目录

一、二叉搜索树简介 - 小结

二、数据流中的第K大元素

2.1 题目要求

2.2 解决过程

三、二叉搜索树的最近公共祖先

3.1 题目要求

3.2 解决过程

四、存在重复元素 III

4.1 题目要求

4.2 解决过程

---

一、二叉搜索树简介 - 小结

参考文献：https://leetcode-cn.com/leetbook/read/introduction-to-data-structure-binary-search-tree/xpbmwd/

---

二、数据流中的第K大元素

2.1 题目要求

2.2 解决过程

测试用例

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13913 more chars

---

验证方法

本方法旨在证明题意要求输出的是位置上逆序排列的第 k 个元素 (索引为 k-1)，而非算术上的第 k 大元素。由于大量使用 Python 内置函数/方法，不具有学习意义，且时间复杂度过高。

2020/08/11 - 13.50% (1140ms)

class KthLargest:def __init__(self, k: int, nums: List[int]):self.kth = kself.arr = numsdef add(self, val: int) -> int:self.arr.append(val)self.arr.sort(reverse=True)return self.arr[self.kth-1]

期望输出 / 实际输出

[null,4,5,5,8,8]

stdout

[4, 5, 8, 2]
[8, 5, 4, 3, 2]
[8, 5, 5, 4, 3, 2]
[10, 8, 5, 5, 4, 3, 2]
[10, 9, 8, 5, 5, 4, 3, 2]
[10, 9, 8, 5, 5, 4, 4, 3, 2]

---

失败测试

法一：单链表。相比于数组，链表结构虽然在指定索引访问元素方式略逊，但在插入/删除元素方面更优。可惜在最后的超长测试中，超出时间限制了。

2020/08/11 - 超出时间限制

class KthLargest:# ---------------------------------------------------------------------- #class _ListNode:__slots__ = "_val", "_next"def __init__(self, val=None, next=None):self._val = valself._next = next# ---------------------------------------------------------------------- #       def _connectNode(self, head, array):""" 根据数组构造单链表 """cur = headfor i in range(len(array)):cur._next = self._ListNode(array[i])cur = cur._next        def _insertNode(self, head, value):""" 插入指定值节点 """new = self._ListNode(value)  # 待插入新节点cur = headwhile cur._next:if cur._next._val <= new._val:new._next = cur._nextcur._next = newreturnelse:cur = cur._nextcur._next = new  # 接在最后def _searchNode(self, head, key):""" 查找指定位置节点 """cur = headfor _ in range(key):cur = cur._nextresult = cur._val if cur is not None else Nonereturn cur._val# ---------------------------------------------------------------------- #    def __init__(self, k: int, nums: List[int]):self.kth = kself.arr = sorted(nums, reverse=True)  # 逆序排列好数组self.dummy = self._ListNode(float("INF"))  # 哨兵节点if self.arr:self._connectNode(self.dummy, self.arr)def add(self, val: int) -> int:if not self.arr:self.arr.append(val)self.dummy._next = self._ListNode(val)return valelse:self._insertNode(self.dummy, val)return self._searchNode(self.dummy, self.kth)# Your KthLargest object will be instantiated and called as such:
# obj = KthLargest(k, nums)
# param_1 = obj.add(val)

法二：二叉搜索树改。相比于数组，链表结构虽然在指定索引访问元素方式略逊，但在插入/删除元素方面更优。而二乘搜索树相比于线性链表，又可以将时间复杂度从 O(n) 改进到 O(logn)。可惜在最后的超长测试中，还是超出时间限制了。看来基于 Python 强行用二叉搜索树速度还是不理想，如果是 C++ 或可实现。

