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Urban Cummerata

Urban Cummerata

1 year ago

B-tree: A Balanced Search Tree with a Variable Order

B-tree is a special type of self-balancing search tree in which each node can contain more than one key and can have more than two children. It is a generalized form of the binary search tree.

It is also known as a height-balanced m-way tree.

B-tree example

B-tree

Why do you need a B-tree data structure?

The need for B-tree arose with the rise in the need for lesser time in accessing physical storage media like a hard disk. The secondary storage devices are slower with a larger capacity. There was a need for such types of data structures that minimize the disk access.

Other data structures such as a binary search tree, avl tree, red-black tree, etc can store only one key in one node. If you have to store a large number of keys, then the height of such trees becomes very large, and the access time increases.

However, B-tree can store many keys in a single node and can have multiple child nodes. This decreases the height significantly allowing faster disk accesses.

B-tree Properties

  1. For each node x, the keys are stored in increasing order.
  2. In each node, there is a boolean value x.leaf which is true if x is a leaf.
  3. If n is the order of the tree, each internal node can contain at most n - 1 keys along with a pointer to each child.
  4. Each node except root can have at most n children and at least n/2 children.
  5. All leaves have the same depth (i.e. height-h of the tree).
  6. The root has at least 2 children and contains a minimum of 1 key.
  7. If n ≥ 1, then for any n-key B-tree of height h and minimum degree t ≥ 2, h ≥ logt (n+1)/2.

Operations on a B-tree

Searching an element in a B-tree

Searching for an element in a B-tree is the generalized form of searching an element in a Binary Search Tree. The following steps are followed.

  1. Starting from the root node, compare k with the first key of the node.
    If k = the first key of the node, return the node and the index.
  2. If k.leaf = true, return NULL (i.e. not found).
  3. If k < the first key of the root node, search the left child of this key recursively.
  4. If there is more than one key in the current node and k > the first key, compare k with the next key in the node.
    If k < next key, search the left child of this key (ie. k lies in between the first and the second keys).
    Else, search the right child of the key.
  5. Repeat steps 1 to 4 until the leaf is reached.

Searching Example

  1. Let us search key k = 17 in the tree below of degree 3.B-treeB-tree
  2. k is not found in the root so, compare it with the root key.Not found on the root nodek is not found on the root node
  3. Since k > 11, go to the right child of the root node.Go to the right subtreeGo to the right subtree
  4. Compare k with 16. Since k > 16, compare k with the next key 18.Compare with the keys from left to rightCompare with the keys from left to right
  5. Since k < 18, k lies between 16 and 18. Search in the right child of 16 or the left child of 18.k lies in between 16 and 18k lies in between 16 and 18
  6. k is found.k is foundk is found

Algorithm for Searching an Element

BtreeSearch(x, k)
 i = 1
 while i ≤ n[x] and k ≥ keyi[x]        // n[x] means number of keys in x node
    do i = i + 1
if i  n[x] and k = keyi[x]
    then return (x, i)
if leaf [x]
    then return NIL
else
    return BtreeSearch(ci[x], k)

B-tree operations code in Python, Java, and C/C++

 

# Searching a key on a B-tree in Python


# Create a node
class BTreeNode:
  def __init__(self, leaf=False):
    self.leaf = leaf
    self.keys = []
    self.child = []


# Tree
class BTree:
  def __init__(self, t):
    self.root = BTreeNode(True)
    self.t = t

    # Insert node
  def insert(self, k):
    root = self.root
    if len(root.keys) == (2 * self.t) - 1:
      temp = BTreeNode()
      self.root = temp
      temp.child.insert(0, root)
      self.split_child(temp, 0)
      self.insert_non_full(temp, k)
    else:
      self.insert_non_full(root, k)

    # Insert nonfull
  def insert_non_full(self, x, k):
    i = len(x.keys) - 1
    if x.leaf:
      x.keys.append((None, None))
      while i >= 0 and k[0] < x.keys[i][0]:
        x.keys[i + 1] = x.keys[i]
        i -= 1
      x.keys[i + 1] = k
    else:
      while i >= 0 and k[0] < x.keys[i][0]:
        i -= 1
      i += 1
      if len(x.child[i].keys) == (2 * self.t) - 1:
        self.split_child(x, i)
        if k[0] > x.keys[i][0]:
          i += 1
      self.insert_non_full(x.child[i], k)

