# Check whether given a pair of arrays are permutation of each other

Does this implementation guarantee execution of $O(n)$ time? What is the additional space utilization for this implementation by excluding the original array size? Is it $O(1)$?

#include<iostream>
#include<set>
#include<algorithm>

bool ArraysPermute(int  array1[],int size1, int  array2[], int size2)
{
if( size1  != size2)
return false;
else
{
std::set<int> first_set(array1, array1+size1);
std::set<int> second_set(array2, array2+size2);

std::pair<std::set<int>::iterator,std::set<int>::iterator> myPair=std::mismatch(first_set.begin(),first_set.end(),second_set.begin());

if( *myPair.first != *myPair.second)

return false;
else
return true;
}
}

int main()
{
int array1[] ={1,2,3,5};
int array2[]={1,2,4,3};

if(ArraysPermute(array1,sizeof(array1)/sizeof(int), array2, sizeof(array2)/sizeof(int)))
std::cout<< " Arrays are permutation of each other\n";
else
std::cout<< " Arrays are not permutation of each other\n";

return 0;
}

• On a side note, there is always std::is_permutation in the standard library since C++11 :) – Morwenn Mar 24 '15 at 14:30
• @Morwenn: the cpp reference hints that std::is_permutation has a complexity worst case of O(n2). Unless the template is special cased for int, it could be much slower. – chqrlie Mar 24 '15 at 22:40

## Complexity

You suggest that the performance is $O(n)$, but it is not. It is $O(n \log{n})$ because the set constructor is a binary-tree building algorithm that has an additional log(n) operations for each value.

## Style

While your code is all neat, and well laid out, the variable names are good, etc., there are two issues I see with unnecessary else-condiditions on if-statements, and un-braced 1-liners. Additionally, white-space around punctuation and operators is useful for readability. This code:

  if( size1  != size2)
return false;
else
{
std::set<int> first_set(array1, array1+size1);
std::set<int> second_set(array2, array2+size2);

std::pair<std::set<int>::iterator,std::set<int>::iterator> myPair=std::mismatch(first_set.begin(),first_set.end(),second_set.begin());

if( *myPair.first != *myPair.second)

return false;
else
return true;
}


should be written:

if( size1 != size2)
{
return false;
}
std::set<int> first_set(array1, array1 + size1);
std::set<int> second_set(array2, array2 + size2);

std::pair<std::set<int>::iterator, std::set<int>::iterator> myPair
= std::mismatch(first_set.begin(), first_set.end(), second_set.begin());

return *myPair.first == *myPair.second;


## Bugs

The set construct keeps just a single entry for all instances of a given value. Thus, for example, given the input [1,2,3,4,4,4,4] it will have just 4 entries.

This leads to bugs in your code when there are duplicates. For example, your code will identify the following two input arrays as being permutations of each other:

[1,2,3,4,5,5,5]
[1,1,1,2,3,4,5]

• Thank you for the answer. To resolve the bug, I may need to use std::vector in place of std::set and sort it before checking the std::mismatch, I will correct it. – Steephen Mar 24 '15 at 13:00
• Most of the std::sort implementations are in n(logn), so again second version should ends up in O(nlogn) performance – Steephen Mar 24 '15 at 13:01
• Reference from the link you mentioned in your answer. " range constructor : Constructs a container with as many elements as the range [first,last), with each element emplace-constructed from its corresponding element in that range." It is obvious running an operation is O(nlogn) with std:;set, but I think construction is O(n) for a range based constructor of std::set – Steephen Mar 24 '15 at 20:53
• @Steephen - if you read the documentation again, you will see it says (emphasis mine): For all other cases, linear in the distance between the iterators if the elements are already sorted according to the same criterion. For unsorted sequences, linearithmic (NlogN) in that distance ...* – rolfl Mar 24 '15 at 22:55
• @Steephen: Sure if your range is already sorted. But I don't see that as a stipulation about your input arrays. Otherwise it is O(n.log(n)) – Martin York Mar 25 '15 at 0:18

This code runs in $O(n \log{n})$ time because a C++ set is a Red-Black tree, and insertions/deletions/finds run in $O(\log{n})$ time. In addition, since you are not using a multiset, your code is not actually correct for arrays with duplicate elements.

In order to resolve this, a simple solution is to use an std::unordered_map (which is a hash table with $O(1)$ operations) with the element as the key and the number of occurrences as the value. Construct a map for the 1st and 2nd arrays, and then iterate over the keys (or the array, complexity will be the same) and verify that the values are the same in both maps.

• construction of first_set itself will take O(n) complexity since it is using a range based constructor of std::set. And none of the listed operations of std::set , listed in your answer are not using also here. So how did you calculate its order is O(nlogn)? – Steephen Mar 24 '15 at 22:52
• Set construction takes O(nlogn) for unsorted arrays. See cplusplus.com/reference/set/set/set. The construction process could be thought of us a series of n insertions, even if it's not implemented that way. I mostly added the operation information for completeness's sake. – mleyfman Mar 24 '15 at 23:21

## Algorithm and Complexity

As reported in other answers, your program most likely executes in $O(n.log(n))$ time and uses $O(n)$ extra space, probably substantially more than the original space used by the arrays, and may report erroneous results because of a bad choice of class.

## Better algorithm

Forget using anything this sophisticated, this problem can be solved in $O(n)$ time with astute brute force, while still using $O(n)$ extra space:

• make temporary copies of array1 and array2
• sort both temporary arrays
• compare them -> if equal, result is true, otherwise false.
• free the temporary arrays.

You may think the complexity to be $O(n.log(n))$, and it may be so if you use the standard sort method, but keep in mind these arrays are a special case that can be sorted in $O(n)$ time using Radix sort

I am not familiar enough with C++ to tell if you will need to implement radix sort yourself or if the standard array sort will perform correctly.

## Style

• indentation is inconsistent;
• use white space more systematically and consistently;
• keep line length under control: a limit of 79 characters is advisable;
• the number of elements in the arrays should be computed in a safer way with sizeof(array1) / sizeof(array1[0]) instead of sizeof(array1)/sizeof(int)
• How do you propose to sort an array in O(n) time? – 200_success Mar 24 '15 at 22:25
• @200_success: HaHa! look up radix sort: en.wikipedia.org/wiki/Radix_sort – chqrlie Mar 24 '15 at 22:27
• That's an iffy claim, as the problem does not state a limit on the size of the elements. – 200_success Mar 24 '15 at 22:30
• The elements are int. Running sizeof(int) passes of distribution sort over 256 buckets will sort the array in O(n) time – chqrlie Mar 24 '15 at 22:32
• @Steephen: call this defensive programming. If you or the next programmer/maintainer of this piece of code later change the type of array1 and array2 to a type with a different size (short or long for example) or if you cut and paste the code to a different function where these types differ, forgetting to update the sizeof(int) will cause spurious bugs where using an expression independent of the type of array1 will not. – chqrlie Mar 25 '15 at 7:36