2178. Maximum Split of Positive Even Integers

You are given an integer finalSum. Split it into a sum of a maximum number of unique positive even integers.

• For example, given finalSum = 12, the following splits are valid (unique positive even integers summing up to finalSum): (12), (2 + 10), (2 + 4 + 6), and (4 + 8). Among them, (2 + 4 + 6) contains the maximum number of integers. Note that finalSum cannot be split into (2 + 2 + 4 + 4) as all the numbers should be unique.

Return a list of integers that represent a valid split containing a maximum number of integers. If no valid split exists for finalSum, return an empty list. You may return the integers in any order.

Example 1:

Input: finalSum = 12
Output: [2,4,6]
Explanation: The following are valid splits: (12), (2 + 10), (2 + 4 + 6), and (4 + 8).
(2 + 4 + 6) has the maximum number of integers, which is $3$. Thus, we return [2,4,6].
Note that [2,6,4], [6,2,4], etc. are also accepted.

Example 2:

Input: finalSum = 7
Output: []
Explanation: There are no valid splits for the given finalSum.
Thus, we return an empty array.

Example 3:

Input: finalSum = 28
Output: [6,8,2,12]
Explanation: The following are valid splits: (2 + 26), (6 + 8 + 2 + 12), and (4 + 24).
(6 + 8 + 2 + 12) has the maximum number of integers, which is $4$. Thus, we return [6,8,2,12].
Note that [10,2,4,12], [6,2,4,16], etc. are also accepted.

Constraints:

• $1 \leq$ finalSum $\leq 10^{10}$

Full Solution

First, let's think of how to determine if a sum $S$ can have a valid split that contains exactly $K$ integers.

The first case we should consider is whether or not a sum $S$ can have a valid split of any size. Since our split includes only even integers, $S$ will only have a split if it's even and we will return an empty list if $S$ is odd.

One observation we can make is that if some sum $S$ does have a valid split of $K$ integers, then a sum $T$ will also have a valid split of $K$ integers if $T$ is even and $T\geq S$. Why is this true? Let's denote $D$ as the difference between $T$ and $S$. From the split of $K$ integers from the sum $S$, incrementing the largest integer in the split by $D$ results in a valid split of $K$ integers with a total sum of $T$. It can be observed that increasing the greatest integer will always keep the entire list distinct.

Example

For this example, let's use the split $12 = 2 + 4 + 6$ with $S = 12$ and $K = 3$. How will we construct a split of size $K$ with sum $T = 16$?

First, we'll find the difference $D = T - S = 4$. Then, we'll add $D$ to the greatest integer in the split with sum $S$, which is $6$.

Thus, we obtain the split $16 = 2 + 4 + 10$ with sum $T$ and size $K$.

Back to the Original Problem

We are given $S$ and asked to find the maximum possible $K$ and construct a split of size $K$.

If we take the sum of the smallest $K$ positive even integers (2 + 4 + 6 + 8 + ... + 2*K), we'll obtain the least possible sum that has a split of size $K$. Let's denote this sum as low. A sum $S$ will have a valid split of size $K$ if $S \geq$ low.

To solve the problem, we'll first find the maximum $K$ where $S \geq$ low. Starting with the split that sums to low, we'll add $S -$ low to the largest integer to obtain our final split for $S$.

Simulation

Time Complexity

For a sum $S$ with a split of maximum size $K$, low = 2 + 4 + 6 + 8 + ... + 2*k. In the sum, there are $O(K)$ elements and the average element is $O(K)$, resulting in $S = O(K^2)$ and $K = O(\sqrt{S})$. Since our algorithm runs in $O(K)$, our final time complexity is $O(\sqrt{S})$.

Time Complexity: $O(\sqrt{S})$

Space Complexity

Since we construct a list of size $K$, our space complexity is also $O(\sqrt{S})$.

Space Complexity: $O(\sqrt{S})$

C++ Solution

1class Solution {
2   public:
3    vector<long long> maximumEvenSplit(long long finalSum) {
4        vector<long long> ans;    // integers in our split
5        if (finalSum % 2 == 1) {  // odd sum provides no solution
6            return ans;
7        }
8        long long currentSum = 0;  // keep track of the value of low
9        int i = 1;
10        while (currentSum + 2 * i <=
11               finalSum) {  // keep increasing size of split until maximum
12            currentSum += 2 * i;
13            ans.push_back(2 * i);
14            i++;
15        }
16        ans[ans.size() - 1] +=
17            finalSum - currentSum;  // add S - low to largest element
18        return ans;
19    }
20};

Java Solution

1class Solution {
2    public List<Long> maximumEvenSplit(long finalSum) {
3        List<Long> ans = new ArrayList<Long>(); // integers in our split
4        if (finalSum % 2 == 1) { // odd sum provides no solution
5            return ans;
6        }
7        long currentSum = 0; // keep track of the value of low
8        int i = 1;
9        while (currentSum + 2 * i
10            <= finalSum) { // keep increasing size of split until maximum
11            currentSum += 2 * i;
12            ans.add((long) 2 * i);
13            i++;
14        }
15        int idx = ans.size() - 1;
16        ans.set(idx,
17            ans.get(idx) + finalSum
18                - currentSum); // add S - low to largest element
19        return ans;
20    }
21}

Python Solution

1class Solution:
2    def maximumEvenSplit(self, finalSum: int) -> List[int]:
3        ans = []  # integers in our split
4        if finalSum % 2 == 1:  # odd sum provides no solution
5            return ans
6        currentSum = 0
7        i = 1
8        while (
9            currentSum + 2 * i <= finalSum
10        ):  # keep increasing size of split until maximum
11            currentSum += 2 * i
12            ans.append(2 * i)
13            i += 1
14        ans[len(ans) - 1] += finalSum - currentSum  # add S - low to largest element
15        return ans
16

Javascript Solution

1/**
2 * @param {number} finalSum
3 * @return {number[]}
4 */
5var maximumEvenSplit = function (finalSum) {
6  let ans = []; // integers in our split
7  if (finalSum % 2 === 1) {
8    // odd sum provides no solution
9    return ans;
10  }
11  let currentSum = 0; // keep track of the value of low
12  let i = 1;
13  while (currentSum + 2 * i <= finalSum) {
14    // keep increasing size of split until maximum
15    currentSum += 2 * i;
16    ans.push(2 * i);
17    i++;
18  }
19  ans[ans.length - 1] += finalSum - currentSum; // add S - low to largest element
20  return ans;
21};

Solution Implementation