    # Split the child
  def split_child(self, x, i):
    t = self.t
    y = x.child[i]
    z = BTreeNode(y.leaf)
    x.child.insert(i + 1, z)
    x.keys.insert(i, y.keys[t - 1])
    z.keys = y.keys[t: (2 * t) - 1]
    y.keys = y.keys[0: t - 1]
    if not y.leaf:
      z.child = y.child[t: 2 * t]
      y.child = y.child[0: t - 1]

  # Print the tree
  def print_tree(self, x, l=0):
    print("Level ", l, " ", len(x.keys), end=":")
    for i in x.keys:
      print(i, end=" ")
    print()
    l += 1
    if len(x.child) > 0:
      for i in x.child:
        self.print_tree(i, l)

  # Search key in the tree
  def search_key(self, k, x=None):
    if x is not None:
      i = 0
      while i < len(x.keys) and k > x.keys[i][0]:
        i += 1
      if i < len(x.keys) and k == x.keys[i][0]:
        return (x, i)
      elif x.leaf:
        return None
      else:
        return self.search_key(k, x.child[i])
      
    else:
      return self.search_key(k, self.root)


def main():
  B = BTree(3)

  for i in range(10):
    B.insert((i, 2 * i))

  B.print_tree(B.root)

  if B.search_key(8) is not None:
    print("\nFound")
  else:
    print("\nNot Found")


if __name__ == '__main__':
  main()
  

// Searching a key on a B-tree in Java 

public class BTree {

  private int T;

  // Node creation
  public class Node {
    int n;
    int key[] = new int[2 * T - 1];
    Node child[] = new Node[2 * T];
    boolean leaf = true;

    public int Find(int k) {
      for (int i = 0; i < this.n; i++) {
        if (this.key[i] == k) {
          return i;
        }
      }
      return -1;
    };
  }

  public BTree(int t) {
    T = t;
    root = new Node();
    root.n = 0;
    root.leaf = true;
  }

  private Node root;

  // Search key
  private Node Search(Node x, int key) {
    int i = 0;
    if (x == null)
      return x;
    for (i = 0; i < x.n; i++) {
      if (key < x.key[i]) {
        break;
      }
      if (key == x.key[i]) {
        return x;
      }
    }
    if (x.leaf) {
      return null;
    } else {
      return Search(x.child[i], key);
    }
  }

  // Splitting the node
  private void Split(Node x, int pos, Node y) {
    Node z = new Node();
    z.leaf = y.leaf;
    z.n = T - 1;
    for (int j = 0; j < T - 1; j++) {
      z.key[j] = y.key[j + T];
    }
    if (!y.leaf) {
      for (int j = 0; j < T; j++) {
        z.child[j] = y.child[j + T];
      }
    }
    y.n = T - 1;
    for (int j = x.n; j >= pos + 1; j--) {
      x.child[j + 1] = x.child[j];
    }
    x.child[pos + 1] = z;

    for (int j = x.n - 1; j >= pos; j--) {
      x.key[j + 1] = x.key[j];
    }
    x.key[pos] = y.key[T - 1];
    x.n = x.n + 1;
  }

  // Inserting a value
  public void Insert(final int key) {
    Node r = root;
    if (r.n == 2 * T - 1) {
      Node s = new Node();
      root = s;
      s.leaf = false;
      s.n = 0;
      s.child[0] = r;
      Split(s, 0, r);
      insertValue(s, key);
    } else {
      insertValue(r, key);
    }
  }

  // Insert the node
  final private void insertValue(Node x, int k) {

    if (x.leaf) {
      int i = 0;
      for (i = x.n - 1; i >= 0 && k < x.key[i]; i--) {
        x.key[i + 1] = x.key[i];
      }
      x.key[i + 1] = k;
      x.n = x.n + 1;
    } else {
      int i = 0;
      for (i = x.n - 1; i >= 0 && k < x.key[i]; i--) {
      }
      ;
      i++;
      Node tmp = x.child[i];
      if (tmp.n == 2 * T - 1) {
        Split(x, i, tmp);
        if (k > x.key[i]) {
          i++;
        }
      }
      insertValue(x.child[i], k);
    }

  }

  public void Show() {
    Show(root);
  }

  // Display
  private void Show(Node x) {
    assert (x == null);
    for (int i = 0; i < x.n; i++) {
      System.out.print(x.key[i] + " ");
    }
    if (!x.leaf) {
      for (int i = 0; i < x.n + 1; i++) {
        Show(x.child[i]);
      }
    }
  }

