Python Data Structure and Algorithm Tutorial – Deletion from a B+ Tree

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Deletion from a B+ Tree

 

In this tutorial, you will learn about deletion operation on a B+ tree. Also, you will find working examples of deleting elements from a B+ tree in C, C++, Java and Python.

Deleting an element on a B+ tree consists of three main events: searching the node where the key to be deleted exists, deleting the key and balancing the tree if required.Underflow is a situation when there is less number of keys in a node than the minimum number of keys it should hold.


Deletion Operation

Before going through the steps below, one must know these facts about a B+ tree of degree m.

  1. A node can have a maximum of m children. (i.e. 3)
  2. A node can contain a maximum of m - 1 keys. (i.e. 2)
  3. A node should have a minimum of ⌈m/2⌉ children. (i.e. 2)
  4. A node (except root node) should contain a minimum of ⌈m/2⌉ - 1 keys. (i.e. 1)

 

While deleting a key, we have to take care of the keys present in the internal nodes (i.e. indexes) as well because the values are redundant in a B+ tree. Search the key to be deleted then follow the following steps.

 

Case I

The key to be deleted is present only at the leaf node not in the indexes (or internal nodes). There are two cases for it:

  1. There is more than the minimum number of keys in the node. Simply delete the key.
    Delete a key from a B+tree
    Deleting 40 from B-tree
  2. There is an exact minimum number of keys in the node. Delete the key and borrow a key from the immediate sibling. Add the median key of the sibling node to the parent.
    Delete a key from a B+tree
    Deleting 5 from B-tree

Case II

The key to be deleted is present in the internal nodes as well. Then we have to remove them from the internal nodes as well. There are the following cases for this situation.

  1. If there is more than the minimum number of keys in the node, simply delete the key from the leaf node and delete the key from the internal node as well.
    Fill the empty space in the internal node with the inorder successor.

    Delete a key from a B+tree
    Deleting 45 from B-tree
  2. If there is an exact minimum number of keys in the node, then delete the key and borrow a key from its immediate sibling (through the parent).
    Fill the empty space created in the index (internal node) with the borrowed key.

    Delete a key from a B+tree
    Deleting 35 from B-tree
  3. This case is similar to Case II(1) but here, empty space is generated above the immediate parent node.
    After deleting the key, merge the empty space with its sibling.
    Fill the empty space in the grandparent node with the inorder successor.

    Delete a key from a B+tree
    Deleting 25 from B-tree

Case III

In this case, the height of the tree gets shrinked. It is a little complicated.Deleting 55 from the tree below leads to this condition. It can be understood in the illustrations below.

Delete a key from a B+tree
Deleting 55 from B-tree

Python Examples

/* B+ tee in python */


import math

/* Node creation */
class Node:
    def __init__(self, order):
        self.order = order
        self.values = []
        self.keys = []
        self.nextKey = None
        self.parent = None
        self.check_leaf = False

    /* Insert at the leaf */
    def insert_at_leaf(self, leaf, value, key):
        if (self.values):
            temp1 = self.values
            for i in range(len(temp1)):
                if (value == temp1[i]):
                    self.keys[i].append(key)
                    break
                elif (value < temp1[i]):
                    self.values = self.values[:i] + [value] + self.values[i:]
                    self.keys = self.keys[:i] + [[key]] + self.keys[i:]
                    break
                elif (i + 1 == len(temp1)):
                    self.values.append(value)
                    self.keys.append([key])
                    break
        else:
            self.values = [value]
            self.keys = [[key]]


/* B plus tree */
class BplusTree:
    def __init__(self, order):
        self.root = Node(order)
        self.root.check_leaf = True

    /* Insert operation */
    def insert(self, value, key):
        value = str(value)
        old_node = self.search(value)
        old_node.insert_at_leaf(old_node, value, key)

        if (len(old_node.values) == old_node.order):
            node1 = Node(old_node.order)
            node1.check_leaf = True
            node1.parent = old_node.parent
            mid = int(math.ceil(old_node.order / 2)) - 1
            node1.values = old_node.values[mid + 1:]
            node1.keys = old_node.keys[mid + 1:]
            node1.nextKey = old_node.nextKey
            old_node.values = old_node.values[:mid + 1]
            old_node.keys = old_node.keys[:mid + 1]
            old_node.nextKey = node1
            self.insert_in_parent(old_node, node1.values[0], node1)