2020/08/11 - 超出时间限制

class KthLargest:# -------------------------- nested _TreeNode class -------------------------- #class _TreeNode:""" Lightweight, nonpublic class for storing a tree node """__slots__ = "_val", "_left", "_right"  # straemline memory usagedef __init__(self, val=None, left=None, right=None):self._val = val                    # reference to the node valueself._left = left                  # reference to the left childself._right = right                # reference to the right child# --------------------- nested methods of KthLargest class -------------------- # def _createTree(self, root, nums):"""nums：待构造为树的数组, 只用一次"""for index in range(1, len(nums)):  # 除根节点外的其余子节点self._insertNode(root, nums[index])    def _insertNode(self, root, val):"""root：待插入树根节点val：待插入节点值"""if not root:return self._TreeNode(val)if val >= root._val:  # 注意：加入了等于的情况root._right = self._insertNode(root._right, val)else:root._left = self._insertNode(root._left, val)return root  # 返回根节点     def _searchNode(self, root, times):  # from: https://blog.csdn.net/qq_39478403/article/details/107678587"""  root：待中序遍历树的根节点times：需要遍历次数"""def _leftmost_inorder(root, stack):while root:stack.append(root)root = root._leftdef _next(stack):""" 树节点迭代器 """topmost_node = stack.pop()  # 目标节点if topmost_node._right:_leftmost_inorder(topmost_node._right, stack)return topmost_nodestack = []  # 中序遍历辅助栈_leftmost_inorder(root, stack)cur = stack[-1]  for _ in range(times):cur = _next(stack)return cur._val# --------------------- basic methods of KthLargest class -------------------- #def __init__(self, k: int, nums: List[int]):self.kth = k                                # 待查索引self.arr = nums                             # 待构造数组self.size = len(nums)                       # 当前节点数self.root = None                            # 默认初始为空树# 非空树if self.arr:  self.root = self._TreeNode(self.arr[0]) # 创建根节点self._createTree(self.root, self.arr)   # 根据数组 nums 构造树def add(self, val: int) -> int:# 插入值为 val 的节点if self.root is None:                       # 专门应对空树, 补充初始化self.root = self._TreeNode(val)else:self._insertNode(self.root, val)        # 将值为 val 的节点插入树self.size += 1                              # 节点数 +1# 查找 kth 节点值times = self.size - self.kth + 1            # 需要遍历次数return self._searchNode(self.root, times)   # 重新在树中找到第 k 个节点# Your KthLargest object will be instantiated and called as such:
# obj = KthLargest(k, nums)
# param_1 = obj.add(val)

---

其他实现 A

// C++ implementation
class KthLargest {int K;multiset<int> st;
public:KthLargest(int k, vector<int>& nums) {for (int n : nums) {st.insert(n);if (st.size() > k) st.erase(st.begin());}       K = k;}int add(int val) {st.insert(val);if (st.size() > K) st.erase(st.begin());return *st.begin();}
};

// C++ implementation
class KthLargest {int K;priority_queue<int, vector<int>, greater<int>> pq;
public:KthLargest(int k, vector<int>& nums) {for (int n : nums) {pq.push(n);if (pq.size() > k) pq.pop();}       K = k;}int add(int val) {pq.push(val);if (pq.size() > K) pq.pop();return pq.top();}
};

---

其他实现 B

求一个数列前 K 大数的问题经常会遇到，在程序中一般用小顶堆可以解决。对于 heapy 模块而言，例如：

>>> heap
[0, 1, 3, 6, 2, 8, 4, 7, 9, 5]
>>> heappush(heap, 0.5)
>>> heap
[0, 0.5, 3, 6, 1, 8, 4, 7, 9, 5, 2]# Python 的 heapq 在实现时，不同于 C++ STL 或 Java 可以传入比较函数
# 元素的排列顺序必须保证一点：位置i处的元素总是大于位置 i//2 处的元素（即小于位置 2*i 和 2*i+1 处的元素）
# 作为底层堆算法的基础，这被称为堆特征（heap property）
# 函数 heappop 弹出最小的元素（总是位于索引0处），并确保剩余元素中最小的那个位于索引 0 处（保持堆特征）

2020/08/11 - 98.23% (108ms) - 最佳 

# Python implementation
class KthLargest:def __init__(self, k: int, nums: List[int]):self.k = kself.nums = numsheapq.heapify(self.nums)  # 原地把 list 调整成 小顶堆 (类似递增列表)while len(self.nums) > k:  heapq.heappop(self.nums)  # 弹出堆顶元素/堆中最小值, 并重新调整 (超出索引 k 的更小而无用部分) def add(self, val: int) -> int:heapq.heappush(self.nums, val)  # 往小顶堆压入新元素, 并重新调整if len(self.nums) > self.k :heapq.heappop(self.nums)  # 弹出堆顶元素/堆中最小值, 并重新调整 (超出索引 k 的更小而无用部分) return self.nums[0]  # 直接弹出堆顶元素即为目标元素

原来研究了半天，是没用对数据结构。看来还需要更多地学习才行！(说不定面试让手写最小堆)  T T ...