  // Check if present
  public boolean Contain(int k) {
    if (this.Search(root, k) != null) {
      return true;
    } else {
      return false;
    }
  }

  public static void main(String[] args) {
    BTree b = new BTree(3);
    b.Insert(8);
    b.Insert(9);
    b.Insert(10);
    b.Insert(11);
    b.Insert(15);
    b.Insert(20);
    b.Insert(17);

    b.Show();

    if (b.Contain(12)) {
      System.out.println("\nfound");
    } else {
      System.out.println("\nnot found");
    }
    ;
  }
}
// Searching a key on a B-tree in C

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

#define MAX 3
#define MIN 2

struct BTreeNode {
  int val[MAX + 1], count;
  struct BTreeNode *link[MAX + 1];
};

struct BTreeNode *root;

// Create a node
struct BTreeNode *createNode(int val, struct BTreeNode *child) {
  struct BTreeNode *newNode;
  newNode = (struct BTreeNode *)malloc(sizeof(struct BTreeNode));
  newNode->val[1] = val;
  newNode->count = 1;
  newNode->link[0] = root;
  newNode->link[1] = child;
  return newNode;
}

// Insert node
void insertNode(int val, int pos, struct BTreeNode *node,
        struct BTreeNode *child) {
  int j = node->count;
  while (j > pos) {
    node->val[j + 1] = node->val[j];
    node->link[j + 1] = node->link[j];
    j--;
  }
  node->val[j + 1] = val;
  node->link[j + 1] = child;
  node->count++;
}

// Split node
void splitNode(int val, int *pval, int pos, struct BTreeNode *node,
         struct BTreeNode *child, struct BTreeNode **newNode) {
  int median, j;

  if (pos > MIN)
    median = MIN + 1;
  else
    median = MIN;

  *newNode = (struct BTreeNode *)malloc(sizeof(struct BTreeNode));
  j = median + 1;
  while (j <= MAX) {
    (*newNode)->val[j - median] = node->val[j];
    (*newNode)->link[j - median] = node->link[j];
    j++;
  }
  node->count = median;
  (*newNode)->count = MAX - median;

  if (pos <= MIN) {
    insertNode(val, pos, node, child);
  } else {
    insertNode(val, pos - median, *newNode, child);
  }
  *pval = node->val[node->count];
  (*newNode)->link[0] = node->link[node->count];
  node->count--;
}

// Set the value
int setValue(int val, int *pval,
           struct BTreeNode *node, struct BTreeNode **child) {
  int pos;
  if (!node) {
    *pval = val;
    *child = NULL;
    return 1;
  }

  if (val < node->val[1]) {
    pos = 0;
  } else {
    for (pos = node->count;
       (val < node->val[pos] && pos > 1); pos--)
      ;
    if (val == node->val[pos]) {
      printf("Duplicates are not permitted\n");
      return 0;
    }
  }
  if (setValue(val, pval, node->link[pos], child)) {
    if (node->count < MAX) {
      insertNode(*pval, pos, node, *child);
    } else {
      splitNode(*pval, pval, pos, node, *child, child);
      return 1;
    }
  }
  return 0;
}

// Insert the value
void insert(int val) {
  int flag, i;
  struct BTreeNode *child;

  flag = setValue(val, &i, root, &child);
  if (flag)
    root = createNode(i, child);
}

// Search node
void search(int val, int *pos, struct BTreeNode *myNode) {
  if (!myNode) {
    return;
  }

  if (val < myNode->val[1]) {
    *pos = 0;
  } else {
    for (*pos = myNode->count;
       (val < myNode->val[*pos] && *pos > 1); (*pos)--)
      ;
    if (val == myNode->val[*pos]) {
      printf("%d is found", val);
      return;
    }
  }
  search(val, pos, myNode->link[*pos]);

  return;
}

// Traverse then nodes
void traversal(struct BTreeNode *myNode) {
  int i;
  if (myNode) {
    for (i = 0; i < myNode->count; i++) {
      traversal(myNode->link[i]);
      printf("%d ", myNode->val[i + 1]);
    }
    traversal(myNode->link[i]);
  }
}

int main() {
  int val, ch;

  insert(8);
  insert(9);
  insert(10);
  insert(11);
  insert(15);
  insert(16);
  insert(17);
  insert(18);
  insert(20);
  insert(23);

  traversal(root);

  printf("\n");
  search(11, &ch, root);
}
// Searching a key on a B-tree in C++

#include <iostream>
using namespace std;

class TreeNode {
  int *keys;
  int t;
  TreeNode **C;
  int n;
  bool leaf;

   public:
  TreeNode(int temp, bool bool_leaf);

  void insertNonFull(int k);
  void splitChild(int i, TreeNode *y);
  void traverse();

