了解B+树的删除

其他 2021-03-20 23:43:00 阅读次数: 0

文章目录

    在本教程中，您将了解有关B+树的删除操作。此外，您还将找到C语言的示例。
    删除B+树上的元素包括三个主要事件：搜索要删除的键所在的节点、删除键和平衡树（如果有必要）。下溢是指节点中的键数少于其应持有的最小键数的情况。

删除操作

    在完成下面的步骤之前，必须了解关于m阶B+树的这些事实。

  1. 一个节点最多可以有m个子节点。
  2. 一个节点最多可以包含m-1个键。
  3. 一个节点至少应有⌈m/2⌉个子节点。
  4. 节点（根节点除外）至少应包含⌈m/2⌉-1个键。

    在删除键时，我们还必须注意内部节点（即索引）中的键，因为这些值在B+树中是冗余的。搜索要删除的键，然后执行以下步骤。

第一种情况

    要删除的键仅存在于叶节点，而不存在于索引（或内部节点）中。有两种情况：

  1. 节点中的键数超过了最小值。只需删除键。
    在这里插入图片描述
  2. 节点中有确切的最小键数。删除键并从直接同级借用键。将同级节点的中间键添加到父节点。
    在这里插入图片描述
第二种情况

    要删除的键存在于内部节点中，然后我们必须从内部节点中删除它们。这种情况有以下几种情况。

  1. 如果节点中的键数超过了最小值，只需从叶节点中删除键，并从内部节点中删除键。然后用顺序后继节点填充内部节点中的空白。
    在这里插入图片描述
  2. 如果节点中有确切的最小键数，则删除该键并从其直接同级（通过父级）借用一个键。然后用借用的键填充索引中的空白空间。
    在这里插入图片描述
  3. 这种情况与第二种情况（1）类似，但在这里，直连的父节点上方会生成空白空间。删除键后，将空白空间与其同级空间合并。然后用顺序后置节点填充祖父母节点中的空白。
    在这里插入图片描述
第三种情况

    在这种情况下，树的高度会缩小。有点复杂。删除下图树上的55导致了这种情况。可以在下面的插图中理解。
在这里插入图片描述

C示例
// Deletion on a B+ Tree in C

#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>

// Default order
#define ORDER 3

typedef struct record {
    
    
  int value;
} record;

// Node
typedef struct node {
    
    
  void **pointers;
  int *keys;
  struct node *parent;
  bool is_leaf;
  int num_keys;
  struct node *next;
} node;

int order = ORDER;
node *queue = NULL;
bool verbose_output = false;

// Enqueue
void enqueue(node *new_node);

// Dequeue
node *dequeue(void);
int height(node *const root);
int pathToLeaves(node *const root, node *child);
void printLeaves(node *const root);
void printTree(node *const root);
void findAndPrint(node *const root, int key, bool verbose);
void findAndPrintRange(node *const root, int range1, int range2, bool verbose);
int findRange(node *const root, int key_start, int key_end, bool verbose,
        int returned_keys[], void *returned_pointers[]);
node *findLeaf(node *const root, int key, bool verbose);
record *find(node *root, int key, bool verbose, node **leaf_out);
int cut(int length);

record *makeRecord(int value);
node *makeNode(void);
node *makeLeaf(void);
int getLeftIndex(node *parent, node *left);
node *insertIntoLeaf(node *leaf, int key, record *pointer);
node *insertIntoLeafAfterSplitting(node *root, node *leaf, int key,
                   record *pointer);
node *insertIntoNode(node *root, node *parent,
           int left_index, int key, node *right);
node *insertIntoNodeAfterSplitting(node *root, node *parent,
                   int left_index,
                   int key, node *right);
node *insertIntoParent(node *root, node *left, int key, node *right);
node *insertIntoNewRoot(node *left, int key, node *right);
node *startNewTree(int key, record *pointer);
node *insert(node *root, int key, int value);

// Enqueue
void enqueue(node *new_node) {
    
    
  node *c;
  if (queue == NULL) {
    
    
    queue = new_node;
    queue->next = NULL;
  } else {
    
    
    c = queue;
    while (c->next != NULL) {
    
    
      c = c->next;
    }
    c->next = new_node;
    new_node->next = NULL;
  }
}

// Dequeue
node *dequeue(void) {
    
    
  node *n = queue;
  queue = queue->next;
  n->next = NULL;
  return n;
}

// Print the leaves
void printLeaves(node *const root) {
    
    
  if (root == NULL) {
    
    
    printf("Empty tree.\n");
    return;
  }
  int i;
  node *c = root;
  while (!c->is_leaf)
    c = c->pointers[0];
  while (true) {
    
    
    for (i = 0; i < c->num_keys; i++) {
    
    
      if (verbose_output)
        printf("%p ", c->pointers[i]);
      printf("%d ", c->keys[i]);
    }
    if (verbose_output)
      printf("%p ", c->pointers[order - 1]);
    if (c->pointers[order - 1] != NULL) {
    
    
      printf(" | ");
      c = c->pointers[order - 1];
    } else
      break;
  }
  printf("\n");
}

