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leetcode/2178. Maximum Split of Positive Even Integers/README.md

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+# [2178. Maximum Split of Positive Even Integers (Medium)](https://leetcode.com/problems/maximum-split-of-positive-even-integers/)
+
+<p>You are given an integer <code>finalSum</code>. Split it into a sum of a <strong>maximum</strong> number of <strong>unique</strong> positive even integers.</p>
+
+<ul>
+	<li>For example, given <code>finalSum = 12</code>, the following splits are <strong>valid</strong> (unique positive even integers summing up to <code>finalSum</code>): <code>(2 + 10)</code>, <code>(2 + 4 + 6)</code>, and <code>(4 + 8)</code>. Among them, <code>(2 + 4 + 6)</code> contains the maximum number of integers. Note that <code>finalSum</code> cannot be split into <code>(2 + 2 + 4 + 4)</code> as all the numbers should be unique.</li>
+</ul>
+
+<p>Return <em>a list of integers that represent a valid split containing a <strong>maximum</strong> number of integers</em>. If no valid split exists for <code>finalSum</code>, return <em>an <strong>empty</strong> list</em>. You may return the integers in <strong>any</strong> order.</p>
+
+<p>&nbsp;</p>
+<p><strong>Example 1:</strong></p>
+
+<pre><strong>Input:</strong> finalSum = 12
+<strong>Output:</strong> [2,4,6]
+<strong>Explanation:</strong> The following are some valid splits: <code>(2 + 10)</code>, <code>(2 + 4 + 6)</code>, and <code>(4 + 8)</code>.
+(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.
+</pre>
+
+<p><strong>Example 2:</strong></p>
+
+<pre><strong>Input:</strong> finalSum = 7
+<strong>Output:</strong> []
+<strong>Explanation:</strong> There are no valid splits for the given finalSum.
+Thus, we return an empty array.
+</pre>
+
+<p><strong>Example 3:</strong></p>
+
+<pre><strong>Input:</strong> finalSum = 28
+<strong>Output:</strong> [6,8,2,12]
+<strong>Explanation:</strong> The following are some valid splits: <code>(2 + 26)</code>, <code>(6 + 8 + 2 + 12)</code>, and <code>(4 + 24)</code>. 
+<code>(6 + 8 + 2 + 12)</code> 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.
+</pre>
+
+<p>&nbsp;</p>
+<p><strong>Constraints:</strong></p>
+
+<ul>
+	<li><code>1 &lt;= finalSum &lt;= 10<sup>10</sup></code></li>
+</ul>
+
+## Solution 1. Backtracking
+
+If `finalSum` is odd, return empty array.
+
+Otherwise, we keep trying subtracting `2, 4, 6, 8, ...` from `finalSum` via backtracking.
+
+```cpp
+// OJ: https://leetcode.com/problems/maximum-split-of-positive-even-integers/
+// Author: github.com/lzl124631x
+// Time: O(sqrt(N))
+// Space: O(sqrt(N))
+class Solution {
+public:
+    vector<long long> maximumEvenSplit(long long s) {
+        if (s % 2) return {};
+        vector<long long> ans;
+        function<bool(long, long)>dfs = [&](long i, long target) {
+            if (target == 0) return true;
+            if (target < i) return false;
+            ans.push_back(i);
+            if (dfs(i + 2, target - i)) return true; // use `i`
+            ans.pop_back();
+            return dfs(i + 2, target); // skip `i`
+        };
+        dfs(2, s);
+        return ans;
+    }
+};
+```
+
+
+## Solution 2. Greedy
+
+In solution 1, in fact we only backtrack at most once.
+
+We can keep trying subtracting `2, 4, 6, 8, ...` from `finalSum`. We stop the loop when subtracting the current number `i` is invalid -- `s - i < i + 2` (the reminder after the subtraction is less than the next even number). And we push the reminder into the answer.
+
+```cpp
+// OJ: https://leetcode.com/problems/maximum-split-of-positive-even-integers/
+// Author: github.com/lzl124631x
+// Time: O(sqrt(N))
+// Space: O(1)
+class Solution {
+public:
+    vector<long long> maximumEvenSplit(long long s) {
+        if (s % 2) return {};
+        vector<long long> ans;
+        for (int i = 2; s - i >= i + 2; i += 2) {
+            ans.push_back(i);
+            s -= i;
+        }
+        ans.push_back(s);
+        return ans;
+    }
+};
+```
+
+## Discuss
+
+https://leetcode.com/problems/maximum-split-of-positive-even-integers/discuss/1783586