feat: add solutions to lc problem: No.0456

No.0456.132 Pattern

Author: yanglbme

solution/0400-0499/0456.132 Pattern/README.md

@@ -50,7 +50,22 @@
 
 <!-- 这里可写通用的实现逻辑 -->
 
-单调栈实现。
+**方法一：单调栈**
+
+**方法二：树状数组**
+
+树状数组，也称作“二叉索引树”（Binary Indexed Tree）或 Fenwick 树。 它可以高效地实现如下两个操作：
+
+1. **单点更新** `update(x, delta)`： 把序列 x 位置的数加上一个值 delta；
+1. **前缀和查询** `query(x)`：查询序列 `[1,...x]` 区间的区间和，即位置 x 的前缀和。
+
+这两个操作的时间复杂度均为 `O(log n)`。
+
+树状数组最基本的功能就是求比某点 x 小的点的个数（这里的比较是抽象的概念，可以是数的大小、坐标的大小、质量的大小等等）。
+
+比如给定数组 `a[5] = {2, 5, 3, 4, 1}`，求 `b[i] = 位置 i 左边小于等于 a[i] 的数的个数`。对于此例，`b[5] = {0, 1, 1, 2, 0}`。
+
+解决方案是直接遍历数组，每个位置先求出 `query(a[i])`，然后再修改树状数组 `update(a[i], 1)` 即可。当数的范围比较大时，需要进行离散化，即先进行去重并排序，然后对每个数字进行编号。
 
 <!-- tabs:start -->
 
@@ -72,6 +87,47 @@ class Solution:
         return False
 ```
 
+```python
+class BinaryIndexedTree:
+    def __init__(self, n):
+        self.n = n
+        self.c = [0] * (n + 1)
+    
+    @staticmethod
+    def lowbit(x):
+        return x & -x
+
+    def update(self, x, delta):
+        while x <= self.n:
+            self.c[x] += delta
+            x += BinaryIndexedTree.lowbit(x)
+    
+    def query(self, x):
+        s = 0
+        while x:
+            s += self.c[x]
+            x -= BinaryIndexedTree.lowbit(x)
+        return s
+
+
+class Solution:
+    def find132pattern(self, nums: List[int]) -> bool:
+        s = sorted(set(nums))
+        m = {v: i for i, v in enumerate(s, 1)}
+        n = len(m)
+        tree = BinaryIndexedTree(n)
+        for v in nums:
+            tree.update(m[v], 1)
+        mi = nums[0]
+        for v in nums:
+            tree.update(m[v], -1)
+            # v 右侧存在 (mi, v - 1] 范围内的数字，说明符合 132
+            if tree.query(m[v] - 1) - tree.query(m[mi]) > 0:
+                return True
+            mi = min(mi, v)
+        return False
+```
+
 ### **Java**
 
 <!-- 这里可写当前语言的特殊实现逻辑 -->
@@ -95,6 +151,67 @@ class Solution {
 }
 ```
 
+```java
+class BinaryIndexedTree {
+    private int n;
+    private int[] c;
+
+    public BinaryIndexedTree(int n) {
+        this.n = n;
+        c = new int[n + 1];
+    }
+
+    public void update(int x, int delta) {
+        while (x <= n) {
+            c[x] += delta;
+            x += lowbit(x);
+        }
+    }
+
+    public int query(int x) {
+        int s = 0;
+        while (x > 0) {
+            s += c[x];
+            x -= lowbit(x);
+        }
+        return s;
+    }
+
+    public static int lowbit(int x) {
+        return x & -x;
+    }
+}
+
+class Solution {
+    public boolean find132pattern(int[] nums) {
+        TreeSet<Integer> ts = new TreeSet();
+        for (int v : nums) {
+            ts.add(v);
+        }
+        int idx = 1;
+        Map<Integer, Integer> m = new HashMap<>();
+        for (int v : ts) {
+            m.put(v, idx++);
+        }
+        int n = m.size();
+        BinaryIndexedTree tree = new BinaryIndexedTree(n);
+        for (int v : nums) {
+            tree.update(m.get(v), 1);
+        }
+        int mi = nums[0];
+        for (int v : nums) {
+            int x = m.get(v);
+            tree.update(x, -1);
+            if (tree.query(x - 1) - tree.query(m.get(mi)) > 0) {
+                return true;
+            }
+            mi = Math.min(mi, v);
+        }
+        return false;
+    }
+}
+```
+
 ### **TypeScript**
 
 ```ts
@@ -144,6 +261,127 @@ impl Solution {
 }
 ```
 
+### **C++**
+
+```cpp
+class BinaryIndexedTree {
+public:
+    int n;
+    vector<int> c;
+
+    BinaryIndexedTree(int _n): n(_n), c(_n + 1){}
+
+    void update(int x, int delta) {
+        while (x <= n)
+        {
+            c[x] += delta;
+            x += lowbit(x);
+        }
+    }
+
+    int query(int x) {
+        int s = 0;
+        while (x > 0)
+        {
+            s += c[x];
+            x -= lowbit(x);
+        }
+        return s;
+    }
+
+    int lowbit(int x) {
+        return x & -x;
+    }
+};
+
+class Solution {
+public:
+    bool find132pattern(vector<int>& nums) {
+        unordered_set<int> s(nums.begin(), nums.end());
+        vector<int> alls(s.begin(), s.end());
+        sort(alls.begin(), alls.end());
+        unordered_map<int, int> m;
+        int n = alls.size();
+        for (int i = 0; i < n; ++i) m[alls[i]] = i + 1;
+        BinaryIndexedTree* tree = new BinaryIndexedTree(n);
+        for (int v : nums) tree->update(m[v], 1);
+        int mi = nums[0];
+        for (int v : nums)
+        {
+            tree->update(m[v], -1);
+            if (tree->query(m[v] - 1) - tree->query(m[mi]) > 0) return true;
+            mi = min(mi, v);
+        }
+        return false;
+    }
+};
+```
+
+### **Go**
+
+```go
+type BinaryIndexedTree struct {
+	n int
+	c []int
+}
+
+func newBinaryIndexedTree(n int) *BinaryIndexedTree {
+	c := make([]int, n+1)
+	return &BinaryIndexedTree{n, c}
+}
+
+func (this *BinaryIndexedTree) lowbit(x int) int {
+	return x & -x
+}
+
+func (this *BinaryIndexedTree) update(x, delta int) {
+	for x <= this.n {
+		this.c[x] += delta
+		x += this.lowbit(x)
+	}
+}
+
+func (this *BinaryIndexedTree) query(x int) int {
+	s := 0
+	for x > 0 {
+		s += this.c[x]
+		x -= this.lowbit(x)
+	}
+	return s
+}
+
+func find132pattern(nums []int) bool {
+	s := make(map[int]bool)
+	for _, v := range nums {
+		s[v] = true
+	}
+	var alls []int
+	for v := range s {
+		alls = append(alls, v)
+	}
+	sort.Ints(alls)
+	m := make(map[int]int)
+	for i, v := range alls {
+		m[v] = i + 1
+	}
+	tree := newBinaryIndexedTree(len(m))
+	for _, v := range nums {
+		tree.update(m[v], 1)
+	}
+	mi := nums[0]
+	for _, v := range nums {
+		tree.update(m[v], -1)
+		if tree.query(m[v]-1)-tree.query(m[mi]) > 0 {
+			return true
+		}
+		if v < mi {
+			mi = v
+		}
+	}
+	return false
+}
+```
+
 ### **...**
 
 ```