// Calculate height
int height(node *const root) {
    
    
  int h = 0;
  node *c = root;
  while (!c->is_leaf) {
    
    
    c = c->pointers[0];
    h++;
  }
  return h;
}

// Get path to root
int pathToLeaves(node *const root, node *child) {
    
    
  int length = 0;
  node *c = child;
  while (c != root) {
    
    
    c = c->parent;
    length++;
  }
  return length;
}

// Print the tree
void printTree(node *const root) {
    
    
  node *n = NULL;
  int i = 0;
  int rank = 0;
  int new_rank = 0;

  if (root == NULL) {
    
    
    printf("Empty tree.\n");
    return;
  }
  queue = NULL;
  enqueue(root);
  while (queue != NULL) {
    
    
    n = dequeue();
    if (n->parent != NULL && n == n->parent->pointers[0]) {
    
    
      new_rank = pathToLeaves(root, n);
      if (new_rank != rank) {
    
    
        rank = new_rank;
        printf("\n");
      }
    }
    if (verbose_output)
      printf("(%p)", n);
    for (i = 0; i < n->num_keys; i++) {
    
    
      if (verbose_output)
        printf("%p ", n->pointers[i]);
      printf("%d ", n->keys[i]);
    }
    if (!n->is_leaf)
      for (i = 0; i <= n->num_keys; i++)
        enqueue(n->pointers[i]);
    if (verbose_output) {
    
    
      if (n->is_leaf)
        printf("%p ", n->pointers[order - 1]);
      else
        printf("%p ", n->pointers[n->num_keys]);
    }
    printf("| ");
  }
  printf("\n");
}

// Find the node and print it
void findAndPrint(node *const root, int key, bool verbose) {
    
    
  node *leaf = NULL;
  record *r = find(root, key, verbose, NULL);
  if (r == NULL)
    printf("Record not found under key %d.\n", key);
  else
    printf("Record at %p -- key %d, value %d.\n",
         r, key, r->value);
}

// Find and print the range
void findAndPrintRange(node *const root, int key_start, int key_end,
             bool verbose) {
    
    
  int i;
  int array_size = key_end - key_start + 1;
  int returned_keys[array_size];
  void *returned_pointers[array_size];
  int num_found = findRange(root, key_start, key_end, verbose,
                returned_keys, returned_pointers);
  if (!num_found)
    printf("None found.\n");
  else {
    
    
    for (i = 0; i < num_found; i++)
      printf("Key: %d   Location: %p  Value: %d\n",
           returned_keys[i],
           returned_pointers[i],
           ((record *)
            returned_pointers[i])
             ->value);
  }
}

// Find the range
int findRange(node *const root, int key_start, int key_end, bool verbose,
        int returned_keys[], void *returned_pointers[]) {
    
    
  int i, num_found;
  num_found = 0;
  node *n = findLeaf(root, key_start, verbose);
  if (n == NULL)
    return 0;
  for (i = 0; i < n->num_keys && n->keys[i] < key_start; i++)
    ;
  if (i == n->num_keys)
    return 0;
  while (n != NULL) {
    
    
    for (; i < n->num_keys && n->keys[i] <= key_end; i++) {
    
    
      returned_keys[num_found] = n->keys[i];
      returned_pointers[num_found] = n->pointers[i];
      num_found++;
    }
    n = n->pointers[order - 1];
    i = 0;
  }
  return num_found;
}

// Find the leaf
node *findLeaf(node *const root, int key, bool verbose) {
    
    
  if (root == NULL) {
    
    
    if (verbose)
      printf("Empty tree.\n");
    return root;
  }
  int i = 0;
  node *c = root;
  while (!c->is_leaf) {
    
    
    if (verbose) {
    
    
      printf("[");
      for (i = 0; i < c->num_keys - 1; i++)
        printf("%d ", c->keys[i]);
      printf("%d] ", c->keys[i]);
    }
    i = 0;
    while (i < c->num_keys) {
    
    
      if (key >= c->keys[i])
        i++;
      else
        break;
    }
    if (verbose)
      printf("%d ->\n", i);
    c = (node *)c->pointers[i];
  }
  if (verbose) {
    
    
    printf("Leaf [");
    for (i = 0; i < c->num_keys - 1; i++)
      printf("%d ", c->keys[i]);
    printf("%d] ->\n", c->keys[i]);
  }
  return c;
}

record *find(node *root, int key, bool verbose, node **leaf_out) {
    
    
  if (root == NULL) {
    
    
    if (leaf_out != NULL) {
    
    
      *leaf_out = NULL;
    }
    return NULL;
  }

  int i = 0;
  node *leaf = NULL;

  leaf = findLeaf(root, key, verbose);

  for (i = 0; i < leaf->num_keys; i++)
    if (leaf->keys[i] == key)
      break;
  if (leaf_out != NULL) {
    
