feat: add solutions to lc problem: No.2926 (doocs#1942)

No.2926.Maximum Balanced Subsequence Sum

Author: yanglbme

solution/2600-2699/2609.Find the Longest Balanced Substring of a Binary String/README.md

@@ -66,7 +66,7 @@
 遍历字符串 $s$，对于当前字符 $c$：
 
 -   如果当前字符为 `'0'`，我们判断此时 $one$ 是否大于 $0$，是则将 $zero$ 和 $one$ 重置为 $0$，接下来将 $zero$ 加 $1$。
--   如果当前字符为 `'1'`，则将 $one$ 加 $1$，并更新答案为 $ans = max(ans, 2 \times min(one, zero))$。
+-   如果当前字符为 `'1'`，则将 $one$ 加 $1$，并更新答案为 $ans = \max(ans, 2 \times \min(one, zero))$。
 
 遍历结束后，即可得到最长的平衡子串的长度。
 
@@ -326,7 +326,7 @@ impl Solution {
                 if s.as_bytes()[k] == b'1' {
                     cnt += 1;
                 } else if cnt > 0 {
-                    return false
+                    return false;
                 }
             }
 
@@ -369,7 +369,7 @@ impl Solution {
                 zero += 1;
             } else {
                 one += 1;
-                ans = std::cmp::max(ans, std::cmp::min(zero, one) * 2)
+                ans = std::cmp::max(ans, std::cmp::min(zero, one) * 2);
             }
         }
 

solution/2900-2999/2926.Maximum Balanced Subsequence Sum/README.md

@@ -67,34 +67,310 @@ nums[3] - nums[0] >= 3 - 0 。
 
 <!-- 这里可写通用的实现逻辑 -->
 
+**方法一：动态规划 + 树状数组**
+
+根据题目描述，我们可以将不等式 $nums[i] - nums[j] \ge i - j$ 转化为 $nums[i] - i \ge nums[j] - j$，因此，我们考虑定义一个新数组 $arr$，其中 $arr[i] = nums[i] - i$，那么平衡子序列满足对于任意 $j \lt i$，都有 $arr[j] \le arr[i]$。即题目转换为求在 $arr$ 中选出一个递增子序列，使得对应的 $nums$ 的和最大。
+
+假设 $i$ 是子序列中最后一个元素的下标，那么我们考虑子序列倒数第二个元素的下标 $j$，如果 $arr[j] \le arr[i]$，我们可以考虑是否要将 $j$ 加入到子序列中。
+
+因此，我们定义 $f[i]$ 表示子序列最后一个元素的下标为 $i$ 时，对应的 $nums$ 的最大和，那么答案为 $\max_{i=0}^{n-1} f[i]$。
+
+状态转移方程为：
+
+$$
+f[i] = \max(\max_{j=0}^{i-1} f[j], 0) + nums[i]
+$$
+
+其中 $j$ 满足 $arr[j] \le arr[i]$。
+
+我们可以使用树状数组来维护前缀的最大值，即对于每个 $arr[i]$，我们维护前缀 $arr[0..i]$ 中 $f[i]$ 的最大值。
+
+时间复杂度 $O(n \times \log n)$，空间复杂度 $O(n)$。其中 $n$ 为数组 $nums$ 的长度。
+
 <!-- tabs:start -->
 
 ### **Python3**
 
 <!-- 这里可写当前语言的特殊实现逻辑 -->
 
 ```python
-
+class BinaryIndexedTree:
+    def __init__(self, n: int):
+        self.n = n
+        self.c = [-inf] * (n + 1)
+
+    def update(self, x: int, v: int):
+        while x <= self.n:
+            self.c[x] = max(self.c[x], v)
+            x += x & -x
+
+    def query(self, x: int) -> int:
+        mx = -inf
+        while x:
+            mx = max(mx, self.c[x])
+            x -= x & -x
+        return mx
+
+
+class Solution:
+    def maxBalancedSubsequenceSum(self, nums: List[int]) -> int:
+        arr = [x - i for i, x in enumerate(nums)]
+        s = sorted(set(arr))
+        tree = BinaryIndexedTree(len(s))
+        for i, x in enumerate(nums):
+            j = bisect_left(s, x - i) + 1
+            v = max(tree.query(j), 0) + x
+            tree.update(j, v)
+        return tree.query(len(s))
 ```
 
 ### **Java**
 
 <!-- 这里可写当前语言的特殊实现逻辑 -->
 
 ```java
-
+class BinaryIndexedTree {
+    private int n;
+    private long[] c;
+    private final long inf = 1L << 60;
+
+    public BinaryIndexedTree(int n) {
+        this.n = n;
+        c = new long[n + 1];
+        Arrays.fill(c, -inf);
+    }
+
+    public void update(int x, long v) {
+        while (x <= n) {
+            c[x] = Math.max(c[x], v);
+            x += x & -x;
+        }
+    }
+
+    public long query(int x) {
+        long mx = -inf;
+        while (x > 0) {
+            mx = Math.max(mx, c[x]);
+            x -= x & -x;
+        }
+        return mx;
+    }
+}
+
+class Solution {
+    public long maxBalancedSubsequenceSum(int[] nums) {
+        int n = nums.length;
+        int[] arr = new int[n];
+        for (int i = 0; i < n; ++i) {
+            arr[i] = nums[i] - i;
+        }
+        Arrays.sort(arr);
+        int m = 0;
+        for (int i = 0; i < n; ++i) {
+            if (i == 0 || arr[i] != arr[i - 1]) {
+                arr[m++] = arr[i];
+            }
+        }
+        BinaryIndexedTree tree = new BinaryIndexedTree(m);
+        for (int i = 0; i < n; ++i) {
+            int j = search(arr, nums[i] - i, m) + 1;
+            long v = Math.max(tree.query(j), 0) + nums[i];
+            tree.update(j, v);
+        }
+        return tree.query(m);
+    }
+
+    private int search(int[] nums, int x, int r) {
+        int l = 0;
+        while (l < r) {
+            int mid = (l + r) >> 1;
+            if (nums[mid] >= x) {
+                r = mid;
+            } else {
+                l = mid + 1;
+            }
+        }
+        return l;
+    }
+}
 ```
 
