From a46b4829518779b91581971dda5014926cde2da3 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Thu, 21 Sep 2023 04:43:15 -0500 Subject: [PATCH] Refactor the hash_map_open_addressing implementation with lazy reallocation. (#776) --- .../hash_map_open_addressing.cpp | 177 ++++++++++-------- .../hash_map_open_addressing.py | 96 +++++----- .../hash_table_open_addressing_deletion.png | Bin 0 -> 59604 bytes docs/chapter_hashing/hash_collision.md | 62 ++++-- 4 files changed, 190 insertions(+), 145 deletions(-) create mode 100644 docs/chapter_hashing/hash_collision.assets/hash_table_open_addressing_deletion.png diff --git a/codes/cpp/chapter_hashing/hash_map_open_addressing.cpp b/codes/cpp/chapter_hashing/hash_map_open_addressing.cpp index eac382f32..b331fffb2 100644 --- a/codes/cpp/chapter_hashing/hash_map_open_addressing.cpp +++ b/codes/cpp/chapter_hashing/hash_map_open_addressing.cpp @@ -6,26 +6,28 @@ #include "./array_hash_map.cpp" -/* 开放寻址哈希表 */ class HashMapOpenAddressing { private: - int size; // 键值对数量 - int capacity; // 哈希表容量 - double loadThres; // 触发扩容的负载因子阈值 - int extendRatio; // 扩容倍数 - vector buckets; // 桶数组 - Pair *removed; // 删除标记 + int size; // 键值对数量 + int capacity = 4; // 哈希表容量 + const double loadThres = 2.0 / 3; // 触发扩容的负载因子阈值 + const int extendRatio = 2; // 扩容倍数 + vector buckets; // 桶数组 + Pair *TOMBSTONE = new Pair(-1, "-1"); // 删除标记 public: /* 构造方法 */ - HashMapOpenAddressing() { - // 构造方法 - size = 0; - capacity = 4; - loadThres = 2.0 / 3.0; - extendRatio = 2; - buckets = vector(capacity, nullptr); - removed = new Pair(-1, "-1"); + HashMapOpenAddressing() : size(0), buckets(capacity, nullptr) { + } + + /* 析构方法 */ + ~HashMapOpenAddressing() { + for (Pair *pair : buckets) { + if (pair != nullptr && pair != TOMBSTONE) { + delete pair; + } + } + delete TOMBSTONE; } /* 哈希函数 */ @@ -35,67 +37,75 @@ class HashMapOpenAddressing { /* 负载因子 */ double loadFactor() { - return static_cast(size) / capacity; + return (double)size / capacity; } - /* 查询操作 */ - string get(int key) { + /* 搜索 key 对应的桶索引 */ + int findBucket(int key) { int index = hashFunc(key); - // 线性探测，从 index 开始向后遍历 - for (int i = 0; i < capacity; i++) { + int firstTombstone = -1; + // 线性探测，当遇到空桶时跳出 + while (buckets[index] != nullptr) { + // 若遇到 key ，返回对应桶索引 + if (buckets[index]->key == key) { + // 若之前遇到了删除标记，则将键值对移动至该索引 + if (firstTombstone != -1) { + buckets[firstTombstone] = buckets[index]; + buckets[index] = TOMBSTONE; + return