参考文献

https://leetcode-cn.com/leetbook/read/introduction-to-data-structure-binary-search-tree/xpjovh/

https://leetcode-cn.com/problems/kth-largest-element-in-a-stream/solution/703-shu-ju-liu-zhong-de-di-kda-yuan-su-liang-chong/

https://leetcode-cn.com/problems/kth-largest-element-in-a-stream/solution/703-pythonshi-xian-shu-ju-liu-zhong-di-kda-xiao-di/

---

三、二叉搜索树的最近公共祖先

3.1 题目要求

3.2 解决过程

个人实现

法一：DFS 递归实现。采用和 剑指 Offer 68 - II. 二叉树的最近公共祖先 完全一致的解决方式 (详见参考文献)。缺点是虽然可以应用于任意类型的二叉树，但是未能充分运用到 BST 的特性。空间复杂度 O(n)，时间复杂度 O(n)。

2020/08/12 - 59.65% (100ms)

class Solution:def lowestCommonAncestor(self, root: 'TreeNode', p: 'TreeNode', q: 'TreeNode') -> 'TreeNode':if (not root) or (root == p) or (root == q):  # 只有两种情况：没找到 / 找到目标节点return root  # left 和 right 保存左、右子树查找结果, 只有两种情况：None 或 目标节点 p / qleft = self.lowestCommonAncestor(root.left, p, q)right = self.lowestCommonAncestor(root.right, p, q)if not left:  # 如果在左子树中没找到目标节点, 返回右子树查找结果return rightif not right:  # 如果在右子树中没找到目标节点, 返回左子树查找结果return leftreturn root  #  如果左、右子树均找到了目标节点, 即得所求的最近公共祖先

法一改：BFS 递归实现。通过 BST 性质先通过循环缩小搜索范围，再进行上述递归搜索。然而，结果并没有得到优化，看来是优化的方式不恰当。

2020/08/12 - 42.87% (104ms)

class Solution:def lowestCommonAncestor(self, root: 'TreeNode', p: 'TreeNode', q: 'TreeNode') -> 'TreeNode':larger = max(p.val, q.val)smaller = min(p.val, q.val)# 根据 BST 性质, p 和 q 的最小公共祖先数值范围必属于 [smaller, larger]while True:if root.val < smaller:root = root.rightelif root.val > larger:root = root.left        else:breakdef dfs(root):if (not root) or (root == p) or (root == q):  # 只有两种情况：没找到 / 找到 目标节点return root  left = dfs(root.left)right = dfs(root.right)   if not left:  # 如果在左子树中没找到目标节点, 返回右子树查找结果return rightif not right:  # 如果在右子树中没找到目标节点, 返回左子树查找结果return leftreturn root  #  如果左、右子树均找到了目标节点, 即得所求的最近公共祖先return dfs(root)

官方实现与说明

class Solution:def lowestCommonAncestor(self, root, p, q):""":type root: TreeNode:type p: TreeNode:type q: TreeNode:rtype: TreeNode"""# Value of current node or parent node.parent_val = root.val# Value of pp_val = p.val# Value of qq_val = q.val# If both p and q are greater than parentif p_val > parent_val and q_val > parent_val:    return self.lowestCommonAncestor(root.right, p, q)# If both p and q are lesser than parentelif p_val < parent_val and q_val < parent_val:    return self.lowestCommonAncestor(root.left, p, q)# We have found the split point, i.e. the LCA node.else:return root

2020/08/12 - 42.87% (104ms)

简化书写，删除冗余变量，增加注释：

class Solution:def lowestCommonAncestor(self, root, p, q):if p.val > root.val and q.val > root.val:               # root.val < min(p.val, q.val)return self.lowestCommonAncestor(root.right, p, q)  # 往右搜索elif p.val < root.val and q.val < root.val:             # root.val > max(p.val, q.val)return self.lowestCommonAncestor(root.left, p, q)   # 往左搜索else:  return root                                         # 一旦到达范围内, 即为目标 LCA

2020/08/12 - 74.77% (96ms)

---

class Solution:def lowestCommonAncestor(self, root, p, q):""":type root: TreeNode:type p: TreeNode:type q: TreeNode:rtype: TreeNode"""# Value of pp_val = p.val# Value of qq_val = q.val# Start from the root node of the treenode = root# Traverse the treewhile node:# Value of current node or parent node.parent_val = node.valif p_val > parent_val and q_val > parent_val:    # If both p and q are greater than parentnode = node.rightelif p_val < parent_val and q_val < parent_val:# If both p and q are lesser than parentnode = node.leftelse:# We have found the split point, i.e. the LCA node.return node

