feat: add solutions to lc problem: No.2607

No.2607.Make K-Subarray Sums Equal

Author: yanglbme

solution/1000-1099/1039.Minimum Score Triangulation of Polygon/Solution.py

@@ -4,6 +4,9 @@ def minScoreTriangulation(self, values: List[int]) -> int:
         def dfs(i: int, j: int) -> int:
             if i + 1 == j:
                 return 0
-            return min(dfs(i, k) + dfs(k, j) + values[i] * values[k] * values[j] for k in range(i + 1, j))
+            return min(
+                dfs(i, k) + dfs(k, j) + values[i] * values[k] * values[j]
+                for k in range(i + 1, j)
+            )
 
         return dfs(0, len(values) - 1)

solution/1600-1699/1637.Widest Vertical Area Between Two Points Containing No Points/Solution.py

@@ -17,4 +17,4 @@ def maxWidthOfVerticalArea(self, points: List[List[int]]) -> int:
                 continue
             ans = max(ans, curmin - prev)
             prev = curmax
-        return ans
+        return ans

solution/1600-1699/1644.Lowest Common Ancestor of a Binary Tree II/Solution.py

@@ -5,8 +5,11 @@
 #         self.left = None
 #         self.right = None
 
+
 class Solution:
-    def lowestCommonAncestor(self, root: 'TreeNode', p: 'TreeNode', q: 'TreeNode') -> 'TreeNode':
+    def lowestCommonAncestor(
+        self, root: 'TreeNode', p: 'TreeNode', q: 'TreeNode'
+    ) -> 'TreeNode':
         def dfs(root, p, q):
             if root is None:
                 return False

solution/1800-1899/1868.Product of Two Run-Length Encoded Arrays/Solution.py

@@ -1,5 +1,7 @@
 class Solution:
-    def findRLEArray(self, encoded1: List[List[int]], encoded2: List[List[int]]) -> List[List[int]]:
+    def findRLEArray(
+        self, encoded1: List[List[int]], encoded2: List[List[int]]
+    ) -> List[List[int]]:
         ans = []
         j = 0
         for vi, fi in encoded1:

solution/2400-2499/2428.Maximum Sum of an Hourglass/Solution.py

@@ -5,7 +5,8 @@ def maxSum(self, grid: List[List[int]]) -> int:
         for i in range(1, m - 1):
             for j in range(1, n - 1):
                 s = -grid[i][j - 1] - grid[i][j + 1]
-                s += sum(grid[x][y] for x in range(i - 1, i + 2)
-                         for y in range(j - 1, j + 2))
+                s += sum(
+                    grid[x][y] for x in range(i - 1, i + 2) for y in range(j - 1, j + 2)
+                )
                 ans = max(ans, s)
         return ans

solution/2600-2699/2600.K Items With the Maximum Sum/Solution.py

@@ -1,5 +1,7 @@
 class Solution:
-    def kItemsWithMaximumSum(self, numOnes: int, numZeros: int, numNegOnes: int, k: int) -> int:
+    def kItemsWithMaximumSum(
+        self, numOnes: int, numZeros: int, numNegOnes: int, k: int
+    ) -> int:
         if numOnes >= k:
             return k
         k -= numOnes

solution/2600-2699/2607.Make K-Subarray Sums Equal/README.md

@@ -57,6 +57,36 @@
 
 <!-- 这里可写通用的实现逻辑 -->
 
+**方法一：数学（裴蜀定理） + 中位数贪心**
+
+题目要求每个长度为 $k$ 的子数组的元素总和相等，那么有以下等式成立：
+
+$$
+arr_i + arr_{i + 1} + \cdots + arr_{i + k - 1} = arr_{i + 1} + arr_{i + 2} + \cdots + arr_{i + k}
+$$
+
+化简得：
+
+$$
+arr_i = arr_{i + k}
+$$
+
+也即是说，数组 $arr$ 有一个大小为 $k$ 的循环节，而由于数组 $arr$ 是一个循环数组，那么数组 $arr$ 也有一个长度为 $n$ 的循环节。换句话说，数组 $arr$ 上间隔为 $k$，以及间隔为 $n$ 的元素均相等。即有：
+
+$$
+arr_i = arr_{i + k \times x + n \times y}
+$$
+
+根据裴蜀定理，有 $a \times x + b \times y = gcd(a, b)$，因此，有：
+
+$$
+arr_i = arr_{i + k \times x + n \times y} = arr_{i + gcd(k, n)}
+$$
+
+因此，数组 $arr$ 上的元素可以分为 $gcd(k, n)$ 组，每组的元素间隔为 $gcd(k, n)$，且每一组中的所有元素均相等。对于每一组，我们可以将其元素按照大小排序，然后取中位数，即可将该组中的所有元素变为中位数。对于所有组，我们将其中位数之差的绝对值求和，即为答案。
+
+时间复杂度 $O(n \times \log n)$，空间复杂度 $O(n)$。其中 $n$ 为数组 $arr$ 的长度。
+
 <!-- tabs:start -->
 
