CMU15445 (Fall 2019) 数据库系统 Project#2 - Hash Table 详解

前言

该实验要求实现一个基于线性探测法的哈希表，但是与直接放在内存中的哈希表不同的是，该实验假设哈希表非常大，无法整个放入内存中，因此需要将哈希表进行分割，将多个键值对放在一个 Page 中，然后搭配上一个实验实现的 Buffer Pool Manager 一起食用。哈希表的大致结构如下图所示：

下面介绍如何实现一个线程安全的哈希表。

代码实现

Page 布局

从上图可以看出，多个键值对被放在 Page 里面，作为 Page 的数据存在磁盘中。为了更好地组织和管理这些键值对，实验任务一要求我们实现两个类：HashTableHeaderPage 和 HashTableBlockPage，HashTableHeaderPage 保存着 block index 到 page id 的映射关系以及其他哈希表元数据，每个哈希表只有一个 HashTableHeaderPage，而 HashTableBlockPage 可以有多个。

Hash Table Header Page

HashTableHeaderPage 有以下几个字段：

字段	大小	描述
lsn_	4 bytes	Log sequence number (Used in Project 4)
size_	4 bytes	Number of Key & Value pairs the hash table can hold
page_id_	4 bytes	Self Page Id
next_ind_	4 bytes	The next index to add a new entry to block_page_ids_
block_page_ids_	4080 bytes	Array of block page_id_t

这些字段总共 4096 字节，正好是一个 Page 的大小，在 src/include/common/config.h 中可以修改 PAGE_SIZE 的大小。该类的实现代码如下：

namespace bustub {
page_id_t HashTableHeaderPage::GetBlockPageId(size_t index) {
  assert(index < next_ind_);
  return block_page_ids_[index];
}

page_id_t HashTableHeaderPage::GetPageId() const { return page_id_; }

void HashTableHeaderPage::SetPageId(bustub::page_id_t page_id) { page_id_ = page_id; }

lsn_t HashTableHeaderPage::GetLSN() const { return lsn_; }

void HashTableHeaderPage::SetLSN(lsn_t lsn) { lsn_ = lsn; }

void HashTableHeaderPage::AddBlockPageId(page_id_t page_id) { block_page_ids_[next_ind_++] = page_id; }

size_t HashTableHeaderPage::NumBlocks() { return next_ind_; }

void HashTableHeaderPage::SetSize(size_t size) { size_ = size; }

size_t HashTableHeaderPage::GetSize() const { return size_; }

}  // namespace bustub

Hash Table Block Page

HashTableBlockPage 包含多个 slot，用于保存键值对，所以该类定义了查询、插入和删除键值对的函数。为了跟踪每个 slot 的使用情况，该类包含以下三个数据成员：

• occupied_ : 第 i 位置 1 表示 Page 的第 i 个 slot 上存储了键值对或者之前存了键值对但之后被删除了（起到墓碑的作用）
• readable_ : 第 i 位置 1 表示 Page 的第 i 个 slot 上存储了键值对
• array_ : 用于保存键值对的数组

每个键值对的大小为 sizeof(std::pair<KeyType, ValueType>) 字节（记为 PS），每个键值对对应两个 bit（occupied 和 readable）即 1/4 个字节，所以一个 Page 最多能保存 BLOCK_ARRAY_SIZE = PAGE_SIZE / (PS + 1/4) 个键值对，即每个 Page 有 BLOCK_ARRAY_SIZE 个 slot。

由于 occupied_ 和 readable_ 被定义为 char 数组，所以需要两个辅助函数 GetBit 和 SetBit 来访问第 i 位的比特。

namespace bustub {
/**
 * Store indexed key and and value together within block page. Supports
 * non-unique keys.
 *
 * Block page format (keys are stored in order):
 *  ----------------------------------------------------------------
 * | KEY(1) + VALUE(1) | KEY(2) + VALUE(2) | ... | KEY(n) + VALUE(n)
 *  ----------------------------------------------------------------
 *
 *  Here '+' means concatenation.
 *
 */
template <typename KeyType, typename ValueType, typename KeyComparator>
class HashTableBlockPage {
 public:
  // Delete all constructor / destructor to ensure memory safety
  HashTableBlockPage() = delete;

