Data Structures and Algorithm: Sample Problems with Solution

DATA STRUCTURES AND
ALGORITHM: SAMPLE
PROBLEMS WITH
SOLUTION
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INTODUCTION:
Welcome to the "Data Structures and Algorithm: Sample Problems
with Solution" presentation. This presentation aims to provide a
comprehensive set of sample problems along with detailed solutions
to help you understand and master fundamental concepts in data
structures and algorithms.
In this presentation, you will find:
 A variety of sample problems ranging from basic to advanced
levels.
 Step-by-step solutions to each problem, making complex
concepts easy to understand.
 Practical examples to enhance your problem-solving skills and
prepare you for real-world applications and exams.
Our goal is to equip you with the necessary tools and knowledge to
excel in your studies and professional career in programming and
software development. Should you have any queries or need further
assistance with your assignments, feel free to reach out to us at
support@programminghomeworkhelp.com or visit our website at
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PROBLEM 1 - DEPTH OF FIBONACCI HEAPS
Unlike regular heaps, Fibonacci heaps do not
achieve their good performance by keeping the
depth of the heap small. Demonstrate this by
exhibiting a sequence of Fibonacci heap operations
on n items that produce a heap
ordered tree of
depth Ω(n).

SOLUTION:
To demonstrate that Fibonacci heaps can have a heap-
ordered tree of depth Ω(n)Omega(n)Ω(n), we can construct a
sequence of operations that produces such a deep tree. Here
is a sequence of Fibonacci heap operations that will produce a
tree of depth Ω(n)Omega(n)Ω(n).
 Sequence of Operations
 Insert n elements: Insert n elements into the
Fibonacci heap. Each insertion operation creates a new
singleton tree, resulting in n separate trees initially.
 Operations: Insert 1, Insert 2, ..., Insert n
 Perform n−1n-1n−1 decrease-key operations:
After all elements are inserted, we will perform a
sequence of n−1n-1n−1 decrease-key operations that
ensure each decrease-key operation links the trees,
creating a single long path.

 For example, suppose the elements inserted are
1,2,3,...,n1, 2, 3, ..., n1,2,3,...,n. We will perform
decrease-key operations that progressively decrease
the keys such that the i-th element's key is decreased
to −i. This way, each decrease-key operation will
trigger a cut and potential cascading cuts that link
the trees into a single path.
 Operations:
 Decrease-key the key of the element n to -n
 Decrease-key the key of the element n-1 to –n-1
 Decrease-key the key of the element 222 to −2-2−2
 Explanation
 Insert Operations:
 After inserting n elements, we have n trees each with
a single node.

 Decrease-Key Operations:
 Each decrease-key operation may potentially link two
trees, resulting in one of the trees being made a child
of the other. By choosing the keys appropriately, we
can ensure that the trees are linked in such a way
that a single long path (a degenerate tree) is formed.
 Example
 Let's go through a small example with n=4n :
 Insert 1, 2, 3, 4
 Heap contains 4 separate trees.
 Decrease-key 4 to -4
 Decrease-key 3 to -5

 After each decrease-key, the trees are linked as follows:
 After Decrease-key(4, -4):
 Node 4 (key -4) is linked to one of the other nodes,
say node 3.
 Node 3 (key -5) is linked to node 2.
 Node 2 (key -6) is linked to node 1.
 In the end, we have a single tree with a path from node
4 to node 1:
4 -> 3 -> 2 -> 1
 This forms a degenerate tree (a single path) of depth
n−1=4−1=3n-1 = 4-1 = 3n−1=4−1=3.

