Problem Description

The problem presents an array of integers arr with an even number of elements n and an integer k. The task is to determine whether arr can be partitioned into n/2 pairs, where the sum of the integers in each pair is divisible by k. If such a partitioning is possible, the function should return true; otherwise, it should return false.

Intuition

The solution is built on the property of divisibility by an integer k. If two numbers a and b have a sum that is divisible by k, it means that (a + b) % k == 0. In other words, the remainder when a + b is divided by k is 0.

For any integer x, x % k will give a remainder in the range [0, k-1]. If x % k is 0, then x is divisible by k. To make a pair with a sum divisible by k, each number x in the array that has a non-zero remainder x % k requires a corresponding number y such that (x + y) % k == 0.

We can think of the remainders as forming pairs themselves. For a non-zero remainder i, there must be an equal number of elements with remainder k - i.

To arrive at the solution, we use a counter to track the frequency of each remainder in the array. We then check two conditions:

1. The count of elements with a remainder of 0 must be even because they can only be paired with other elements with a remainder of 0.
2. For each non-zero remainder i, there should be an equal count of elements with remainder i and k - i.

If both these conditions are met, the array can be rearranged into n/2 pairs, where the sum of each pair is divisible by k.

Solution Approach

The implementation of the solution involves using the Counter class from Python's collections module to efficiently count the occurrences of each remainder when elements of the array are divided by k. The Counter object will map each unique remainder to the number of times it appears in the array.

Here's a step-by-step walkthrough of the solution:

1. Calculate the remainder of each element in the array when divided by k. This is done using a list comprehension: [x % k for x in arr].
2. Feed this list of remainders into Counter to get the frequency count of each remainder: Counter(x % k for x in arr).
3. Verify the first condition for elements with zero remainders:
   • cnt[0] % 2 == 0. This checks if the count of elements that are exactly divisible by k is even. These elements can only be paired with themselves, so an odd count would make it impossible to form pairs where the sum is divisible by k.
4. Verify the second condition for elements with non-zero remainders:
   • Loop through the range from 1 to k - 1 and for each i, check if the count of elements with a remainder of i is equal to the count of elements with a remainder of k - i.
   • This check is performed by the list comprehension: all(cnt[i] == cnt[k - i] for i in range(1, k)).
   • The all function ensures that this condition must hold true for all remainders in the specified range for the function to return true.

The use of the Counter and the all function makes the algorithm efficient and concise. The time complexity of this algorithm is primarily dependent on the time it takes to traverse the array and calculate the remainders, which is ( O(n) ), and the time to check the pair frequencies, which is ( O(k) ). This makes the overall time complexity ( O(n + k) ).

Here's the relevant snippet from the Python solution for reference:

class Solution:
    def canArrange(self, arr: List[int], k: int) -> bool:
        cnt = Counter(x % k for x in arr)
        return cnt[0] % 2 == 0 and all(cnt[i] == cnt[k - i] for i in range(1, k))

By adhering to these steps, the algorithm efficiently solves the problem while respecting the constraints of even length arrays and the divisibility requirement.

Example Walkthrough

Let's illustrate the solution approach with a small example:

Suppose we have an array arr = [2, 4, 1, 5] and an integer k = 3. We need to check if we can partition this array into pairs such that the sum of each pair is divisible by k. Since there are 4 elements in arr, we are looking for n/2 = 2 pairs.

Following the solution steps:

1. Calculate the remainders of each element when divided by k = 3:

   • 2 % 3 = 2
   • 4 % 3 = 1
   • 1 % 3 = 1
   • 5 % 3 = 2 The remainders are [2, 1, 1, 2].

2. Count the frequency of each remainder using the Counter:

   • Remainder 1 appears 2 times.
   • Remainder 2 appears 2 times.

3. Verify the first condition for elements with zero remainders:

   • Our example does not have any element with a remainder of 0, so we can skip this.

4. Verify the second condition for elements with non-zero remainders:

   • Check each i from 1 to k - 1, which in this case is just 1 and 2 since k = 3.
   • For i = 1: Count is 2.
   • For k - i = 3 - 1 = 2: Count is 2.
   • Since the counts are equal for i and k - i, the condition is met.

Since the array only has non-zero remainders and each remainder i has an equal count to k - i, we have successfully verified that it is possible to partition the array into n/2 pairs where the sum of each pair is divisible by k. Thus, the function would return true.

The corresponding Python function would process the information as follows:

from collections import Counter

def canArrange(arr: List[int], k: int) -> bool:
    cnt = Counter(x % k for x in arr)
    # Since there is no remainder of 0, we skip checking cnt[0] % 2
    # Validate the second condition using a list comprehension and all()
    return all(cnt[i] == cnt[k - i] for i in range(1, k))

# Example array and k
arr = [2, 4, 1, 5]
k = 3

# Call the function with the example inputs
print(canArrange(arr, k))  # Output: True

By adhering to the solution steps, we are given a working example of the algorithm applied to this simple array, confirming the algorithm's correctness in this scenario.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code can be analyzed by looking at the operations performed:

• The list comprehension x % k for x in arr iterates over each element in arr, resulting in O(n) time complexity, where n is the length of arr.
• The Counter from the standard collections module constructs the frequency dictionary in once again O(n) time complexity.
• The return statement consists of two parts:
  • Checking if cnt[0] % 2 == 0 is a constant time O(1) operation because we access a dictionary key and then check for evenness.
  • The list comprehension all(cnt[i] == cnt[k - i] for i in range(1, k)) could iterate up to k times. In the worst case, this is O(k). However, since there are only 'n' unique modulo results possible (all elements x in arr are used to compute x % k), we are effectively iterating up to min(n, k), hence O(min(n, k)).

Based on these observations, the total time complexity is O(n) + O(1) + O(min(n, k)) which simplifies to O(n + min(n, k)). Since n and k are independent variables, we cannot simplify this further without knowing the relationship between n and k.

Space Complexity

The space complexity is determined by the auxiliary space used by the program. In this case:

• The frequency dictionary cnt may contain up to min(n, k) different keys (since these are the unique modulo results) and therefore has a space complexity of O(min(n, k)).
• The space complexity for the list comprehension is O(1) since it's not stored but used directly in the all function.

Consequently, the space complexity of the whole code is dominated by the space requirement of the frequency dictionary, which is O(min(n, k)).

Learn more about how to find time and space complexity quickly using problem constraints.