    /* Search operation for different operations */
    def search(self, value):
        current_node = self.root
        while(current_node.check_leaf == False):
            temp2 = current_node.values
            for i in range(len(temp2)):
                if (value == temp2[i]):
                    current_node = current_node.keys[i + 1]
                    break
                elif (value < temp2[i]):
                    current_node = current_node.keys[i]
                    break
                elif (i + 1 == len(current_node.values)):
                    current_node = current_node.keys[i + 1]
                    break
        return current_node

    /* Find the node */
    def find(self, value, key):
        l = self.search(value)
        for i, item in enumerate(l.values):
            if item == value:
                if key in l.keys[i]:
                    return True
                else:
                    return False
        return False

    /* Inserting at the parent */
    def insert_in_parent(self, n, value, ndash):
        if (self.root == n):
            rootNode = Node(n.order)
            rootNode.values = [value]
            rootNode.keys = [n, ndash]
            self.root = rootNode
            n.parent = rootNode
            ndash.parent = rootNode
            return

        parentNode = n.parent
        temp3 = parentNode.keys
        for i in range(len(temp3)):
            if (temp3[i] == n):
                parentNode.values = parentNode.values[:i] + 
                    [value] + parentNode.values[i:]
                parentNode.keys = parentNode.keys[:i +
                                                  1] + [ndash] + parentNode.keys[i + 1:]
                if (len(parentNode.keys) > parentNode.order):
                    parentdash = Node(parentNode.order)
                    parentdash.parent = parentNode.parent
                    mid = int(math.ceil(parentNode.order / 2)) - 1
                    parentdash.values = parentNode.values[mid + 1:]
                    parentdash.keys = parentNode.keys[mid + 1:]
                    value_ = parentNode.values[mid]
                    if (mid == 0):
                        parentNode.values = parentNode.values[:mid + 1]
                    else:
                        parentNode.values = parentNode.values[:mid]
                    parentNode.keys = parentNode.keys[:mid + 1]
                    for j in parentNode.keys:
                        j.parent = parentNode
                    for j in parentdash.keys:
                        j.parent = parentdash
                    self.insert_in_parent(parentNode, value_, parentdash)

    /* Delete a node */
    def delete(self, value, key):
        node_ = self.search(value)

        temp = 0
        for i, item in enumerate(node_.values):
            if item == value:
                temp = 1

                if key in node_.keys[i]:
                    if len(node_.keys[i]) > 1:
                        node_.keys[i].pop(node_.keys[i].index(key))
                    elif node_ == self.root:
                        node_.values.pop(i)
                        node_.keys.pop(i)
                    else:
                        node_.keys[i].pop(node_.keys[i].index(key))
                        del node_.keys[i]
                        node_.values.pop(node_.values.index(value))
                        self.deleteEntry(node_, value, key)
                else:
                    print("Value not in Key")
                    return
        if temp == 0:
            print("Value not in Tree")
            return

    /* Delete an entry */
    def deleteEntry(self, node_, value, key):

        if not node_.check_leaf:
            for i, item in enumerate(node_.keys):
                if item == key:
                    node_.keys.pop(i)
                    break
            for i, item in enumerate(node_.values):
                if item == value:
                    node_.values.pop(i)
                    break

        if self.root == node_ and len(node_.keys) == 1:
            self.root = node_.keys[0]
            node_.keys[0].parent = None
            del node_
            return
        elif (len(node_.keys) < int(math.ceil(node_.order / 2)) and node_.check_leaf == False) or (len(node_.values) < int(math.ceil((node_.order - 1) / 2)) and node_.check_leaf == True):

            is_predecessor = 0
            parentNode = node_.parent
            PrevNode = -1
            NextNode = -1
            PrevK = -1
            PostK = -1
            for i, item in enumerate(parentNode.keys):