  KeyType KeyAt(slot_offset_t bucket_ind) const;
  ValueType ValueAt(slot_offset_t bucket_ind) const;
  bool Insert(slot_offset_t bucket_ind, const KeyType &key, const ValueType &value);
  void Remove(slot_offset_t bucket_ind);
  bool IsOccupied(slot_offset_t bucket_ind) const;
  bool IsReadable(slot_offset_t bucket_ind) const;

 private:
  bool GetBit(const std::atomic_char *array, slot_offset_t bucket_ind) const;
  void SetBit(std::atomic_char *array, slot_offset_t bucket_ind, bool value);

  std::atomic_char occupied_[(BLOCK_ARRAY_SIZE - 1) / 8 + 1];

  // 0 if tombstone/brand new (never occupied), 1 otherwise.
  std::atomic_char readable_[(BLOCK_ARRAY_SIZE - 1) / 8 + 1];
  MappingType array_[0];
};

}  // namespace bustub

实现代码如下：

namespace bustub {

template <typename KeyType, typename ValueType, typename KeyComparator>
KeyType HASH_TABLE_BLOCK_TYPE::KeyAt(slot_offset_t bucket_ind) const {
  return array_[bucket_ind].first;
}

template <typename KeyType, typename ValueType, typename KeyComparator>
ValueType HASH_TABLE_BLOCK_TYPE::ValueAt(slot_offset_t bucket_ind) const {
  return array_[bucket_ind].second;
}

template <typename KeyType, typename ValueType, typename KeyComparator>
bool HASH_TABLE_BLOCK_TYPE::Insert(slot_offset_t bucket_ind, const KeyType &key, const ValueType &value) {
  if (IsReadable(bucket_ind)) {
    return false;
  }

  array_[bucket_ind] = {key, value};
  SetBit(readable_, bucket_ind, true);
  SetBit(occupied_, bucket_ind, true);
  return true;
}

template <typename KeyType, typename ValueType, typename KeyComparator>
void HASH_TABLE_BLOCK_TYPE::Remove(slot_offset_t bucket_ind) {
  SetBit(readable_, bucket_ind, false);
}

template <typename KeyType, typename ValueType, typename KeyComparator>
bool HASH_TABLE_BLOCK_TYPE::IsOccupied(slot_offset_t bucket_ind) const {
  return GetBit(occupied_, bucket_ind);
}

template <typename KeyType, typename ValueType, typename KeyComparator>
bool HASH_TABLE_BLOCK_TYPE::IsReadable(slot_offset_t bucket_ind) const {
  return GetBit(readable_, bucket_ind);
}

template <typename KeyType, typename ValueType, typename KeyComparator>
bool HASH_TABLE_BLOCK_TYPE::GetBit(const std::atomic_char *array, slot_offset_t bucket_ind) const {
  return array[bucket_ind / 8] & (1 << bucket_ind % 8);
}

template <typename KeyType, typename ValueType, typename KeyComparator>
void HASH_TABLE_BLOCK_TYPE::SetBit(std::atomic_char *array, slot_offset_t bucket_ind, bool value) {
  if (value) {
    array[bucket_ind / 8] |= (1 << bucket_ind % 8);
  } else {
    array[bucket_ind / 8] &= ~(1 << bucket_ind % 8);
  }
}

// DO NOT REMOVE ANYTHING BELOW THIS LINE
template class HashTableBlockPage<int, int, IntComparator>;
template class HashTableBlockPage<GenericKey<4>, RID, GenericComparator<4>>;
template class HashTableBlockPage<GenericKey<8>, RID, GenericComparator<8>>;
template class HashTableBlockPage<GenericKey<16>, RID, GenericComparator<16>>;
template class HashTableBlockPage<GenericKey<32>, RID, GenericComparator<32>>;
template class HashTableBlockPage<GenericKey<64>, RID, GenericComparator<64>>;