 ### **Python3**
@@ -67,15 +97,12 @@
 class Solution:
     def makeSubKSumEqual(self, arr: List[int], k: int) -> int:
         n = len(arr)
-        if n == k:
-            return 0
         g = gcd(n, k)
         ans = 0
         for i in range(g):
-            t = sorted(arr[j] for j in range(i, n, g))
+            t = sorted(arr[i:n:g])
             mid = t[len(t) >> 1]
-            s = sum(abs(x - mid) for x in t)
-            ans += s
+            ans += sum(abs(x - mid) for x in t)
         return ans
 ```
 
@@ -87,9 +114,6 @@ class Solution:
 class Solution {
     public long makeSubKSumEqual(int[] arr, int k) {
         int n = arr.length;
-        if (n == k) {
-            return 0;
-        }
         int g = gcd(n, k);
         long ans = 0;
         for (int i = 0; i < g; ++i) {
@@ -99,11 +123,9 @@ class Solution {
             }
             t.sort((a, b) -> a - b);
             int mid = t.get(t.size() >> 1);
-            long s = 0;
             for (int x : t) {
-                s += Math.abs(x - mid);
+                ans += Math.abs(x - mid);
             }
-            ans += s;
         }
         return ans;
     }
@@ -121,9 +143,6 @@ class Solution {
 public:
     long long makeSubKSumEqual(vector<int>& arr, int k) {
         int n = arr.size();
-        if (n == k) {
-            return 0;
-        }
         int g = gcd(n, k);
         long long ans = 0;
         for (int i = 0; i < g; ++i) {
@@ -133,11 +152,9 @@ public:
             }
             sort(t.begin(), t.end());
             int mid = t[t.size() / 2];
-            long long s = 0;
             for (int x : t) {
-                s += abs(x - mid);
+                ans += abs(x - mid);
             }
-            ans += s;
         }
         return ans;
     }
@@ -149,9 +166,6 @@ public:
 ```go
 func makeSubKSumEqual(arr []int, k int) (ans int64) {
 	n := len(arr)
-	if n == k {
-		return 0
-	}
 	g := gcd(n, k)
 	for i := 0; i < g; i++ {
 		t := []int{}

solution/2600-2699/2607.Make K-Subarray Sums Equal/README_EN.md

@@ -61,15 +61,12 @@ The array after the operations is [5,5,5,5]
 class Solution:
     def makeSubKSumEqual(self, arr: List[int], k: int) -> int:
         n = len(arr)
-        if n == k:
-            return 0
         g = gcd(n, k)
         ans = 0
         for i in range(g):
-            t = sorted(arr[j] for j in range(i, n, g))
+            t = sorted(arr[i:n:g])
             mid = t[len(t) >> 1]
-            s = sum(abs(x - mid) for x in t)
-            ans += s
+            ans += sum(abs(x - mid) for x in t)
         return ans
 ```
 
@@ -79,9 +76,6 @@ class Solution:
 class Solution {
     public long makeSubKSumEqual(int[] arr, int k) {
         int n = arr.length;
-        if (n == k) {
-            return 0;
-        }
         int g = gcd(n, k);
         long ans = 0;
         for (int i = 0; i < g; ++i) {
@@ -91,11 +85,9 @@ class Solution {
             }
             t.sort((a, b) -> a - b);
             int mid = t.get(t.size() >> 1);
-            long s = 0;
             for (int x : t) {
-                s += Math.abs(x - mid);
+                ans += Math.abs(x - mid);
             }
-            ans += s;
         }
         return ans;
     }
@@ -113,9 +105,6 @@ class Solution {
 public:
     long long makeSubKSumEqual(vector<int>& arr, int k) {
         int n = arr.size();
-        if (n == k) {
-            return 0;
-        }
         int g = gcd(n, k);
         long long ans = 0;
         for (int i = 0; i < g; ++i) {
@@ -125,11 +114,9 @@ public:
             }
             sort(t.begin(), t.end());
             int mid = t[t.size() / 2];
-            long long s = 0;
             for (int x : t) {
-                s += abs(x - mid);
+                ans += abs(x - mid);
             }
-            ans += s;
         }
         return ans;
     }
@@ -141,9 +128,6 @@ public:
 ```go
 func makeSubKSumEqual(arr []int, k int) (ans int64) {
 	n := len(arr)
-	if n == k {
-		return 0
-	}
 	g := gcd(n, k)
 	for i := 0; i < g; i++ {
 		t := []int{}

solution/2600-2699/2607.Make K-Subarray Sums Equal/Solution.cpp

@@ -2,9 +2,6 @@ class Solution {
 public:
     long long makeSubKSumEqual(vector<int>& arr, int k) {
         int n = arr.size();
-        if (n == k) {
-            return 0;
-        }
         int g = gcd(n, k);
         long long ans = 0;
         for (int i = 0; i < g; ++i) {
@@ -14,11 +11,9 @@ class Solution {
             }
             sort(t.begin(), t.end());
             int mid = t[t.size() / 2];
-            long long s = 0;
             for (int x : t) {
-                s += abs(x - mid);
+                ans += abs(x - mid);
             }
-            ans += s;
         }
         return ans;
     }

solution/2600-2699/2607.Make K-Subarray Sums Equal/Solution.go

@@ -1,8 +1,5 @@
 func makeSubKSumEqual(arr []int, k int) (ans int64) {
 	n := len(arr)
-	if n == k {
-		return 0
-	}
 	g := gcd(n, k)
 	for i := 0; i < g; i++ {
 		t := []int{}