                if item == node_:
                    if i > 0:
                        PrevNode = parentNode.keys[i - 1]
                        PrevK = parentNode.values[i - 1]

                    if i < len(parentNode.keys) - 1:
                        NextNode = parentNode.keys[i + 1]
                        PostK = parentNode.values[i]

            if PrevNode == -1:
                ndash = NextNode
                value_ = PostK
            elif NextNode == -1:
                is_predecessor = 1
                ndash = PrevNode
                value_ = PrevK
            else:
                if len(node_.values) + len(NextNode.values) < node_.order:
                    ndash = NextNode
                    value_ = PostK
                else:
                    is_predecessor = 1
                    ndash = PrevNode
                    value_ = PrevK

            if len(node_.values) + len(ndash.values) < node_.order:
                if is_predecessor == 0:
                    node_, ndash = ndash, node_
                ndash.keys += node_.keys
                if not node_.check_leaf:
                    ndash.values.append(value_)
                else:
                    ndash.nextKey = node_.nextKey
                ndash.values += node_.values

                if not ndash.check_leaf:
                    for j in ndash.keys:
                        j.parent = ndash

                self.deleteEntry(node_.parent, value_, node_)
                del node_
            else:
                if is_predecessor == 1:
                    if not node_.check_leaf:
                        ndashpm = ndash.keys.pop(-1)
                        ndashkm_1 = ndash.values.pop(-1)
                        node_.keys = [ndashpm] + node_.keys
                        node_.values = [value_] + node_.values
                        parentNode = node_.parent
                        for i, item in enumerate(parentNode.values):
                            if item == value_:
                                p.values[i] = ndashkm_1
                                break
                    else:
                        ndashpm = ndash.keys.pop(-1)
                        ndashkm = ndash.values.pop(-1)
                        node_.keys = [ndashpm] + node_.keys
                        node_.values = [ndashkm] + node_.values
                        parentNode = node_.parent
                        for i, item in enumerate(p.values):
                            if item == value_:
                                parentNode.values[i] = ndashkm
                                break
                else:
                    if not node_.check_leaf:
                        ndashp0 = ndash.keys.pop(0)
                        ndashk0 = ndash.values.pop(0)
                        node_.keys = node_.keys + [ndashp0]
                        node_.values = node_.values + [value_]
                        parentNode = node_.parent
                        for i, item in enumerate(parentNode.values):
                            if item == value_:
                                parentNode.values[i] = ndashk0
                                break
                    else:
                        ndashp0 = ndash.keys.pop(0)
                        ndashk0 = ndash.values.pop(0)
                        node_.keys = node_.keys + [ndashp0]
                        node_.values = node_.values + [ndashk0]
                        parentNode = node_.parent
                        for i, item in enumerate(parentNode.values):
                            if item == value_:
                                parentNode.values[i] = ndash.values[0]
                                break

                if not ndash.check_leaf:
                    for j in ndash.keys:
                        j.parent = ndash
                if not node_.check_leaf:
                    for j in node_.keys:
                        j.parent = node_
                if not parentNode.check_leaf:
                    for j in parentNode.keys:
                        j.parent = parentNode


/* Print the tree */
def printTree(tree):
    lst = [tree.root]
    level = [0]
    leaf = None
    flag = 0
    lev_leaf = 0

    node1 = Node(str(level[0]) + str(tree.root.values))

    while (len(lst) != 0):
        x = lst.pop(0)
        lev = level.pop(0)
        if (x.check_leaf == False):
            for i, item in enumerate(x.keys):
                print(item.values)
        else:
            for i, item in enumerate(x.keys):
                print(item.values)
            if (flag == 0):
                lev_leaf = lev
                leaf = x
                flag = 1


record_len = 3
bplustree = BplusTree(record_len)
bplustree.insert('5', '33')
bplustree.insert('15', '21')
bplustree.insert('25', '31')
bplustree.insert('35', '41')
bplustree.insert('45', '10')

printTree(bplustree)

if(bplustree.find('5', '34')):
    print("Found")
else:
    print("Not found")

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