SAMPLE CODE:
def print_heap(node, depth=0):
if node is not None:
print(" " * depth + f"Node({node.key})")
child = node.child
while child is not None:
print_heap(child, depth + 2)
child = child.right
if child == node.child:
break
# Create a Fibonacci heap
fib_heap = FibonacciHeap()
# Insert n elements into the Fibonacci heap
n = 10
nodes = []
for i in range(n):
node = fib_heap.insert(i)
nodes.append(node)

print("Heap after insertions:")
for root in fib_heap.iterate():
print_heap(root)
# Perform decrease-key operations to form a single long path
for i in range(n - 1, 0, -1):
fib_heap.decrease_key(nodes[i], nodes[i].key - n - i)
print("nHeap after decrease-key operations:")
print_heap(root)
# To visualize the tree structure, let's use a function to print the heap
def visualize_heap(fib_heap):
print_heap(root)
visualize_heap(fib_heap)

 Explanation:
 Insert n Elements:
 We insert n elements into the Fibonacci heap. Each
insertion creates a new singleton tree.
 Decrease-Key Operations:
 We perform decrease-key operations in such a way that
each decrease-key operation links the trees into a single
path.
 Printing and Visualization:
 The print_heap function helps us visualize the tree
structure to verify the depth.
 Output:
 The output will show the structure of the Fibonacci heap after
the insertions and the decrease-key operations, demonstrating
the depth of the heap.
 You can adjust the value of n to see how the depth increases
as the number of elements grows. This script helps to visualize
and verify that the Fibonacci heap can indeed have a tree of
depth Ω(n)Omega(n)Ω(n).

PROBLEM 2 - MODIFIED FIBONACCI HEAPS
Suppose that Fibonacci heaps were modified so
that a node was cut only after losing k children.
Show that this will improve the amortized cost of
decrease key (to a better constant) at the cost of a
worse cost for delete
mine (by a constant factor).

SOLUTION:
To analyze the impact of modifying Fibonacci heaps so that a
node is cut only after losing kkk children, let's examine how
this change affects the amortized cost of the decrease-key and
delete-min operations.
 Fibonacci Heaps Review
In a traditional Fibonacci heap:
 Decrease-key has an amortized cost of O(1)).
 Delete-min has an amortized cost of O(log⁡
), where n is
the number of nodes in the heap.
 Modification: Node Cut After Losing k Children
In the modified Fibonacci heap, a node is cut only after
losing k children. This change impacts both the decrease-
key and delete-min operations. We'll analyze the effect of
this modification on both operations.

 Decrease-key Operation
 Decrease-key and Cut Operations:
 In a standard Fibonacci heap, when a node's key is
decreased, it may trigger a cut of that node if its
parent loses a child.
 With the modification, a node is cut only after
losing k children.
 Impact on Amortized Cost:
 Standard Fibonacci Heap: The amortized cost of
a decrease-key operation is O(1)O(1)O(1) because
the number of cuts and cascading cuts is bounded
and amortized over the sequence of operations.
 Modified Fibonacci Heap: Since a node is cut
only after losing k children, the cost of the
decrease-key operation can increase because it
requires maintaining additional structure to
ensure nodes are not cut prematurely. However, if
k is fixed, the impact on the decrease-key operation
is

minimal compared to the potential reduction in the
number of cascading cuts.
In practice, the amortized cost of decrease-key will still be
constant, but the constant factor might be better (improved)
because:
 Fewer cascading cuts will be needed as nodes are cut
less frequently.
 The reduction in the frequency of cuts and cascading
cuts leads to fewer structural changes and thus lower
amortized cost per operation.
Delete-min Operation
 Delete-min Operation and Tree Consolidation:
 In a standard Fibonacci heap, delete-min involves
removing the minimum node and then consolidating
trees in the root list. This consolidation involves linking
trees together, which can be done efficiently.
 With the modification, nodes are not cut until they lose
k children, which changes the structure of the heap.

 Impact on Amortized Cost:
 Standard Fibonacci Heap: The amortized cost of delete-min is
O(log⁡
n), which includes both removing the minimum node and the
tree consolidation step.
 Modified Fibonacci Heap: As nodes are cut less frequently, the
heap can have a larger number of nodes with fewer cuts. This
leads to a larger number of trees in the root list, increasing the
number of links needed during consolidation. Consequently, the
consolidation process becomes more costly.
 The result is that the amortized cost of delete-min increases,
typically by a constant factor related to k, due to the increased
number of trees and the more complex tree consolidation process.

