Data Structures and Algorithms in Python :-

This tutorial is a beginner-friendly guide for learning data structures and algorithms using Python. In this article, we will discuss the in-built data structures such as lists, tuples, dictionaries, etc, and some user-defined data structures such as linked lists, trees, graphs, etc, and traversal as well as searching and sorting algorithms with the help of good and well-explained examples and practice questions.

Lists

Python Lists are ordered collections of data just like arrays in other programming languages. It allows different types of elements in the list. The implementation of Python List is similar to Vhttps://datavalley.ai/category/python/page/5/ectors in C++ or ArrayList in JAVA. The costly operation is inserting or deleting the element from the beginning of the List as all the elements are needed to be shifted. Insertion and deletion at the end of the list can also become costly in the case where the preallocated memory becomes full.

Example: Creating Python List

• Python3

List = [1, 2, 3, "GFG", 2.3]
print(List)

Output

[1, 2, 3, 'GFG', 2.3]

List elements can be accessed by the assigned index. In python starting index of the list, a sequence is 0 and the ending index is (if N elements are there) N-1.

Example: Python List Operations

• Python3

# Creating a List with
# the use of multiple values
List = ["Geeks", "For", "Geeks"]
print("\nList containing multiple values: ")
print(List)
  
# Creating a Multi-Dimensional List
# (By Nesting a list inside a List)
List2 = [['Geeks', 'For'], ['Geeks']]
print("\nMulti-Dimensional List: ")
print(List2)
  
# accessing a element from the
# list using index number
print("Accessing element from the list")
print(List[0])
print(List[2])
  
# accessing a element using
# negative indexing
print("Accessing element using negative indexing")
      
# print the last element of list
print(List[-1])
      
# print the third last element of list
print(List[-3])

Output

List containing multiple values:
['Geeks', 'For', 'Geeks']


Multi-Dimensional List:
[['Geeks', 'For'], ['Geeks']]
Accessing element from the list
Geeks
Geeks
Accessing element using negative indexing
Geeks
Geeks

Tuple

Python tuples are similar to lists but Tuples are immutable in nature i.e. once created it cannot be modified. Just like a List, a Tuple can also contain elements of various types.

In Python, tuples are created by placing a sequence of values separated by ‘comma’ with or without the use of parentheses for grouping of the data sequence. 

Note: To create a tuple of one element there must be a trailing comma. For example, (8,) will create a tuple containing 8 as the element.

Example: Python Tuple Operations

• Python3

# Creating a Tuple with
# the use of Strings
Tuple = ('Geeks', 'For')
print("\nTuple with the use of String: ")
print(Tuple)
      
# Creating a Tuple with
# the use of list
list1 = [1, 2, 4, 5, 6]
print("\nTuple using List: ")
Tuple = tuple(list1)
  
# Accessing element using indexing
print("First element of tuple")
print(Tuple[0])
  
# Accessing element from last
# negative indexing
print("\nLast element of tuple")
print(Tuple[-1])
  
print("\nThird last element of tuple")
print(Tuple[-3])

Output

Tuple with the use of String:
('Geeks', 'For')


Tuple using List:
First element of tuple
1


Last element of tuple
6


Third last element of tuple

Set

Python set is a mutable collection of data that does not allow any duplication. Sets are basically used to include membership testing and eliminating duplicate entries. The data structure used in this is Hashing, a popular technique to perform insertion, deletion, and traversal in O(1) on average. 

If Multiple values are present at the same index position, then the value is appended to that index position, to form a Linked List. In, CPython Sets are implemented using a dictionary with dummy variables, where key beings the members set with greater optimizations to the time complexity.

Set Implementation:

Sets with Numerous operations on a single HashTable:

Example: Python Set Operations

• Python3

# Creating a Set with
# a mixed type of values
# (Having numbers and strings)
Set = set([1, 2, 'Geeks', 4, 'For', 6, 'Geeks'])
print("\nSet with the use of Mixed Values")
print(Set)
  
# Accessing element using
# for loop
print("\nElements of set: ")
for i in Set:
    print(i, end =" ")
print()
  
# Checking the element
# using in keyword
print("Geeks" in Set)

Output

Set with the use of Mixed Values
{1, 2, 4, 6, 'For', 'Geeks'}


Elements of set:
1 2 4 6 For Geeks
True

Frozen Sets

Frozen sets in Python are immutable objects that only support methods and operators that produce a result without affecting the frozen set or sets to which they are applied. While elements of a set can be modified at any time, elements of the frozen set remain the same after creation.