    
    *leaf_out = leaf;
  }
  if (i == leaf->num_keys)
    return NULL;
  else
    return (record *)leaf->pointers[i];
}

int cut(int length) {
    
    
  if (length % 2 == 0)
    return length / 2;
  else
    return length / 2 + 1;
}

record *makeRecord(int value) {
    
    
  record *new_record = (record *)malloc(sizeof(record));
  if (new_record == NULL) {
    
    
    perror("Record creation.");
    exit(EXIT_FAILURE);
  } else {
    
    
    new_record->value = value;
  }
  return new_record;
}

node *makeNode(void) {
    
    
  node *new_node;
  new_node = malloc(sizeof(node));
  if (new_node == NULL) {
    
    
    perror("Node creation.");
    exit(EXIT_FAILURE);
  }
  new_node->keys = malloc((order - 1) * sizeof(int));
  if (new_node->keys == NULL) {
    
    
    perror("New node keys array.");
    exit(EXIT_FAILURE);
  }
  new_node->pointers = malloc(order * sizeof(void *));
  if (new_node->pointers == NULL) {
    
    
    perror("New node pointers array.");
    exit(EXIT_FAILURE);
  }
  new_node->is_leaf = false;
  new_node->num_keys = 0;
  new_node->parent = NULL;
  new_node->next = NULL;
  return new_node;
}

node *makeLeaf(void) {
    
    
  node *leaf = makeNode();
  leaf->is_leaf = true;
  return leaf;
}

int getLeftIndex(node *parent, node *left) {
    
    
  int left_index = 0;
  while (left_index <= parent->num_keys &&
       parent->pointers[left_index] != left)
    left_index++;
  return left_index;
}

node *insertIntoLeaf(node *leaf, int key, record *pointer) {
    
    
  int i, insertion_point;

  insertion_point = 0;
  while (insertion_point < leaf->num_keys && leaf->keys[insertion_point] < key)
    insertion_point++;

  for (i = leaf->num_keys; i > insertion_point; i--) {
    
    
    leaf->keys[i] = leaf->keys[i - 1];
    leaf->pointers[i] = leaf->pointers[i - 1];
  }
  leaf->keys[insertion_point] = key;
  leaf->pointers[insertion_point] = pointer;
  leaf->num_keys++;
  return leaf;
}

node *insertIntoLeafAfterSplitting(node *root, node *leaf, int key, record *pointer) {
    
    
  node *new_leaf;
  int *temp_keys;
  void **temp_pointers;
  int insertion_index, split, new_key, i, j;

  new_leaf = makeLeaf();

  temp_keys = malloc(order * sizeof(int));
  if (temp_keys == NULL) {
    
    
    perror("Temporary keys array.");
    exit(EXIT_FAILURE);
  }

  temp_pointers = malloc(order * sizeof(void *));
  if (temp_pointers == NULL) {
    
    
    perror("Temporary pointers array.");
    exit(EXIT_FAILURE);
  }

  insertion_index = 0;
  while (insertion_index < order - 1 && leaf->keys[insertion_index] < key)
    insertion_index++;

  for (i = 0, j = 0; i < leaf->num_keys; i++, j++) {
    
    
    if (j == insertion_index)
      j++;
    temp_keys[j] = leaf->keys[i];
    temp_pointers[j] = leaf->pointers[i];
  }

  temp_keys[insertion_index] = key;
  temp_pointers[insertion_index] = pointer;

  leaf->num_keys = 0;

  split = cut(order - 1);

  for (i = 0; i < split; i++) {
    
    
    leaf->pointers[i] = temp_pointers[i];
    leaf->keys[i] = temp_keys[i];
    leaf->num_keys++;
  }

  for (i = split, j = 0; i < order; i++, j++) {
    
    
    new_leaf->pointers[j] = temp_pointers[i];
    new_leaf->keys[j] = temp_keys[i];
    new_leaf->num_keys++;
  }

  free(temp_pointers);
  free(temp_keys);

  new_leaf->pointers[order - 1] = leaf->pointers[order - 1];
  leaf->pointers[order - 1] = new_leaf;

  for (i = leaf->num_keys; i < order - 1; i++)
    leaf->pointers[i] = NULL;
  for (i = new_leaf->num_keys; i < order - 1; i++)
    new_leaf->pointers[i] = NULL;

  new_leaf->parent = leaf->parent;
  new_key = new_leaf->keys[0];

