Python Data Structure and Algorithm Tutorial – Decrease Key and Delete Node Operations on a Fibonacci Heap

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Decrease Key and Delete Node Operations on a Fibonacci Heap

 

In this tutorial, you will learn how decrease key and delete node operations work. Also, you will find working examples of these operations on a fibonacci heap in Python.

In a fibonacci heap, decrease-key and delete-node are important operations. These operations are discussed below.


Decreasing a Key

In decreasing a key operation, the value of a key is decreased to a lower value.

Following functions are used for decreasing the key.

Decrease-Key

  1. Select the node to be decreased, x, and change its value to the new value k.
  2. If the parent of xy, is not null and the key of parent is greater than that of the k then call Cut(x) and Cascading-Cut(y) subsequently.
  3. If the key of x is smaller than the key of min, then mark x as min.

 

Cut

  1. Remove x from the current position and add it to the root list.
  2. If x is marked, then mark it as false.

 

Cascading-Cut

  1. If the parent of y is not null then follow the following steps.
  2. If y is unmarked, then mark y.
  3. Else, call Cut(y) and Cascading-Cut(parent of y).

 


Decrease Key Example

The above operations can be understood in the examples below.

Example: Decreasing 46 to 15.

  1. Decrease the value 46 to 15.
    Decrease 46 to 15
    Decrease 46 to 15
  2. Cut part: Since 24 ≠ nill and 15 < its parent, cut it and add it to the root list. Cascading-Cut part: mark 24.
    Cut and Cascading part
    Add 15 to root list and mark 24

Example: Decreasing 35 to 5

  1. Decrease the value 35 to 5.
    Decrease 35 to 5
    Decrease 35 to 5
  2. Cut part: Since 26 ≠ nill and 5<its parent, cut it and add it to the root list.
    Cut 5 and add it to root list
    Cut 5 and add it to root list
  3. Cascading-Cut part: Since 26 is marked, the flow goes to Cut and Cascading-Cut.
    Cut(26): Cut 26 and add it to the root list and mark it as false.

    Cut 26 and add it to root list
    Cut 26 and add it to root list

    Cascading-Cut(24):
    Since the 24 is also marked, again call Cut(24) and Cascading-Cut(7). These operations result in the tree below.

    Cut 24 and add it to root list
    Cut 24 and add it to root list
  4. Since 5 < 7, mark 5 as min.
    Mark the min
    Mark 5 as min

Deleting a Node

This process makes use of decrease-key and extract-min operations. The following steps are followed for deleting a node.

  1. Let k be the node to be deleted.
  2. Apply decrease-key operation to decrease the value of k to the lowest possible value (i.e. -∞).
  3. Apply extract-min operation to remove this node.

 


Python Examples

/* Fibonacci Heap in python */

import math

class FibonacciTree:
    def __init__(self, key):
        self.key = key
        self.children = []
        self.order = 0

    def add_at_end(self, t):
        self.children.append(t)
        self.order = self.order + 1


class FibonacciHeap:
    def __init__(self):
        self.trees = []
        self.least = None
        self.count = 0

    def insert(self, key):
        new_tree = FibonacciTree(key)
        self.trees.append(new_tree)
        if (self.least is None or key < self.least.key):
            self.least = new_tree
        self.count = self.count + 1

    def get_min(self):
        if self.least is None:
            return None
        return self.least.key

    def extract_min(self):
        smallest = self.least
        if smallest is not None:
            for child in smallest.children:
                self.trees.append(child)
            self.trees.remove(smallest)
            if self.trees == []:
                self.least = None
            else:
                self.least = self.trees[0]
                self.consolidate()
            self.count = self.count - 1
            return smallest.key

    def consolidate(self):
        aux = (floor_log2(self.count) + 1) * [None]

        while self.trees != []:
            x = self.trees[0]
            order = x.order
            self.trees.remove(x)
            while aux[order] is not None:
                y = aux[order]
                if x.key > y.key:
                    x, y = y, x
                x.add_at_end(y)
                aux[order] = None
                order = order + 1
            aux[order] = x

        self.least = None
        for k in aux:
            if k is not None:
                self.trees.append(k)
                if (self.least is None
                        or k.key < self.least.key):
                    self.least = k


def floor_log2(x):
    return math.frexp(x)[1] - 1


fheap = FibonacciHeap()

fheap.insert(11)
fheap.insert(10)
fheap.insert(39)
fheap.insert(26)
fheap.insert(24)

print('Minimum value: {}'.format(fheap.get_min()))

print('Minimum value removed: {}'.format(fheap.extract_min()))

Complexities

Decrease Key O(1)
Delete Node O(log n)

 

 

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