Example: Python Frozen set

• Python3

# Same as {"a", "b","c"}
normal_set = set(["a", "b","c"])
  
print("Normal Set")
print(normal_set)
  
# A frozen set
frozen_set = frozenset(["e", "f", "g"])
  
print("\nFrozen Set")
print(frozen_set)
  
# Uncommenting below line would cause error as
# we are trying to add element to a frozen set
# frozen_set.add("h")

Output

Normal Set
{'a', 'b', 'c'}


Frozen Set
frozenset({'f', 'g', 'e'})

String

Python Strings is the immutable array of bytes representing Unicode characters. Python does not have a character data type, a single character is simply a string with a length of 1.

Note: As strings are immutable, modifying a string will result in creating a new copy.

Example: Python Strings Operations

• Python3

String = "Welcome to GeeksForGeeks"
print("Creating String: ")
print(String)
      
# Printing First character
print("\nFirst character of String is: ")
print(String[0])
      
# Printing Last character
print("\nLast character of String is: ")
print(String[-1])

Output

Creating String:
Welcome to GeeksForGeeks


First character of String is:
W


Last character of String is:

Dictionary

Python dictionary is an unordered collection of data that stores data in the format of key:value pair. It is like hash tables in any other language with the time complexity of O(1). Indexing of Python Dictionary is done with the help of keys. These are of any hashable type i.e. an object whose can never change like strings, numbers, tuples, etc. We can create a dictionary by using curly braces ({}) or dictionary comprehension.

Example: Python Dictionary Operations

• Python3

# Creating a Dictionary
Dict = {'Name': 'Geeks', 1: [1, 2, 3, 4]}
print("Creating Dictionary: ")
print(Dict)
  
# accessing a element using key
print("Accessing a element using key:")
print(Dict['Name'])
  
# accessing a element using get()
# method
print("Accessing a element using get:")
print(Dict.get(1))
  
# creation using Dictionary comprehension
myDict = {x: x**2 for x in [1,2,3,4,5]}
print(myDict)

Output

Creating Dictionary:
{'Name': 'Geeks', 1: [1, 2, 3, 4]}
Accessing a element using key:
Geeks
Accessing a element using get:
[1, 2, 3, 4]
{1: 1, 2: 4, 3: 9, 4: 16, 5: 25}

Matrix

A matrix is a 2D array where each element is of strictly the same size. To create a matrix we will be using the NumPy package.

Example: Python NumPy Matrix Operations

• Python3

import numpy as np
  
a = np.array([[1,2,3,4],[4,55,1,2],
              [8,3,20,19],[11,2,22,21]])
m = np.reshape(a,(4, 4))
print(m)
  
# Accessing element
print("\nAccessing Elements")
print(a[1])
print(a[2][0])
  
# Adding Element
m = np.append(m,[[1, 15,13,11]],0)
print("\nAdding Element")
print(m)
  
# Deleting Element
m = np.delete(m,[1],0)
print("\nDeleting Element")
print(m)

Output

[[ 1  2  3  4]
 [ 4 55  1  2]
 [ 8  3 20 19]
 [11  2 22 21]]


Accessing Elements
[ 4 55  1  2]
8


Adding Element
[[ 1  2  3  4]
 [ 4 55  1  2]
 [ 8  3 20 19]
 [11  2 22 21]
 [ 1 15 13 11]]


Deleting Element
[[ 1  2  3  4]
 [ 8  3 20 19]
 [11  2 22 21]
 [ 1 15 13 11

Bytearray

Python Bytearray gives a mutable sequence of integers in the range 0 <= x < 256.

Example: Python Bytearray Operations

• Python3

# Creating bytearray
a = bytearray((12, 8, 25, 2))
print("Creating Bytearray:")
print(a)
  
# accessing elements
print("\nAccessing Elements:", a[1])
  
# modifying elements
a[1] = 3
print("\nAfter Modifying:")
print(a)
  
# Appending elements
a.append(30)
print("\nAfter Adding Elements:")
print(a)

Output

Creating Bytearray:
bytearray(b'\x0c\x08\x19\x02')


Accessing Elements: 8


After Modifying:
bytearray(b'\x0c\x03\x19\x02')


After Adding Elements:
bytearray(b'\x0c\x03\x19\x02\x1e

Linked List

A linked list is a linear data structure, in which the elements are not stored at contiguous memory locations. The elements in a linked list are linked using pointers as shown in the below image:

A linked list is represented by a pointer to the first node of the linked list. The first node is called the head. If the linked list is empty, then the value of the head is NULL. Each node in a list consists of at least two parts:

• Data
• Pointer (Or Reference) to the next node

Example: Defining Linked List in Python

• Python3

# Node class
class Node:
  
    # Function to initialize the node object
    def __init__(self, data):
        self.data = data # Assign data
        self.next = None # Initialize
                        # next as null
  
# Linked List class
class LinkedList:
      
    # Function to initialize the Linked
    # List object
    def __init__(self):
        self.head = None

Let us create a simple linked list with 3 nodes.