  TreeNode *search(int k);

  friend class BTree;
};

class BTree {
  TreeNode *root;
  int t;

   public:
  BTree(int temp) {
    root = NULL;
    t = temp;
  }

  void traverse() {
    if (root != NULL)
      root->traverse();
  }

  TreeNode *search(int k) {
    return (root == NULL) ? NULL : root->search(k);
  }

  void insert(int k);
};

TreeNode::TreeNode(int t1, bool leaf1) {
  t = t1;
  leaf = leaf1;

  keys = new int[2 * t - 1];
  C = new TreeNode *[2 * t];

  n = 0;
}

void TreeNode::traverse() {
  int i;
  for (i = 0; i < n; i++) {
    if (leaf == false)
      C[i]->traverse();
    cout << " " << keys[i];
  }

  if (leaf == false)
    C[i]->traverse();
}

TreeNode *TreeNode::search(int k) {
  int i = 0;
  while (i < n && k > keys[i])
    i++;

  if (keys[i] == k)
    return this;

  if (leaf == true)
    return NULL;

  return C[i]->search(k);
}

void BTree::insert(int k) {
  if (root == NULL) {
    root = new TreeNode(t, true);
    root->keys[0] = k;
    root->n = 1;
  } else {
    if (root->n == 2 * t - 1) {
      TreeNode *s = new TreeNode(t, false);

      s->C[0] = root;

      s->splitChild(0, root);

      int i = 0;
      if (s->keys[0] < k)
        i++;
      s->C[i]->insertNonFull(k);

      root = s;
    } else
      root->insertNonFull(k);
  }
}

void TreeNode::insertNonFull(int k) {
  int i = n - 1;

  if (leaf == true) {
    while (i >= 0 && keys[i] > k) {
      keys[i + 1] = keys[i];
      i--;
    }

    keys[i + 1] = k;
    n = n + 1;
  } else {
    while (i >= 0 && keys[i] > k)
      i--;

    if (C[i + 1]->n == 2 * t - 1) {
      splitChild(i + 1, C[i + 1]);

      if (keys[i + 1] < k)
        i++;
    }
    C[i + 1]->insertNonFull(k);
  }
}

void TreeNode::splitChild(int i, TreeNode *y) {
  TreeNode *z = new TreeNode(y->t, y->leaf);
  z->n = t - 1;

  for (int j = 0; j < t - 1; j++)
    z->keys[j] = y->keys[j + t];

  if (y->leaf == false) {
    for (int j = 0; j < t; j++)
      z->C[j] = y->C[j + t];
  }

  y->n = t - 1;
  for (int j = n; j >= i + 1; j--)
    C[j + 1] = C[j];

  C[i + 1] = z;

  for (int j = n - 1; j >= i; j--)
    keys[j + 1] = keys[j];

  keys[i] = y->keys[t - 1];
  n = n + 1;
}

int main() {
  BTree t(3);
  t.insert(8);
  t.insert(9);
  t.insert(10);
  t.insert(11);
  t.insert(15);
  t.insert(16);
  t.insert(17);
  t.insert(18);
  t.insert(20);
  t.insert(23);

  cout << "The B-tree is: ";
  t.traverse();

  int k = 10;
  (t.search(k) != NULL) ? cout << endl
                 << k << " is found"
              : cout << endl
                 << k << " is not Found";

  k = 2;
  (t.search(k) != NULL) ? cout << endl
                 << k << " is found"
              : cout << endl
                 << k << " is not Found\n";
}

Searching Complexity on B Tree

Worst case Time complexity: Θ(log n)

Average case Time complexity: Θ(log n)

Best case Time complexity: Θ(log n)

Average case Space complexity: Θ(n)

Worst case Space complexity: Θ(n)

B Tree Applications

  • databases and file systems
  • to store blocks of data (secondary storage media)
  • multilevel indexing
  • This blog post was originally published at:https://www.programiz.com/

#data  #dsa  #algorithm 

B-tree: A Balanced Search Tree with a Variable Order
1.05 GEEK