firstTombstone; // 返回移动后的桶索引 + } + return index; // 返回桶索引 + } + // 记录遇到的首个删除标记 + if (firstTombstone == -1 && buckets[index] == TOMBSTONE) { + firstTombstone = index; + } // 计算桶索引，越过尾部返回头部 - int j = (index + i) % capacity; - // 若遇到空桶，说明无此 key ，则返回 nullptr - if (buckets[j] == nullptr) - return nullptr; - // 若遇到指定 key ，则返回对应 val - if (buckets[j]->key == key && buckets[j] != removed) - return buckets[j]->val; + index = (index + 1) % capacity; + } + // 若 key 不存在，则返回添加点的索引 + return firstTombstone == -1 ? index : firstTombstone; + } + + /* 查询操作 */ + string get(int key) { + // 搜索 key 对应的桶索引 + int index = findBucket(key); + // 若找到键值对，则返回对应 val + if (buckets[index] != nullptr && buckets[index] != TOMBSTONE) { + return buckets[index]->val; } - return nullptr; + // 若键值对不存在，则返回空字符串 + return ""; } /* 添加操作 */ void put(int key, string val) { // 当负载因子超过阈值时，执行扩容 - if (loadFactor() > loadThres) + if (loadFactor() > loadThres) { extend(); - int index = hashFunc(key); - // 线性探测，从 index 开始向后遍历 - for (int i = 0; i < capacity; i++) { - // 计算桶索引，越过尾部返回头部 - int j = (index + i) % capacity; - // 若遇到空桶、或带有删除标记的桶，则将键值对放入该桶 - if (buckets[j] == nullptr || buckets[j] == removed) { - buckets[j] = new Pair(key, val); - size += 1; - return; - } - // 若遇到指定 key ，则更新对应 val - if (buckets[j]->key == key) { - buckets[j]->val = val; - return; - } } + // 搜索 key 对应的桶索引 + int index = findBucket(key); + // 若找到键值对，则覆盖 val 并返回 + if (buckets[index] != nullptr && buckets[index] != TOMBSTONE) { + buckets[index]->val = val; + return; + } + // 若键值对不存在，则添加该键值对 + buckets[index] = new Pair(key, val); + size++; } /* 删除操作 */ void remove(int key) { - int index = hashFunc(key); - // 线性探测，从 index 开始向后遍历 - for (int i = 0; i < capacity; i++) { - // 计算桶索引，越过尾部返回头部 - int j = (index + i) % capacity; - // 若遇到空桶，说明无此 key ，则直接返回 - if (buckets[j] == nullptr) - return; - // 若遇到指定 key ，则标记删除并返回 - if (buckets[j]->key == key) { - delete buckets[j]; // 释放内存 - buckets[j] = removed; - size -= 1; - return; - } + // 搜索 key 对应的桶索引 + int index = findBucket(key); + // 若找到键值对，则用删除标记覆盖它 + if (buckets[index] != nullptr && buckets[index] != TOMBSTONE) { + delete buckets[index]; + buckets[index] = TOMBSTONE; + size--; } } @@ -109,19 +119,22 @@ class HashMapOpenAddressing { size = 0; // 将键值对从原哈希表搬运至新哈希表 for (Pair *pair : bucketsTmp) { - if (pair != nullptr && pair != removed) { + if (pair != nullptr && pair != TOMBSTONE) { put(pair->key, pair->val); + delete pair; } } } /* 打印哈希表 */ void print() { - for (auto &pair : buckets) { - if (pair != nullptr) { - cout << pair->key << " -> " << pair->val << endl; - } else { + for (Pair *pair : buckets) { + if (pair == nullptr) { cout << "nullptr" << endl; + } else if (pair == TOMBSTONE) { + cout << "TOMBSTONE" << endl; + } else { + cout << pair->key << " -> " << pair->val << endl; } } } @@ -129,29 +142,29 @@ class HashMapOpenAddressing { /* Driver Code */ int main() { - /* 初始化哈希表 */ - HashMapOpenAddressing map = HashMapOpenAddressing(); + // 初始化哈希表 + HashMapOpenAddressing hashmap; - /* 添加操作 */ - // 在哈希表中添加键值对 (key, value) - map.put(12836, "小哈"); - map.put(15937, "小啰"); - map.put(16750, "小算"); - map.put(13276, "小法"); - map.put(10583, "小鸭"); + // 添加操作 + // 在哈希表中添加键值对 (key, val) + hashmap.put(12836, "小哈"); + hashmap.put(15937, "小啰"); + hashmap.put(16750, "小算"); + hashmap.put(13276, "小法"); + hashmap.put(10583, "小鸭"); cout << "\n添加完成后，哈希表为\nKey -> Value" << endl; - map.print(); + hashmap.print(); - /* 查询操作 */ - // 向哈希表输入键 key ，得到值 value - string name = map.get(13276); + // 查询操作 + // 向哈希表输入键 key ，得到值 val + string name = hashmap.get(13276); cout << "\n输入学号 13276 ，查询到姓名 " << name << endl; - /* 删除操作 */ - // 在哈希表中删除键值对 (key, value) - map.remove(16750); + // 删除操作 + // 在哈希表中删除键值对 (key, val) + hashmap.remove(16750); cout << "\n删除 16750 后，哈希表为\nKey -> Value" << endl; - map.print(); + hashmap.print(); return 0; } diff --git a/codes/python/chapter_hashing/hash_map_open_addressing.py b/codes/python/chapter_hashing/hash_map_open_addressing.py index 7dbc6015c..9ecefe863 100644 --- a/codes/python/chapter_hashing/hash_map_open_addressing.py +++ b/codes/python/chapter_hashing/hash_map_open_addressing.py @@ -20,7 +20,7 @@ class HashMapOpenAddressing: self.load_thres = 2 / 3 # 触发扩容的负载因子阈值 self.extend_ratio = 2 # 扩容倍数 self.buckets: list[Pair | None] = [None] * self.capacity # 桶数组 - self.removed = Pair(-1, "-1") # 删除标记 + self.TOMBSTONE = Pair(-1, "-1") # 删除标记 def hash_func(self, key: int) -> int: """哈希函数""" @@ -30,55 +30,61 @@ class HashMapOpenAddressing: """负载因子""" return self.size / self.capacity - def get(self, key: int) -> str: - """查询操作""" + def find_bucket(self, key: int) -> int: + """搜索 key 对应的桶索引""" index = self.hash_func(key) - # 线性探测，从 index 开始向后遍历 - for i in range(self.capacity): + first_tombstone = -1 + # 线性探测，当遇到空桶时跳出 + while self.buckets[index] is not