feat: add solutions to lc problem: No.0493

No.0493.Reverse Pairs

Author: yanglbme

solution/0400-0499/0493.Reverse Pairs/README.md

@@ -35,14 +35,31 @@
 
 <!-- 这里可写通用的实现逻辑 -->
 
-归并排序实现逆序对统计。
+**方法一：归并排序**
+
+**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 -->
 
 ### **Python3**
 
 <!-- 这里可写当前语言的特殊实现逻辑 -->
 
+归并排序：
+
 ```python
 class Solution:
     def reversePairs(self, nums: List[int]) -> int:
@@ -81,10 +98,53 @@ class Solution:
         return merge_sort(nums, 0, len(nums) - 1)
 ```
 
+树状数组：
+
+```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 > 0:
+            s += self.c[x]
+            x -= BinaryIndexedTree.lowbit(x)
+        return s
+
+
+class Solution:
+    def reversePairs(self, nums: List[int]) -> int:
+        s = set()
+        for num in nums:
+            s.add(num)
+            s.add(num * 2)
+        alls = sorted(s)
+        m = {v: i for i, v in enumerate(alls, 1)}
+        ans = 0
+        tree = BinaryIndexedTree(len(m))
+        for num in nums[::-1]:
+            ans += tree.query(m[num] - 1)
+            tree.update(m[num * 2], 1)
+        return ans
+```
+
 ### **Java**
 
 <!-- 这里可写当前语言的特殊实现逻辑 -->
 
+归并排序：
+
 ```java
 class Solution {
     private static int[] tmp = new int[50010];
@@ -131,8 +191,67 @@ class Solution {
 }
 ```
 
+树状数组：
+
+```java
+class Solution {
+    public int reversePairs(int[] nums) {
+        TreeSet<Long> ts = new TreeSet<>();
+        for (int num : nums) {
+            ts.add((long) num);
+            ts.add((long) num * 2);
+        }
+        Map<Long, Integer> m = new HashMap<>();
+        int idx = 0;
+        for (long num : ts) {
+            m.put(num, ++idx);
+        }
+        BinaryIndexedTree tree = new BinaryIndexedTree(m.size());
+        int ans = 0;
+        for (int i = nums.length - 1; i >= 0; --i) {
+            int x = m.get((long) nums[i]);
+            ans += tree.query(x - 1);
+            tree.update(m.get((long) nums[i] * 2), 1);
+        }
+        return ans;
+    }
+}
+
+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;
+    }
+}
+```
+
 ### **C++**
 
+归并排序：
+
 ```cpp
 class Solution {
 public:
@@ -175,8 +294,67 @@ private:
 };
 ```
 
+树状数组：
+
+```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:
+    int reversePairs(vector<int>& nums) {
+        set<long long> s;
+        for (int num : nums)
+        {
+            s.insert(num);
+            s.insert(num * 2ll);
+        }
+        unordered_map<long long, int> m;
+        int idx = 0;
+        for (long long num : s) m[num] = ++idx;
+        BinaryIndexedTree* tree = new BinaryIndexedTree(m.size());
+        int ans = 0;
+        for (int i = nums.size() - 1; i >= 0; --i)
+        {
+            ans += tree->query(m[nums[i]] - 1);
+            tree->update(m[nums[i] * 2ll], 1);
+        }
+        return ans;
+    }
+};
+```
+
 ### **Go**
 
+归并排序：
+
 ```go
 func reversePairs(nums []int) int {
 	return mergeSort(nums, 0, len(nums)-1)
@@ -223,6 +401,64 @@ func mergeSort(nums []int, left, right int) int {
 }
 ```
 
+树状数组：
+
+```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 reversePairs(nums []int) int {
+	s := make(map[int]bool)
+	for _, num := range nums {
+		s[num] = true
+		s[num*2] = true
+	}
+	var alls []int
+	for num := range s {
+		alls = append(alls, num)
+	}
+	sort.Ints(alls)
+	m := make(map[int]int)
+	for i, num := range alls {
+		m[num] = i + 1
+	}
+	tree := newBinaryIndexedTree(len(m))
+	ans := 0
+	for i := len(nums) - 1; i >= 0; i-- {
+		ans += tree.query(m[nums[i]] - 1)
+		tree.update(m[nums[i]*2], 1)
+	}
+	return ans
+}
+```
+
 ### **...**
 
 ```