 ### **C++**
 
 ```cpp
-
+class BinaryIndexedTree {
+private:
+    int n;
+    vector<long long> c;
+    const long long inf = 1e18;
+
+public:
+    BinaryIndexedTree(int n) {
+        this->n = n;
+        c.resize(n + 1, -inf);
+    }
+
+    void update(int x, long long v) {
+        while (x <= n) {
+            c[x] = max(c[x], v);
+            x += x & -x;
+        }
+    }
+
+    long long query(int x) {
+        long long mx = -inf;
+        while (x > 0) {
+            mx = max(mx, c[x]);
+            x -= x & -x;
+        }
+        return mx;
+    }
+};
+
+class Solution {
+public:
+    long long maxBalancedSubsequenceSum(vector<int>& nums) {
+        int n = nums.size();
+        vector<int> arr(n);
+        for (int i = 0; i < n; ++i) {
+            arr[i] = nums[i] - i;
+        }
+        sort(arr.begin(), arr.end());
+        arr.erase(unique(arr.begin(), arr.end()), arr.end());
+        int m = arr.size();
+        BinaryIndexedTree tree(m);
+        for (int i = 0; i < n; ++i) {
+            int j = lower_bound(arr.begin(), arr.end(), nums[i] - i) - arr.begin() + 1;
+            long long v = max(tree.query(j), 0LL) + nums[i];
+            tree.update(j, v);
+        }
+        return tree.query(m);
+    }
+};
 ```
 
 ### **Go**
 
 ```go
+const inf int = 1e18
+
+type BinaryIndexedTree struct {
+	n int
+	c []int
+}
+
+func NewBinaryIndexedTree(n int) BinaryIndexedTree {
+	c := make([]int, n+1)
+	for i := range c {
+		c[i] = -inf
+	}
+	return BinaryIndexedTree{n: n, c: c}
+}
+
+func (bit *BinaryIndexedTree) update(x, v int) {
+	for x <= bit.n {
+		bit.c[x] = max(bit.c[x], v)
+		x += x & -x
+	}
+}
+
+func (bit *BinaryIndexedTree) query(x int) int {
+	mx := -inf
+	for x > 0 {
+		mx = max(mx, bit.c[x])
+		x -= x & -x
+	}
+	return mx
+}
+
+func maxBalancedSubsequenceSum(nums []int) int64 {
+	n := len(nums)
+	arr := make([]int, n)
+	for i, x := range nums {
+		arr[i] = x - i
+	}
+	sort.Ints(arr)
+	m := 0
+	for i, x := range arr {
+		if i == 0 || x != arr[i-1] {
+			arr[m] = x
+			m++
+		}
+	}
+	arr = arr[:m]
+	tree := NewBinaryIndexedTree(m)
+	for i, x := range nums {
+		j := sort.SearchInts(arr, x-i) + 1
+		v := max(tree.query(j), 0) + x
+		tree.update(j, v)
+	}
+	return int64(tree.query(m))
+}
+```
 
+### **TypeScript**
+
+```ts
+class BinaryIndexedTree {
+    private n: number;
+    private c: number[];
+
+    constructor(n: number) {
+        this.n = n;
+        this.c = Array(n + 1).fill(-Infinity);
+    }
+
+    update(x: number, v: number): void {
+        while (x <= this.n) {
+            this.c[x] = Math.max(this.c[x], v);
+            x += x & -x;
+        }
+    }
+
+    query(x: number): number {
+        let mx = -Infinity;
+        while (x > 0) {
+            mx = Math.max(mx, this.c[x]);
+            x -= x & -x;
+        }
+        return mx;
+    }
+}
+
+function maxBalancedSubsequenceSum(nums: number[]): number {
+    const n = nums.length;
+    const arr = Array(n).fill(0);
+    for (let i = 0; i < n; ++i) {
+        arr[i] = nums[i] - i;
+    }
+    arr.sort((a, b) => a - b);
+    let m = 0;
+    for (let i = 0; i < n; ++i) {
+        if (i === 0 || arr[i] !== arr[i - 1]) {
+            arr[m++] = arr[i];
+        }
+    }
+    arr.length = m;
+    const tree = new BinaryIndexedTree(m);
+    const search = (nums: number[], x: number): number => {
+        let [l, r] = [0, nums.length];
+        while (l < r) {
+            const mid = (l + r) >> 1;
+            if (nums[mid] >= x) {
+                r = mid;
+            } else {
+                l = mid + 1;
+            }
+        }
+        return l;
+    };
+    for (let i = 0; i < n; ++i) {
+        const j = search(arr, nums[i] - i) + 1;
+        const v = Math.max(tree.query(j), 0) + nums[i];
+        tree.update(j, v);
+    }
+    return tree.query(m);
+}
 ```
 
 ### **...**