• Python3

# A simple Python program to introduce a linked list
  
# Node class
class Node:
  
    # Function to initialise the node object
    def __init__(self, data):
        self.data = data # Assign data
        self.next = None # Initialize next as null
  
  
# Linked List class contains a Node object
class LinkedList:
  
    # Function to initialize head
    def __init__(self):
        self.head = None
  
  
# Code execution starts here
if __name__=='__main__':
  
    # Start with the empty list
    llist = LinkedList()
  
    llist.head = Node(1)
    second = Node(2)
    third = Node(3)
  
    '''
    Three nodes have been created.
    We have references to these three blocks as head,
    second and third
  
    llist.head     second             third
        |             |                 |
        |             |                 |
    +----+------+     +----+------+     +----+------+
    | 1 | None |     | 2 | None |     | 3 | None |
    +----+------+     +----+------+     +----+------+
    '''
  
    llist.head.next = second; # Link first node with second
  
    '''
    Now next of first Node refers to second. So they
    both are linked.
  
    llist.head     second             third
        |             |                 |
        |             |                 |
    +----+------+     +----+------+     +----+------+
    | 1 | o-------->| 2 | null |     | 3 | null |
    +----+------+     +----+------+     +----+------+
    '''
  
    second.next = third; # Link second node with the third node
  
    '''
    Now next of second Node refers to third. So all three
    nodes are linked.
  
    llist.head     second             third
        |             |                 |
        |             |                 |
    +----+------+     +----+------+     +----+------+
    | 1 | o-------->| 2 | o-------->| 3 | null |
    +----+------+     +----+------+     +----+------+
    '''

Linked List Traversal 

In the previous program, we have created a simple linked list with three nodes. Let us traverse the created list and print the data of each node. For traversal, let us write a general-purpose function printList() that prints any given list.

• Python3

# A simple Python program for traversal of a linked list
  
# Node class
class Node:
  
    # Function to initialise the node object
    def __init__(self, data):
        self.data = data # Assign data
        self.next = None # Initialize next as null
  
  
# Linked List class contains a Node object
class LinkedList:
  
    # Function to initialize head
    def __init__(self):
        self.head = None
  
    # This function prints contents of linked list
    # starting from head
    def printList(self):
        temp = self.head
        while (temp):
            print (temp.data)
            temp = temp.next
  
  
# Code execution starts here
if __name__=='__main__':
  
    # Start with the empty list
    llist = LinkedList()
  
    llist.head = Node(1)
    second = Node(2)
    third = Node(3)
  
    llist.head.next = second; # Link first node with second
    second.next = third; # Link second node with the third node
  
    llist.printList()

Output

1
2
3

Stack

A stack is a linear data structure that stores items in a Last-In/First-Out (LIFO) or First-In/Last-Out (FILO) manner. In stack, a new element is added at one end and an element is removed from that end only. The insert and delete operations are often called push and pop.

The functions associated with stack are:

• empty() – Returns whether the stack is empty – Time Complexity: O(1)
• size() – Returns the size of the stack – Time Complexity: O(1)
• top() – Returns a reference to the topmost element of the stack – Time Complexity: O(1)
• push(a) – Inserts the element ‘a’ at the top of the stack – Time Complexity: O(1)
• pop() – Deletes the topmost element of the stack – Time Complexity: O(1)
• Python3

stack = []
  
# append() function to push
# element in the stack
stack.append('g')
stack.append('f')
stack.append('g')
  
print('Initial stack')
print(stack)
  
# pop() function to pop
# element from stack in
# LIFO order
print('\nElements popped from stack:')
print(stack.pop())
print(stack.pop())
print(stack.pop())
  
print('\nStack after elements are popped:')
print(stack)
  
# uncommenting print(stack.pop())
# will cause an IndexError
# as the stack is now empty

Output

Initial stack
['g', 'f', 'g']


Elements popped from stack:
g
f
g


Stack after elements are popped:
[

Queue

As a stack, the queue is a linear data structure that stores items in a First In First Out (FIFO) manner. With a queue, the least recently added item is removed first. A good example of the queue is any queue of consumers for a resource where the consumer that came first is served first.

Operations associated with queue are:

• Enqueue: Adds an item to the queue. If the queue is full, then it is said to be an Overflow condition – Time Complexity: O(1)
• Dequeue: Removes an item from the queue. The items are popped in the same order in which they are pushed. If the queue is empty, then it is said to be an Underflow condition – Time Complexity: O(1)
• Front: Get the front item from queue – Time Complexity: O(1)
• Rear: Get the last item from queue – Time Complexity: O(1)
• Python3

# Initializing a queue
queue = []
  
# Adding elements to the queue
queue.append('g')
queue.append('f')
queue.append('g')
  
print("Initial queue")
print(queue)
  
# Removing elements from the queue
print("\nElements dequeued from queue")
print(queue.pop(0))
print(queue.pop(0))
print(queue.pop(0))
  
print("\nQueue after removing elements")
print(queue)
  
# Uncommenting print(queue.pop(0))
# will raise and IndexError
# as the queue is now empty

Output

Initial queue
['g', 'f', 'g']


Elements dequeued from queue
g
f
g


Queue after removing elements
[

Priority Queue

Priority Queues are abstract data structures where each data/value in the queue has a certain priority. For example, In airlines, baggage with the title “Business” or “First-class” arrives earlier than the rest. Priority Queue is an extension of the queue with the following properties.