}  // namespace bustub

哈希表

声明

实验要求我们实现哈希表的插入、查找、删除和调整大小的的操作，对应的类声明如下，为了完成这些操作，我们多定义了几个私有的辅助函数和成员变量：

namespace bustub {

#define HASH_TABLE_TYPE LinearProbeHashTable<KeyType, ValueType, KeyComparator>

template <typename KeyType, typename ValueType, typename KeyComparator>
class LinearProbeHashTable : public HashTable<KeyType, ValueType, KeyComparator> {
 public:

  explicit LinearProbeHashTable(const std::string &name, BufferPoolManager *buffer_pool_manager,
                                const KeyComparator &comparator, size_t num_buckets, HashFunction<KeyType> hash_fn);

  bool Insert(Transaction *transaction, const KeyType &key, const ValueType &value) override;
  bool Remove(Transaction *transaction, const KeyType &key, const ValueType &value) override;
  bool GetValue(Transaction *transaction, const KeyType &key, std::vector<ValueType> *result) override;
  void Resize(size_t initial_size);
  size_t GetSize();

 private:
  using slot_index_t = size_t;
  using block_index_t = size_t;
  enum class LockType { READ = 0, WRITE = 1 };

  /**
   * initialize header page and allocate block pages for it
   * @param page the hash table header page
   */
  void InitHeaderPage(HashTableHeaderPage *page);

  /**
   * get index according to key
   * @param key the key to be hashed
   * @return a tuple contains slot index, block page index and bucket index
   */
  std::tuple<slot_index_t, block_index_t, slot_offset_t> GetIndex(const KeyType &key);

  /**
   * linear probe step forward
   * @param bucket_index the bucket index
   * @param block_index the hash table block page index
   * @param header_page hash table header page
   * @param raw_block_page raw hash table block page
   * @param block_page hash table block page
   * @param lock_type lock type of block page
   */
  void StepForward(slot_offset_t &bucket_index, block_index_t &block_index, Page *&raw_block_page,
                   HASH_TABLE_BLOCK_TYPE *&block_page, LockType lockType);

  bool InsertImpl(Transaction *transaction, const KeyType &key, const ValueType &value);

  inline bool IsMatch(HASH_TABLE_BLOCK_TYPE *block_page, slot_offset_t bucket_index, const KeyType &key,
                      const ValueType &value) {
    return !comparator_(key, block_page->KeyAt(bucket_index)) && value == block_page->ValueAt(bucket_index);
  }

  inline HashTableHeaderPage *HeaderPageCast(Page *page) {
    return reinterpret_cast<HashTableHeaderPage *>(page->GetData());
  }

  inline HASH_TABLE_BLOCK_TYPE *BlockPageCast(Page *page) {
    return reinterpret_cast<HASH_TABLE_BLOCK_TYPE *>(page->GetData());
  }

  /**
   * get the slot number of hash table block page
   * @param block_index the index of hash table block page
   * @return the slot number of block page
   */
  inline size_t GetBlockArraySize(block_index_t block_index){
    return block_index < num_pages_ - 1 ? BLOCK_ARRAY_SIZE : last_block_array_size_;
  }

  // member variable
  page_id_t header_page_id_;
  BufferPoolManager *buffer_pool_manager_;
  KeyComparator comparator_;
  std::vector<page_id_t> page_ids_;
  size_t num_buckets_;
  size_t num_pages_;
  size_t last_block_array_size_;

  // Readers includes inserts and removes, writer is only resize
  ReaderWriterLatch table_latch_;

  // Hash function
  HashFunction<KeyType> hash_fn_;
};