None: + # 若遇到 key ，返回对应桶索引 + if self.buckets[index].key == key: + # 若之前遇到了删除标记，则将键值对移动至该索引 + if first_tombstone != -1: + self.buckets[first_tombstone] = self.buckets[index] + self.buckets[index] = self.TOMBSTONE + return first_tombstone # 返回移动后的桶索引 + return index # 返回桶索引 + # 记录遇到的首个删除标记 + if first_tombstone == -1 and self.buckets[index] is self.TOMBSTONE: + first_tombstone = index # 计算桶索引，越过尾部返回头部 - j = (index + i) % self.capacity - # 若遇到空桶，说明无此 key ，则返回 None - if self.buckets[j] is None: - return None - # 若遇到指定 key ，则返回对应 val - if self.buckets[j].key == key and self.buckets[j] != self.removed: - return self.buckets[j].val + index = (index + 1) % self.capacity + # 若 key 不存在，则返回添加点的索引 + return index if first_tombstone == -1 else first_tombstone + + def get(self, key: int) -> str: + """查询操作""" + # 搜索 key 对应的桶索引 + index = self.find_bucket(key) + # 若找到键值对，则返回对应 val + if self.buckets[index] not in [None, self.TOMBSTONE]: + return self.buckets[index].val + # 若键值对不存在，则返回 None + return None def put(self, key: int, val: str): """添加操作""" # 当负载因子超过阈值时，执行扩容 if self.load_factor() > self.load_thres: self.extend() - index = self.hash_func(key) - # 线性探测，从 index 开始向后遍历 - for i in range(self.capacity): - # 计算桶索引，越过尾部返回头部 - j = (index + i) % self.capacity - # 若遇到空桶、或带有删除标记的桶，则将键值对放入该桶 - if self.buckets[j] in [None, self.removed]: - self.buckets[j] = Pair(key, val) - self.size += 1 - return - # 若遇到指定 key ，则更新对应 val - if self.buckets[j].key == key: - self.buckets[j].val = val - return + # 搜索 key 对应的桶索引 + index = self.find_bucket(key) + # 若找到键值对，则覆盖 val 并返回 + if self.buckets[index] not in [None, self.TOMBSTONE]: + self.buckets[index].val = val + return + # 若键值对不存在，则添加该键值对 + self.buckets[index] = Pair(key, val) + self.size += 1 def remove(self, key: int): """删除操作""" - index = self.hash_func(key) - # 线性探测，从 index 开始向后遍历 - for i in range(self.capacity): - # 计算桶索引，越过尾部返回头部 - j = (index + i) % self.capacity - # 若遇到空桶，说明无此 key ，则直接返回 - if self.buckets[j] is None: - return - # 若遇到指定 key ，则标记删除并返回 - if self.buckets[j].key == key: - self.buckets[j] = self.removed - self.size -= 1 - return + # 搜索 key 对应的桶索引 + index = self.find_bucket(key) + # 若找到键值对，则用删除标记覆盖它 + if self.buckets[index] not in [None, self.TOMBSTONE]: + self.buckets[index] = self.TOMBSTONE + self.size -= 1 def extend(self): """扩容哈希表""" @@ -90,21 +96,23 @@ class HashMapOpenAddressing: self.size = 0 # 将键值对从原哈希表搬运至新哈希表 for pair in