CMU15445 (Fall 2019) 之 Project#2 - Hash Table 詳解

前言

該實驗要求實現一個基於線性探測法的雜湊表，但是與直接放在記憶體中的雜湊表不同的是，該實驗假設雜湊表非常大，無法整個放入記憶體中，因此需要將雜湊表進行分割，將多個鍵值對放在一個 Page 中，然後搭配上一個實驗實現的 Buffer Pool Manager 一起食用。雜湊表的大致結構如下圖所示：

下面介紹如何實現一個執行緒安全的雜湊表。

程式碼實現

Page 佈局

從上圖可以看出，多個鍵值對被放在 Page 裡面，作為 Page 的資料存在磁碟中。為了更好地組織和管理這些鍵值對，實驗任務一要求我們實現兩個類：HashTableHeaderPage 和 HashTableBlockPage，HashTableHeaderPage 儲存著 block index 到 page id 的對映關係以及其他雜湊表後設資料，每個雜湊表只有一個 HashTableHeaderPage，而 HashTableBlockPage 可以有多個。

Hash Table Header Page

HashTableHeaderPage 有以下幾個欄位：

欄位	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

這些欄位總共 4096 位元組，正好是一個 Page 的大小，在 src/include/common/config.h 中可以修改 PAGE_SIZE 的大小。該類的實現程式碼如下：

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 包含多個 slot，用於儲存鍵值對，所以該類定義了查詢、插入和刪除鍵值對的函式。為了跟蹤每個 slot 的使用情況，該類包含以下三個資料成員：

• occupied_ : 第 i 位置 1 表示 Page 的第 i 個 slot 上儲存了鍵值對或者之前存了鍵值對但之後被刪除了（起到墓碑的作用）
• readable_ : 第 i 位置 1 表示 Page 的第 i 個 slot 上儲存了鍵值對
• array_ : 用於儲存鍵值對的陣列

每個鍵值對的大小為 sizeof(std::pair<KeyType, ValueType>) 位元組（記為 PS），每個鍵值對對應兩個 bit（occupied 和 readable）即 1/4 個位元組，所以一個 Page 最多能儲存 BLOCK_ARRAY_SIZE = PAGE_SIZE / (PS + 1/4) 個鍵值對，即每個 Page 有 BLOCK_ARRAY_SIZE 個 slot。

由於 occupied_ 和 readable_ 被定義為 char 陣列，所以需要兩個輔助函式 GetBit 和 SetBit 來訪問第 i 位的位元。

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

實現程式碼如下：

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

雜湊表

宣告

實驗要求我們實現雜湊表的插入、查詢、刪除和調整大小的的操作，對應的類宣告如下，為了完成這些操作，我們多定義了幾個私有的輔助函式和成員變數：

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

建構函式

在建構函式中負責根據使用者指定的 num_buckets （也就是 slot 的數量）分配 Page，最後一個 Page 的 slot 數量可能少於前面的 Page。這裡還將每個 HashTableBlockPage 對應的 page_id 儲存到 page_ids_ 成員裡面了，這樣之後就不需要僅僅為了知道某個 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);
  }
}

查詢

雜湊表使用線性探測法進行鍵值對的查詢，由於實驗要求雜湊表支援插入同鍵不同值的鍵值對，所以線上性探測過程中需要將所有相同鍵的值插入 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 函式根據 key 計算出對應的 slot_index、block_index 和 bucket_index（就是 slot offset），結合上圖就能理解該函式的工作原理了：

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};
}

線上性探測過程中，我們可能從從一個 HashTableBlockPage 跳到下一個，這時候需要更新 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

實驗要求雜湊表不允許插入已經存在的鍵值對，同時插入過程中如果回到了最初的位置，說明沒有可用的 slot 用於插入鍵值對，這時需要將雜湊表的大小翻倍。由於 Resize 的函式也要用到插入操作，如果直接呼叫 Insert 會出現死鎖，所以這裡使用 InsertImpl 來實現插入：

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);

      Resize(num_pages_);
      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;
}

調整大小

由於實驗假設雜湊表很大，所以我們不能將原本的鍵值對全部儲存到記憶體中，然後調整 HashTableHeaderPage 的大小，複用 HashTableBlockPage 並建立新的 Page，再把鍵值對重新插入。而是應該直接建立新的 HashTableHeaderPage 和 HashTableBlockPage ，並刪除舊的雜湊表頁：

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();
}

刪除

刪除操作和查詢操作很像，不過是將找到的 slot 標上墓碑罷了：

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;
}

獲取大小

最後是獲取雜湊表的大小操作，直接返回 num_buckets_ 就行了：

template <typename KeyType, typename ValueType, typename KeyComparator>
size_t HASH_TABLE_TYPE::GetSize() {
  return num_buckets_;
}

測試

對雜湊表的測試結果如下，6 個測試全部通過了：

總結

該實驗主要考察對線性探測雜湊表、緩衝池管理器和讀寫鎖的理解，難度相比上一個實驗略有提升，但是理解了雜湊表的結構圖之後應該就不難完成該實驗了，以上~~