}  // namespace bustub

构造函数

在构造函数中负责根据用户指定的 num_buckets （也就是 slot 的数量）分配 Page，最后一个 Page 的 slot 数量可能少于前面的 Page。这里还将每个 HashTableBlockPage 对应的 page_id 保存到 page_ids_ 成员里面了，这样之后就不需要仅仅为了知道某个 HashTableBlockPage 的 page_id 而去找 BufferPoolManager 索要 HashTableHeaderPage。

template <typename KeyType, typename ValueType, typename KeyComparator>
HASH_TABLE_TYPE::LinearProbeHashTable(const std::string &name, BufferPoolManager *buffer_pool_manager,
                                      const KeyComparator &comparator, size_t num_buckets,
                                      HashFunction<KeyType> hash_fn)
    : buffer_pool_manager_(buffer_pool_manager),
      comparator_(comparator),
      num_buckets_(num_buckets),
      num_pages_((num_buckets - 1) / BLOCK_ARRAY_SIZE + 1),
      last_block_array_size_(num_buckets - (num_pages_ - 1) * BLOCK_ARRAY_SIZE),
      hash_fn_(std::move(hash_fn)) {
  auto page = buffer_pool_manager->NewPage(&header_page_id_);
  page->WLatch();

  InitHeaderPage(HeaderPageCast(page));

  page->WUnlatch();
  buffer_pool_manager_->UnpinPage(header_page_id_, true);
}

template <typename KeyType, typename ValueType, typename KeyComparator>
void HASH_TABLE_TYPE::InitHeaderPage(HashTableHeaderPage *header_page) {
  header_page->SetPageId(header_page_id_);
  header_page->SetSize(num_buckets_);

  page_ids_.clear();
  for (size_t i = 0; i < num_pages_; ++i) {
    page_id_t page_id;
    buffer_pool_manager_->NewPage(&page_id);
    buffer_pool_manager_->UnpinPage(page_id, false);
    header_page->AddBlockPageId(page_id);
    page_ids_.push_back(page_id);
  }
}

查找

哈希表使用线性探测法进行键值对的查找，由于实验要求哈希表支持插入同键不同值的键值对，所以在线性探测过程中需要将所有相同键的值插入 result 向量中：

template <typename KeyType, typename ValueType, typename KeyComparator>
bool HASH_TABLE_TYPE::GetValue(Transaction *transaction, const KeyType &key, std::vector<ValueType> *result) {
  table_latch_.RLock();

  // get slot index, block page index and bucket index according to key
  auto [slot_index, block_index, bucket_index] = GetIndex(key);

  // get block page that contains the key
  auto raw_block_page = buffer_pool_manager_->FetchPage(page_ids_[block_index]);
  raw_block_page->RLatch();
  auto block_page = BlockPageCast(raw_block_page);

  // linear probe
  while (block_page->IsOccupied(bucket_index)) {
    // find the correct position
    if (block_page->IsReadable(bucket_index) && !comparator_(key, block_page->KeyAt(bucket_index))) {
      result->push_back(block_page->ValueAt(bucket_index));
    }

    StepForward(bucket_index, block_index, raw_block_page, block_page, LockType::READ);

    // break loop if we have returned to original position
    if (block_index * BLOCK_ARRAY_SIZE + bucket_index == slot_index) {
      break;
    }
  }

  // unlock
  raw_block_page->RUnlatch();
  buffer_pool_manager_->UnpinPage(raw_block_page->GetPageId(), false);
  table_latch_.RUnlock();
  return result->size() > 0;
}

GetIndex 函数根据 key 计算出对应的 slot_index、block_index 和 bucket_index（就是 slot offset），结合上图就能理解该函数的工作原理了：

template <typename KeyType, typename ValueType, typename KeyComparator>
auto HASH_TABLE_TYPE::GetIndex(const KeyType &key) -> std::tuple<slot_index_t, block_index_t, slot_offset_t> {
  slot_index_t slot_index = hash_fn_.GetHash(key) % num_buckets_;
  block_index_t block_index = slot_index / BLOCK_ARRAY_SIZE;
  slot_offset_t bucket_index = slot_index % BLOCK_ARRAY_SIZE;
  return {slot_index, block_index, bucket_index};
}

