feat: add solutions to lcci problem: No.17.08

No.17.08.Circus Tower

Author: yanglbme

lcci/17.08.Circus Tower/README.md

@@ -5,6 +5,7 @@
 ## 题目描述
 
 <!-- 这里写题目描述 -->
+
 <p>有个马戏团正在设计叠罗汉的表演节目，一个人要站在另一人的肩膀上。出于实际和美观的考虑，在上面的人要比下面的人矮一点且轻一点。已知马戏团每个人的身高和体重，请编写代码计算叠罗汉最多能叠几个人。</p>
 <p><strong>示例：</strong></p>
 <pre><strong>输入：</strong>height = [65,70,56,75,60,68] weight = [100,150,90,190,95,110]
@@ -18,22 +19,245 @@
 ## 解法
 
 <!-- 这里可写通用的实现逻辑 -->
+
+**方法一：排序 + 离散化 + 树状数组**
+
+我们现将所有人按照身高从小到大排序，若身高相同，则按照体重从大到小排序。这样我们可以将问题转换为求体重数组的最长递增子序列的问题。
+
+最长递增子序列的问题可以使用动态规划求解，时间复杂度 $O(n^2)$。但是我们可以使用树状数组来优化求解过程，时间复杂度 $O(n \log n)$。
+
+空间复杂度 $O(n)$。其中 $n$ 为人数。
+
 <!-- 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):
+        self.n = n
+        self.c = [0] * (n + 1)
+
+    def update(self, x, delta):
+        while x <= self.n:
+            self.c[x] = max(self.c[x], delta)
+            x += x & -x
 
+    def query(self, x):
+        s = 0
+        while x:
+            s = max(s, self.c[x])
+            x -= x & -x
+        return s
+
+
+class Solution:
+    def bestSeqAtIndex(self, height: List[int], weight: List[int]) -> int:
+        arr = list(zip(height, weight))
+        arr.sort(key=lambda x: (x[0], -x[1]))
+        alls = sorted({w for _, w in arr})
+        m = {w: i for i, w in enumerate(alls, 1)}
+        tree = BinaryIndexedTree(len(m))
+        ans = 1
+        for _, w in arr:
+            x = m[w]
+            t = tree.query(x - 1) + 1
+            ans = max(ans, t)
+            tree.update(x, t)
+        return ans
 ```
 
 ### **Java**
 
 <!-- 这里可写当前语言的特殊实现逻辑 -->
 
 ```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 val) {
+        while (x <= n) {
+            this.c[x] = Math.max(this.c[x], val);
+            x += x & -x;
+        }
+    }
+
+    public int query(int x) {
+        int s = 0;
+        while (x > 0) {
+            s = Math.max(s, this.c[x]);
+            x -= x & -x;
+        }
+        return s;
+    }
+}
+
+class Solution {
+    public int bestSeqAtIndex(int[] height, int[] weight) {
+        int n = height.length;
+        int[][] arr = new int[n][2];
+        for (int i = 0; i < n; ++i) {
+            arr[i] = new int[]{height[i], weight[i]};
+        }
+        Arrays.sort(arr, (a, b) -> a[0] == b[0] ? b[1] - a[1] : a[0] - b[0]);
+        Set<Integer> s = new HashSet<>();
+        for (int[] e : arr) {
+            s.add(e[1]);
+        }
+        List<Integer> alls = new ArrayList<>(s);
+        Collections.sort(alls);
+        Map<Integer, Integer> m = new HashMap<>(alls.size());
+        for (int i = 0; i < alls.size(); ++i) {
+            m.put(alls.get(i), i + 1);
+        }
+        BinaryIndexedTree tree = new BinaryIndexedTree(alls.size());
+        int ans = 1;
+        for (int[] e : arr) {
+            int x = m.get(e[1]);
+            int t = tree.query(x - 1) + 1;
+            ans = Math.max(ans, t);
+            tree.update(x, t);
+        }
+        return ans;
+    }
+}
+```
+
+### **C++**
+
+```cpp
+class BinaryIndexedTree {
+public:
+    BinaryIndexedTree(int _n)
+        : n(_n)
+        , c(_n + 1) {}
+
+    void update(int x, int val) {
+        while (x <= n) {
+            c[x] = max(c[x], val);
+            x += x & -x;
+        }
+    }
+
+    int query(int x) {
+        int s = 0;
+        while (x > 0) {
+            s = max(s, c[x]);
+            x -= x & -x;
+        }
+        return s;
+    }
+
+private:
+    int n;
+    vector<int> c;
+};
+
+class Solution {
+public:
+    int bestSeqAtIndex(vector<int>& height, vector<int>& weight) {
+        int n = height.size();
+        vector<pair<int, int>> people;
+        for (int i = 0; i < n; ++i) {
+            people.emplace_back(height[i], weight[i]);
+        }
+        sort(people.begin(), people.end(), [](const pair<int, int>& a, const pair<int, int>& b) {
+            if (a.first == b.first) {
+                return a.second > b.second;
+            }
+            return a.first < b.first;
+        });
+        vector<int> alls = weight;
+        sort(alls.begin(), alls.end());
+        alls.erase(unique(alls.begin(), alls.end()), alls.end());
+        BinaryIndexedTree tree(alls.size());
+        int ans = 1;
+        for (auto& [_, w] : people) {
+            int x = lower_bound(alls.begin(), alls.end(), w) - alls.begin() + 1;
+            int t = tree.query(x - 1) + 1;
+            ans = max(ans, t);
+            tree.update(x, t);
+        }
+        return ans;
+    }
+};
+```
+
+### **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) update(x, val int) {
+	for x <= this.n {
+		if this.c[x] < val {
+			this.c[x] = val
+		}
+		x += x & -x
+	}
+}
+
+func (this *BinaryIndexedTree) query(x int) int {
+	s := 0
+	for x > 0 {
+		if s < this.c[x] {
+			s = this.c[x]
+		}
+		x -= x & -x
+	}
+	return s
+}
+
+func bestSeqAtIndex(height []int, weight []int) int {
+	n := len(height)
+	people := make([][2]int, n)
+	s := map[int]bool{}
+	for i := range people {
+		people[i] = [2]int{height[i], weight[i]}
+		s[weight[i]] = true
+	}
+	sort.Slice(people, func(i, j int) bool {
+		a, b := people[i], people[j]
+		return a[0] < b[0] || a[0] == b[0] && a[1] > b[1]
+	})
+	alls := make([]int, 0, len(s))
+	for k := range s {
+		alls = append(alls, k)
+	}
+	sort.Ints(alls)
+	tree := newBinaryIndexedTree(len(alls))
+	ans := 1
+	for _, p := range people {
+		x := sort.SearchInts(alls, p[1]) + 1
+		t := tree.query(x-1) + 1
+		ans = max(ans, t)
+		tree.update(x, t)
+	}
+	return ans
+}
 
+func max(a, b int) int {
+	if a > b {
+		return a
+	}
+	return b
+}
 ```
 
 ### **...**