2020/08/12 - 88.84% (92ms)

简化书写，删除冗余变量，增加注释：

class Solution:def lowestCommonAncestor(self, root, p, q):node = rootwhile node:if p.val > node.val and q.val > node.val:    # node.val < min(p.val, q.val)    node = node.right                        # 往右搜索elif p.val < node.val and q.val < node.val:  # node.val > max(p.val, q.val)node = node.left                         # 往左搜索else:return node                              # 一旦到达范围内, 即为目标 LCA

 2020/08/12 - 98.42% (84ms) - 最佳

---

虽然已经发现了 关键规律：p 和 q 的最小公共祖先数值范围必属于 [smaller, larger]，但是未能写出最佳解法，因为没有证明或想清楚 自顶向下首个满足数组范围内的节点即为目标 LCA。

参考文献

https://leetcode-cn.com/leetbook/read/introduction-to-data-structure-binary-search-tree/xpf523/

https://blog.csdn.net/qq_39478403/article/details/107613468

https://leetcode-cn.com/problems/lowest-common-ancestor-of-a-binary-search-tree/solution/er-cha-sou-suo-shu-de-zui-jin-gong-gong-zu-xian--2/

---

四、存在重复元素 III

4.1 题目要求

4.2 解决过程

测试用例

[1,2,3,1]
3
0
[1,0,1,1]
1
2
[1,5,9,1,5,9]
2
3

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... 57793 more chars

失败测试

法一：暴力迭代法。空间复杂度 O(1)，时间复杂度 O(n^2)。

2020/08/12 - 超出时间限制

class Solution:def containsNearbyAlmostDuplicate(self, nums: List[int], k: int, t: int) -> bool:n = len(nums)for i in range(n):  # 遍历数组for j in range(k+1):  # 可变范围if (i+j >= n) or (i == i+j):  # 超出范围/索引重合 立即结束本次continueif abs(nums[i]-nums[i+j]) <= t:  # 满足要求return Truereturn False

官方实现与说明

// Java implementation
public boolean containsNearbyAlmostDuplicate(int[] nums, int k, int t) {for (int i = 0; i < nums.length; ++i) {for (int j = Math.max(i - k, 0); j < i; ++j) {if (Math.abs(nums[i] - nums[j]) <= t) return true;}}return false;
}
// Time limit exceeded.

---

# 严重左偏的非平衡二叉搜索树举例6/5/4/3/2/1

# 一棵平衡的二叉搜索树举例4/   \2     6/ \   /1   3  5

// Java implementation
public boolean containsNearbyAlmostDuplicate(int[] nums, int k, int t) {TreeSet<Integer> set = new TreeSet<>();  // 这个 TreeSet 在 Python 标准库没有！for (int i = 0; i < nums.length; ++i) {// Find the successor of current elementInteger s = set.ceiling(nums[i]);if (s != null && s <= nums[i] + t) return true;// Find the predecessor of current elementInteger g = set.floor(nums[i]);if (g != null && nums[i] <= g + t) return true;set.add(nums[i]);if (set.size() > k) {set.remove(nums[i - k]);}}return false;
}

// C++ implementation
class Solution {
public:bool containsNearbyAlmostDuplicate(vector<int>& nums, int k, int t) {set<int> s;for (int i = 0; i < nums.size(); ++i) {auto lower = s.lower_bound(nums[i]);if (lower != s.end() && *lower <= nums[i] + t) return true;auto upper = s.upper_bound(nums[i]);if (upper != s.begin() && nums[i] <= *--upper + t) return true; // upper 为 s.begin() 时说明查找结果为空s.insert(nums[i]);if (s.size() > k) s.erase(nums[i - k]);}return false;}
};

---

// Java implementation
public class Solution {// Get the ID of the bucket from element value x and bucket width w// In Java, `-3 / 5 = 0` and but we need `-3 / 5 = -1`.private long getID(long x, long w) {return x < 0 ? (x + 1) / w - 1 : x / w;}public boolean containsNearbyAlmostDuplicate(int[] nums, int k, int t) {if (t < 0) return false;Map<Long, Long> d = new HashMap<>();long w = (long)t + 1;for (int i = 0; i < nums.length; ++i) {long m = getID(nums[i], w);// check if bucket m is empty, each bucket may contain at most one elementif (d.containsKey(m))return true;// check the nei***or buckets for almost duplicateif (d.containsKey(m - 1) && Math.abs(nums[i] - d.get(m - 1)) < w)return true;if (d.containsKey(m + 1) && Math.abs(nums[i] - d.get(m + 1)) < w)return true;// now bucket m is empty and no almost duplicate in nei***or bucketsd.put(m, (long)nums[i]);if (i >= k) d.remove(getID(nums[i - k], w));}return false;}
}