buckets_tmp: - if pair not in [None, self.removed]: + if pair not in [None, self.TOMBSTONE]: self.put(pair.key, pair.val) def print(self): """打印哈希表""" for pair in self.buckets: - if pair is not None: - print(pair.key, "->", pair.val) - else: + if pair is None: print("None") + elif pair is self.TOMBSTONE: + print("TOMBSTONE") + else: + print(pair.key, "->", pair.val) """Driver Code""" if __name__ == "__main__": - # 测试代码 + # 初始化哈希表 hashmap = HashMapOpenAddressing() # 添加操作 diff --git a/docs/chapter_hashing/hash_collision.assets/hash_table_open_addressing_deletion.png 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QIPgzaQc0pn-1zDL15^~j-T(jq literal 0 HcmV?d00001 diff --git a/docs/chapter_hashing/hash_collision.md b/docs/chapter_hashing/hash_collision.md index dd3522620..3d076926f 100644 --- a/docs/chapter_hashing/hash_collision.md +++ b/docs/chapter_hashing/hash_collision.md @@ -1,6 +1,6 @@ # 哈希冲突 -上节提到，**通常情况下哈希函数的输入空间远大于输出空间**，因此理论上哈希冲突是不可避免的。比如，输入空间为全体整数，输出空间为数组容量大小，则必然有多个整数映射至同一数组索引。 +上节提到，**通常情况下哈希函数的输入空间远大于输出空间**，因此理论上哈希冲突是不可避免的。比如，输入空间为全体整数，输出空间为数组容量大小，则必然有多个整数映射至同一桶索引。 哈希冲突会导致查询结果错误，严重影响哈希表的可用性。为解决该问题，我们可以每当遇到哈希冲突时就进行哈希表扩容，直至冲突消失为止。此方法简单粗暴且有效，但效率太低，因为哈希表扩容需要进行大量的数据搬运与哈希值计算。为了提升效率，我们可以采用以下策略。 @@ -15,9 +15,9 @@ ![链式地址哈希表](hash_collision.assets/hash_table_chaining.png) -哈希表在链式地址下的操作方法发生了一些变化。 +基于链式地址实现的哈希表的操作方法发生了以下变化。 -- **查询元素**：输入 `key` ，经过哈希函数得到数组索引，即可访问链表头节点，然后遍历链表并对比 `key` 以查找目标键值对。 +- **查询元素**：输入 `key` ，经过哈希函数得到桶索引，即可访问链表头节点，然后遍历链表并对比 `key` 以查找目标键值对。 - **添加元素**：先通过哈希函数访问链表头节点，然后将节点（即键值对）添加到链表中。 - **删除元素**：根据哈希函数的结果访问链表头部，接着遍历链表以查找目标节点，并将其删除。 @@ -109,26 +109,32 @@ 「开放寻址 open addressing」不引入额外的数据结构，而是通过“多次探测”来处理哈希冲突，探测方式主要包括线性探测、平方探测、多次哈希等。 +下面将主要以线性探测为例，介绍开放寻址哈希表的工作机制与代码实现。 + ### 线性探测 线性探测采用固定步长的线性搜索来进行探测，其操作方法与普通哈希表有所不同。 -- **插入元素**：通过哈希函数计算数组索引，若发现桶内已有元素，则从冲突位置向后线性遍历（步长通常为 $1$ ），直至找到空位，将元素插入其中。 -- **查找元素**：若发现哈希冲突，则使用相同步长向后线性遍历，直到找到对应元素，返回 `value` 即可；如果遇到空位，说明目标键值对不在哈希表中，返回 $\text{None}$ 。 +- **插入元素**：通过哈希函数计算桶索引，若发现桶内已有元素，则从冲突位置向后线性遍历（步长通常为 $1$ ），直至找到空桶，将元素插入其中。 +- **查找元素**：若发现哈希冲突，则使用相同步长向后线性遍历，直到找到对应元素，返回 `value` 即可；如果遇到空桶，说明目标元素不在哈希表中，返回 $\text{None}$ 。 -下图展示了一个在开放寻址（线性探测）下工作的哈希表。 +下图展示了开放寻址（线性探测）哈希表的键值对分布。根据此哈希函数，最后两位相同的 `key` 都会被映射到相同的桶。而通过线性探测，它们被依次存储在该桶以及之下的桶中。 ![开放寻址和线性探测](hash_collision.assets/hash_table_linear_probing.png) -然而，线性探测存在以下缺陷。 +然而，**线性探测容易产生“聚集现象”**。具体来说，数组中连续被占用的位置越长，这些连续位置发生哈希冲突的可能性越大，从而进一步促使该位置的聚堆生长，形成恶性循环，最终导致增删查改操作效率劣化。 + +值得注意的是，**我们不能在开放寻址哈希表中直接删除元素**。这是因为删除元素会在数组内产生一个空桶 $\text{None}$ ，而当查询元素时，线性探测到该空桶就会返回，因此在该空桶之下的元素都无法再被访问到，程序可能误判这些元素不存在。 + +![在开放寻址中删除元素导致的查询问题](hash_collision.assets/hash_table_open_addressing_deletion.png) + +为了解决该问题，我们可以采用「懒删除 lazy deletion」机制：它不直接从哈希表中移除元素，**而是利用一个常量 `TOMBSTONE` 来标记这个桶**。在该机制下，$\text{None}$ 和 `TOMBSTONE` 都代表空桶，都可以放置键值对。但不同的是，线性探测到 `TOMBSTONE` 时应该继续遍历，因为其之下可能还存在键值对。 -- **不能直接删除元素**。删除元素会在数组内产生一个空位，当查找该空位之后的元素时，该空位可能导致程序误判元素不存在。为此，通常需要借助一个标志位来标记已删除元素。 -- **容易产生聚集**。数组内连续被占用位置越长，这些连续位置发生哈希冲突的可能性越大，进一步促使这一位置的聚堆生长，形成恶性循环，最终导致增删查改操作效率劣化。 +然而，**懒删除可能会加速哈希表的性能退化**。这是因为每次删除操作都会产生一个删除标记，随着 `TOMBSTONE` 的增加，搜索时间也会增加，因为线性探测可能需要跳过多个 `TOMBSTONE` 才能找到目标元素。 -以下代码实现了一个简单的开放寻址（线性探测）哈希表。 +为此，考虑在线性探测中记录遇到的首个 `TOMBSTONE` 的索引，并将搜索到的目标元素与该 `TOMBSTONE` 交换位置。这样做的好处是当每次查询或添加元素时，元素会被移动至距离理想位置（探测起始点）更近的桶，从而优化查询效率。 -- 我们使用一个固定的键值对实例 `removed` 来标记已删除元素。也就是说，当一个桶内的元素为 $\text{None}$ 或 `removed` 时，说明这个桶是空的，可用于放置键值对。 -- 在线性探测时，我们从当前索引 `index` 向后遍历；而当越过数组尾部时，需要回到头部继续遍历。 +以下代码实现了一个包含懒删除的开放寻址（线性探测）哈希表。为了更加充分地使用哈希表的空间，我们将哈希表表看作是一个“环形数组”，当越过数组尾部时，回到头部继续遍历。 === "Python" @@ -202,19 +208,37 @@ [class]{HashMapOpenAddressing}-[func]{} ``` +### 平方探测 + +平方探测与线性探测类似，都是开放寻址的常见策略之一。当发生冲突时，平方探测不是简单地跳过一个固定的步数，而是跳过“探测次数的平方”的步数，即 $1, 4, 9, \dots$ 步。 + +平方探测通主要具有以下优势。 + +- 平方探测通过跳过平方的距离，试图缓解线性探测的聚集效应。 +- 平方探测会跳过更大的距离来寻找空位置，有助于数据分布得更加均匀。 + +然而，平方探测也并不是完美的。 + +- 仍然存在聚集现象，即某些位置比其他位置更容易被占用。 +- 由于平方的增长，平方探测可能不会探测整个哈希表，这意味着即使哈希表中有空桶，平方探测也可能无法访问到它。 + ### 多次哈希 -顾名思义，多次哈希方法是使用多个哈希函数 $f_1(x)$、$f_2(x)$、$f_3(x)$、$\dots$ 进行探测。 +多次哈希使用多个哈希函数 $f_1(x)$、$f_2(x)$、$f_3(x)$、$\dots$ 进行探测。 -- **插入元素**：若哈希函数 $f_1(x)$ 出现冲突，则尝试 $f_2(x)$ ，以此类推，直到找到空位后插入元素。 -- **查找元素**：在相同的哈希函数顺序下进行查找，直到找到目标元素时返回；或遇到空位或已尝试所有哈希函数，说明哈希表中不存在该元素，则返回 $\text{None}$ 。 +- **插入元素**：若哈希函数 $f_1(x)$ 出现冲突，则尝试 $f_2(x)$ ，以此类推，直到找到空桶后插入元素。 +- **查找元素**：在相同的哈希函数顺序下进行查找，直到找到目标元素时返回；或当遇到空桶或已尝试所有哈希函数，说明哈希表中不存在该元素，则返回 $\text{None}$ 。 与线性探测相比，多次哈希方法不易产生聚集，但多个哈希函数会增加额外的计算量。 -## 编程语言的选择 +!!! tip + + 请注意，开放寻址（线性探测、平方探测和多次哈希）哈希表都存在“不能直接删除元素”的问题。 -Java 采用链式地址。自 JDK 1.8 以来，当 HashMap 内数组长度达到 64 且链表长度达到 8 时，链表会被转换为红黑树以提升查找性能。 +## 编程语言的选择 -Python 采用开放寻址。字典 dict 使用伪随机数进行探测。 +各个编程语言采取了不同的哈希表实现策略，以下举几个例子。 -Golang 采用链式地址。Go 规定每个桶最多存储 8 个键值对，超出容量则连接一个溢出桶；当溢出桶过多时，会执行一次特殊的等量扩容操作，以确保性能。 +- Java 采用链式地址。自 JDK 1.8 以来，当 HashMap 内数组长度达到 64 且链表长度达到 8 时，链表会被转换为红黑树以提升查找性能。 +- Python 采用开放寻址。字典 dict 使用伪随机数进行探测。 +- Golang 采用链式地址。Go 规定每个桶最多存储 8 个键值对，超出容量则连接一个溢出桶。当溢出桶过多时，会执行一次特殊的等量扩容操作，以确保性能。