在线性探测过程中，我们可能从从一个 HashTableBlockPage 跳到下一个，这时候需要更新 bucket_index 和 block_index：

template <typename KeyType, typename ValueType, typename KeyComparator>
void HASH_TABLE_TYPE::StepForward(slot_offset_t &bucket_index, block_index_t &block_index, Page *&raw_block_page,
                                  HASH_TABLE_BLOCK_TYPE *&block_page, LockType lockType) {
  if (++bucket_index != GetBlockArraySize(block_index)) {
    return;
  }

  // move to next block page
  if (lockType == LockType::READ) {
    raw_block_page->RUnlatch();
  } else {
    raw_block_page->WUnlatch();
  }
  buffer_pool_manager_->UnpinPage(page_ids_[block_index], false);

  // update index
  bucket_index = 0;
  block_index = (block_index + 1) % num_pages_;

  // update page
  raw_block_page = buffer_pool_manager_->FetchPage(page_ids_[block_index]);
  if (lockType == LockType::READ) {
    raw_block_page->RLatch();
  } else {
    raw_block_page->WLatch();
  }
  block_page = BlockPageCast(raw_block_page);
}

插入

实验要求哈希表不允许插入已经存在的键值对，同时插入过程中如果回到了最初的位置，说明没有可用的 slot 用于插入键值对，这时需要将哈希表的大小翻倍。由于 Resize 的函数也要用到插入操作，如果直接调用 Insert 会出现死锁，所以这里使用 InsertImpl 来实现插入：

template <typename KeyType, typename ValueType, typename KeyComparator>
bool HASH_TABLE_TYPE::Insert(Transaction *transaction, const KeyType &key, const ValueType &value) {
  table_latch_.RLock();
  auto success = InsertImpl(transaction, key, value);
  table_latch_.RUnlock();
  return success;
}

template <typename KeyType, typename ValueType, typename KeyComparator>
bool HASH_TABLE_TYPE::InsertImpl(Transaction *transaction, const KeyType &key, const ValueType &value) {
  // get slot index, block page index and bucket index according to key
  auto [slot_index, block_index, bucket_index] = GetIndex(key);

  // get block page that contains the key
  auto raw_block_page = buffer_pool_manager_->FetchPage(page_ids_[block_index]);
  raw_block_page->WLatch();
  auto block_page = BlockPageCast(raw_block_page);

  bool success = true;
  while (!block_page->Insert(bucket_index, key, value)) {
    // return false if (key, value) pair already exists
    if (block_page->IsReadable(bucket_index) && IsMatch(block_page, bucket_index, key, value)) {
      success = false;
      break;
    }

    StepForward(bucket_index, block_index, raw_block_page, block_page, LockType::WRITE);

    // resize hash table if we have returned to original position
    if (block_index * BLOCK_ARRAY_SIZE + bucket_index == slot_index) {
      raw_block_page->WUnlatch();
      buffer_pool_manager_->UnpinPage(raw_block_page->GetPageId(), false);

      table_latch_.RUnlock();
      Resize(num_buckets_);
      table_latch_.RLock();

      std::tie(slot_index, block_index, bucket_index) = GetIndex(key);

      raw_block_page = buffer_pool_manager_->FetchPage(page_ids_[block_index]);
      raw_block_page->WLatch();
      block_page = BlockPageCast(raw_block_page);
    }
  }

  raw_block_page->WUnlatch();
  buffer_pool_manager_->UnpinPage(raw_block_page->GetPageId(), success);
  return success;
}