SAMPLE CODE:
import math
class Node:
def __init__(self, key):
self.key = key
self.degree = 0
self.parent = None
self.child = None
self.is_marked = False
self.next = self
self.prev = self
class FibonacciHeap:
def __init__(self, k):
self.min_node = None
self.node_count = 0
self.k = k

def insert(self, key):
new_node = Node(key)
if self.min_node is None:
self.min_node = new_node
else:
self._link(self.min_node, new_node)
if key < self.min_node.key:
self.min_node = new_node
self.node_count += 1
def decrease_key(self, node, new_key):
if new_key > node.key:
raise ValueError("New key is greater than current key")
node.key = new_key
if node.parent and node.key < node.parent.key:
self._cut(node)
if node.key < self.min_node.key:
self.min_node = node

def delete_min(self):
min_node = self.min_node
if min_node is not None:
if min_node.child:
child = min_node.child
while True:
next_child = child.next
self._link(self.min_node, child)
child.parent = None
if next_child == min_node.child:
break
child = next_child
self._remove(min_node)
if min_node.next == min_node:
self.min_node = None
else:
self.min_node = min_node.next
self._consolidate()
self.node_count -= 1
return min_node.key

def _cut(self, node):
if node.parent:
node.parent.degree -= 1
if node.parent.degree == self.k:
self._remove(node)
self._link(self.min_node, node)
else:
if node == node.parent.child:
node.parent.child = node.next
self._remove(node)
self._link(self.min_node, node)
node.parent = None
node.is_marked = False
def _consolidate(self):
degree_table = {}
nodes = [node for node in self._iterate_nodes(self.min_node)]
for node in nodes:
degree = node.degree

while degree in degree_table:
other = degree_table[degree]
if node.key > other.key:
node, other = other, node
self._link(node, other)
degree += 1
degree_table[degree] = node
def _link(self, min_node, node):
if min_node is None:
min_node = node
min_node.next = min_node
min_node.prev = min_node
else:
node.prev = min_node
node.next = min_node.next
min_node.next.prev = node
min_node.next = node

def _remove(self, node):
node.prev.next = node.next
node.next.prev = node.prev
def _iterate_nodes(self, start_node):
nodes = []
current = start_node
while True:
nodes.append(current)
current = current.next
if current == start_node:
break
return nodes
# Example usage
k = 3
fib_heap = FibonacciHeap(k)
fib_heap.insert(10)
fib_heap.insert(3)
fib_heap.insert(15)
fib_heap.insert(7)
print("Min:", fib_heap.delete_min()) # Should print 3

 Explanation:
 Node Class:
 Represents a node in the Fibonacci heap. Each node has a key,
degree, parent, child, and pointers for linked list operations.
 FibonacciHeap Class:
 Initialization: Initializes with a k value, which determines how
many children a node must lose before it is cut.
 Insert: Adds a node to the heap.
 Decrease Key: Decreases the key of a given node and performs a
cut if necessary.
 Delete Min: Removes the node with the minimum key and
consolidates the heap.
 _Cut: Handles the cut operation, adjusting the node's position
based on the value of k.
 _Consolidate: Merges trees of the same degree in the root list.
 _Link: Links two nodes together in the circular doubly linked list.
 _Remove: Removes a node from the linked list.
 _Iterate Nodes: Provides an iterator to go through all nodes in the
root list.
This code provides a basic framework and illustrates the core ideas behind
the modified Fibonacci heap with a cut threshold of kkk. You might need
additional methods and refinements to handle all edge cases and ensure
complete functionality.

CONCLUSION
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Data Structures and Algorithm: Sample Problems with Solution

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