---

其他实现与说明 - 桶排序

class Solution:def containsNearbyAlmostDuplicate(self, nums: List[int], k: int, t: int) -> bool:if t < 0 or k < 0:return Falseall_buckets = {}  # 使用 dict 充当 bucketsbucket_size = t + 1  # 桶的大小设成 t+1 更加方便for i in range(len(nums)):# 放入桶号索引, 这样同一桶内数字最多相差 t, 相邻桶内数字最多相差 2*t-1, 其余不考虑bucket_index = nums[i] // bucket_size # 若桶中已经有元素了, 则将同时满足 t 和 k 条件, 返回 Trueif bucket_index in all_buckets:       return True# 把nums[i]放入桶中all_buckets[bucket_index] = nums[i]   # 检查前一个桶, 若有元素则满足 k 条件, 需要再确定 t 条件if (bucket_index-1) in all_buckets and abs(all_buckets[bucket_index-1] - nums[i]) <= t: return True# 检查后一个桶, 若有元素则满足 k 条件, 需要再确定 t 条件if (bucket_index+1) in all_buckets and abs(all_buckets[bucket_index+1] - nums[i]) <= t: return True# 若不构成返回条件, 那么当 i >= k 时就要删除旧桶,# 以维持桶中的元素索引跟下一个 i+1 索引只相差不超过 kif i >= k:all_buckets.pop(nums[i-k]//bucket_size)return False

2020/08/14 - 77.66% (48ms) - 基本思想还是以空间换时间

---

# Python implementation
class Solution:def containsNearbyAlmostDuplicate(self, nums: List[int], k: int, t: int) -> bool:bucket = dict()if t < 0: return Falsefor i in range(len(nums)):nth = nums[i] // (t + 1)if nth in bucket:return Trueif nth - 1 in bucket and abs(nums[i] - bucket[nth - 1]) <= t:return Trueif nth + 1 in bucket and abs(nums[i] - bucket[nth + 1]) <= t:return Truebucket[nth] = nums[i]if i >= k: bucket.pop(nums[i - k] // (t + 1))return False

// C++ implementation
class Solution {
public:bool containsNearbyAlmostDuplicate(vector<int>& nums, int k, int t) {if(t<0) return false;//t+1可能会溢出，所以要+ 1LLlong long mod = t + 1LL;unordered_map<long long,long long> buck;for(int i=0;i<nums.size();i++){long long nth = nums[i] / mod;//可能nums[i]为负数，比如-4 / 5 以及 -4 / 5都等于0，所以负数要向下移动一位if(nums[i] < 0) nth--;//这里要用find 不能直接[],因为可能本身存储的数字就为0if(buck.find(nth)!=buck.end()) return true;else if(buck.find(nth-1)!=buck.end() && abs(nums[i] - buck[nth-1]) <= t)return true;else if(buck.find(nth+1)!=buck.end() && abs(nums[i] - buck[nth+1]) <= t)return true;buck[nth] = nums[i];if(i >= k){buck.erase(nums[i - k] / mod);}}return false;}
};

参考文献

https://leetcode-cn.com/leetbook/read/introduction-to-data-structure-binary-search-tree/xpffam/

https://leetcode-cn.com/problems/contains-duplicate-iii/solution/cun-zai-zhong-fu-yuan-su-iii-by-leetcode/

https://leetcode-cn.com/problems/contains-duplicate-iii/solution/cun-zai-zhong-fu-yuan-su-iii-by-leetcode/311904

https://leetcode-cn.com/problems/contains-duplicate-iii/solution/cun-zai-zhong-fu-yuan-su-iii-by-leetcode/151367

https://leetcode-cn.com/problems/contains-duplicate-iii/solution/li-yong-tong-de-yuan-li-onpython3-by-zhou-pen-chen/

https://leetcode-cn.com/problems/contains-duplicate-iii/solution/220-cun-zai-zhong-fu-yuan-su-iii-cong-on2-dao-on-p/

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