(n(n+1)(2n+1)/6) = O(n^3)

Answer 2

Looks like it would be a big-O of n^2 for the nested loop:

  for(Person lPerson : personList) {
    for(Person sPerson : deepCopyPersonSet) {
      Person uniquePerson = CombinePeople.combine(lPerson, sPerson);
      personSet.add(uniquePerson);
    }
  }

You have to iterate over the each element for each element in the set.

And then the recursive call has a big O of n since it will call your method once for each element in the set.

Combining the two: n * n^2 will result in a big O of n^3

Answer 3

Because you have two nested loops you have the runtime complexity of O(m*n) . It's because for n - Person s in deepCopyPersonSet you iterate m times. n in this example is the quantity of Person s in personList .

Your code is basically:

for(int i = 0, i < m, i++)
  for(int j = 0, j < n, j++)
    //your code 

For every iteration of m, we have n iterations of code