  return insertIntoParent(root, leaf, new_key, new_leaf);
}

node *insertIntoNode(node *root, node *n,
           int left_index, int key, node *right) {
    
    
  int i;

  for (i = n->num_keys; i > left_index; i--) {
    
    
    n->pointers[i + 1] = n->pointers[i];
    n->keys[i] = n->keys[i - 1];
  }
  n->pointers[left_index + 1] = right;
  n->keys[left_index] = key;
  n->num_keys++;
  return root;
}

node *insertIntoNodeAfterSplitting(node *root, node *old_node, int left_index,
                   int key, node *right) {
    
    
  int i, j, split, k_prime;
  node *new_node, *child;
  int *temp_keys;
  node **temp_pointers;

  temp_pointers = malloc((order + 1) * sizeof(node *));
  if (temp_pointers == NULL) {
    
    
    exit(EXIT_FAILURE);
  }
  temp_keys = malloc(order * sizeof(int));
  if (temp_keys == NULL) {
    
    
    exit(EXIT_FAILURE);
  }

  for (i = 0, j = 0; i < old_node->num_keys + 1; i++, j++) {
    
    
    if (j == left_index + 1)
      j++;
    temp_pointers[j] = old_node->pointers[i];
  }

  for (i = 0, j = 0; i < old_node->num_keys; i++, j++) {
    
    
    if (j == left_index)
      j++;
    temp_keys[j] = old_node->keys[i];
  }

  temp_pointers[left_index + 1] = right;
  temp_keys[left_index] = key;

  split = cut(order);
  new_node = makeNode();
  old_node->num_keys = 0;
  for (i = 0; i < split - 1; i++) {
    
    
    old_node->pointers[i] = temp_pointers[i];
    old_node->keys[i] = temp_keys[i];
    old_node->num_keys++;
  }
  old_node->pointers[i] = temp_pointers[i];
  k_prime = temp_keys[split - 1];
  for (++i, j = 0; i < order; i++, j++) {
    
    
    new_node->pointers[j] = temp_pointers[i];
    new_node->keys[j] = temp_keys[i];
    new_node->num_keys++;
  }
  new_node->pointers[j] = temp_pointers[i];
  free(temp_pointers);
  free(temp_keys);
  new_node->parent = old_node->parent;
  for (i = 0; i <= new_node->num_keys; i++) {
    
    
    child = new_node->pointers[i];
    child->parent = new_node;
  }

  return insertIntoParent(root, old_node, k_prime, new_node);
}

node *insertIntoParent(node *root, node *left, int key, node *right) {
    
    
  int left_index;
  node *parent;

  parent = left->parent;

  if (parent == NULL)
    return insertIntoNewRoot(left, key, right);

  left_index = getLeftIndex(parent, left);

  if (parent->num_keys < order - 1)
    return insertIntoNode(root, parent, left_index, key, right);

  return insertIntoNodeAfterSplitting(root, parent, left_index, key, right);
}

node *insertIntoNewRoot(node *left, int key, node *right) {
    
    
  node *root = makeNode();
  root->keys[0] = key;
  root->pointers[0] = left;
  root->pointers[1] = right;
  root->num_keys++;
  root->parent = NULL;
  left->parent = root;
  right->parent = root;
  return root;
}

node *startNewTree(int key, record *pointer) {
    
    
  node *root = makeLeaf();
  root->keys[0] = key;
  root->pointers[0] = pointer;
  root->pointers[order - 1] = NULL;
  root->parent = NULL;
  root->num_keys++;
  return root;
}

node *insert(node *root, int key, int value) {
    
    
  record *record_pointer = NULL;
  node *leaf = NULL;

  record_pointer = find(root, key, false, NULL);
  if (record_pointer != NULL) {
    
    
    record_pointer->value = value;
    return root;
  }

  record_pointer = makeRecord(value);

  if (root == NULL)
    return startNewTree(key, record_pointer);

  leaf = findLeaf(root, key, false);

  if (leaf->num_keys < order - 1) {
    
    
    leaf = insertIntoLeaf(leaf, key, record_pointer);
    return root;
  }

  return insertIntoLeafAfterSplitting(root, leaf, key, record_pointer);
}

int main() {
    
    
  node *root;
  char instruction;

  root = NULL;

  root = insert(root, 5, 33);
  root = insert(root, 15, 21);
  root = insert(root, 25, 31);
  root = insert(root, 35, 41);
  root = insert(root, 45, 10);

  printTree(root);

  findAndPrint(root, 15, instruction = 'a');
}
参考文档

[1]Parewa Labs Pvt. Ltd.Deletion from a B+ Tree[EB/OL].https://www.programiz.com/dsa/deletion-from-a-b-plus-tree,2020-01-01.

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