调整大小

由于实验假设哈希表很大，所以我们不能将原本的键值对全部保存到内存中，然后调整 HashTableHeaderPage 的大小，复用 HashTableBlockPage 并创建新的 Page，再把键值对重新插入。而是应该直接创建新的 HashTableHeaderPage 和 HashTableBlockPage ，并删除旧的哈希表页：

template <typename KeyType, typename ValueType, typename KeyComparator>
void HASH_TABLE_TYPE::Resize(size_t initial_size) {
  table_latch_.WLock();
  num_buckets_ = 2 * initial_size;
  num_pages_ = (num_buckets_ - 1) / BLOCK_ARRAY_SIZE + 1;
  last_block_array_size_ = num_buckets_ - (num_pages_ - 1) * BLOCK_ARRAY_SIZE;

  // save the old header page id
  auto old_header_page_id = header_page_id_;
  std::vector<page_id_t> old_page_ids(page_ids_);

  // get the new header page
  auto raw_header_page = buffer_pool_manager_->NewPage(&header_page_id_);
  raw_header_page->WLatch();
  InitHeaderPage(HeaderPageCast(raw_header_page));

  // move (key, value) pairs to new space
  for (size_t block_index = 0; block_index < num_pages_; ++block_index) {
    auto old_page_id = old_page_ids[block_index];
    auto raw_block_page = buffer_pool_manager_->FetchPage(old_page_id);
    raw_block_page->RLatch();
    auto block_page = BlockPageCast(raw_block_page);

    // move (key, value) pair from each readable slot
    for (slot_offset_t bucket_index = 0; bucket_index < GetBlockArraySize(block_index); ++bucket_index) {
      if (block_page->IsReadable(bucket_index)) {
        InsertImpl(nullptr, block_page->KeyAt(bucket_index), block_page->ValueAt(bucket_index));
      }
    }

    // delete old page
    raw_block_page->RUnlatch();
    buffer_pool_manager_->UnpinPage(old_page_id, false);
    buffer_pool_manager_->DeletePage(old_page_id);
  }

  raw_header_page->WUnlatch();
  buffer_pool_manager_->UnpinPage(header_page_id_, false);
  buffer_pool_manager_->DeletePage(old_header_page_id);
  table_latch_.WUnlock();
}

删除

删除操作和查找操作很像，不过是将找到的 slot 标上墓碑罢了：

template <typename KeyType, typename ValueType, typename KeyComparator>
bool HASH_TABLE_TYPE::Remove(Transaction *transaction, const KeyType &key, const ValueType &value) {
  table_latch_.RLock();

  // get slot index, block page index and bucket index according to key
  auto [slot_index, block_index, bucket_index] = GetIndex(key);

  // get block page that contains the key
  auto raw_block_page = buffer_pool_manager_->FetchPage(page_ids_[block_index]);
  raw_block_page->WLatch();
  auto block_page = BlockPageCast(raw_block_page);

  bool success = false;
  while (block_page->IsOccupied(bucket_index)) {
    // remove the (key, value) pair if find the matched readable one
    if (IsMatch(block_page, bucket_index, key, value)) {
      if (block_page->IsReadable(bucket_index)) {
        block_page->Remove(bucket_index);
        success = true;
      } else {
        success = false;
      }
      break;
    }

    // step forward
    StepForward(bucket_index, block_index, raw_block_page, block_page, LockType::WRITE);

    // break loop if we have returned to original position
    if (block_index * BLOCK_ARRAY_SIZE + bucket_index == slot_index) {
      break;
    }
  }

  raw_block_page->WUnlatch();
  buffer_pool_manager_->UnpinPage(raw_block_page->GetPageId(), success);
  table_latch_.RUnlock();
  return success;
}

获取大小

最后是获取哈希表的大小操作，直接返回 num_buckets_ 就行了：

template <typename KeyType, typename ValueType, typename KeyComparator>
size_t HASH_TABLE_TYPE::GetSize() {
  table_latch_.RLock();
  size_t size = num_buckets_;
  table_latch_.RUnlock();
  return size;
}

测试

对哈希表的测试结果如下，6 个测试全部通过了：

总结

该实验主要考察对线性探测哈希表、缓冲池管理器和读写锁的理解，难度相比上一个实验略有提升，但是理解了哈希表的结构图之后应该就不难完成该实验了，以上~~