• An element with high priority is dequeued before an element with low priority.
• If two elements have the same priority, they are served according to their order in the queue.
• Python3

# A simple implementation of Priority Queue
# using Queue.
class PriorityQueue(object):
    def __init__(self):
        self.queue = []
  
    def __str__(self):
        return ' '.join([str(i) for i in self.queue])
  
    # for checking if the queue is empty
    def isEmpty(self):
        return len(self.queue) == 0
  
    # for inserting an element in the queue
    def insert(self, data):
        self.queue.append(data)
  
    # for popping an element based on Priority
    def delete(self):
        try:
            max = 0
            for i in range(len(self.queue)):
                if self.queue[i] > self.queue[max]:
                    max = i
            item = self.queue[max]
            del self.queue[max]
            return item
        except IndexError:
            print()
            exit()
  
if __name__ == '__main__':
    myQueue = PriorityQueue()
    myQueue.insert(12)
    myQueue.insert(1)
    myQueue.insert(14)
    myQueue.insert(7)
    print(myQueue)        
    while not myQueue.isEmpty():
        print(myQueue.delete())

Output

12 1 14 7
14
12
7
1

Heap

heapq module in Python provides the heap data structure that is mainly used to represent a priority queue. The property of this data structure is that it always gives the smallest element (min heap) whenever the element is popped. Whenever elements are pushed or popped, heap structure is maintained. The heap[0] element also returns the smallest element each time. It supports the extraction and insertion of the smallest element in the O(log n) times.

Generally, Heaps can be of two types:

• Max-Heap: In a Max-Heap the key present at the root node must be greatest among the keys present at all of it’s children. The same property must be recursively true for all sub-trees in that Binary Tree.
• Min-Heap: In a Min-Heap the key present at the root node must be minimum among the keys present at all of it’s children. The same property must be recursively true for all sub-trees in that Binary Tree.

• Python3

# importing "heapq" to implement heap queue
import heapq
  
# initializing list
li = [5, 7, 9, 1, 3]
  
# using heapify to convert list into heap
heapq.heapify(li)
  
# printing created heap
print ("The created heap is : ",end="")
print (list(li))
  
# using heappush() to push elements into heap
# pushes 4
heapq.heappush(li,4)
  
# printing modified heap
print ("The modified heap after push is : ",end="")
print (list(li))
  
# using heappop() to pop smallest element
print ("The popped and smallest element is : ",end="")
print (heapq.heappop(li))

Output

The created heap is : [1, 3, 9, 7, 5]
The modified heap after push is : [1, 3, 4, 7, 5, 9]
The popped and smallest element is : 1

Binary Tree

A tree is a  hierarchical data structure that looks like the below figure –

     tree
    ----
     j    <-- root
   /   \
  f      k
/   \      \
a     h      z    <-- leaves

The topmost node of the tree is called the root whereas the bottommost nodes or the nodes with no children are called the leaf nodes. The nodes that are directly under a node are called its children and the nodes that are directly above something are called its parent.

A binary tree is a tree whose elements can have almost two children. Since each element in a binary tree can have only 2 children, we typically name them the left and right children. A Binary Tree node contains the following parts.

• Data
• Pointer to left child
• Pointer to the right child

Example: Defining Node Class

• Python3

# A Python class that represents an individual node
# in a Binary Tree
class Node:
    def __init__(self,key):
        self.left = None
        self.right = None
        self.val = key

Now let’s create a tree with 4 nodes in Python. Let’s assume the tree structure looks like below –

     tree
    ----
     1    <-- root
   /   \
  2     3
 /
4

Example: Adding data to the tree

• Python3

# Python program to introduce Binary Tree
  
# A class that represents an individual node in a
# Binary Tree
class Node:
    def __init__(self,key):
        self.left = None
        self.right = None
        self.val = key
  
  
# create root
root = Node(1)
''' following is the tree after above statement
        1
    / \
    None None'''
  
root.left     = Node(2);
root.right     = Node(3);
  
''' 2 and 3 become left and right children of 1
        1
        / \
        2     3
    / \ / \
None None None None'''
  
  
root.left.left = Node(4);
'''4 becomes left child of 2
        1
    /     \
    2         3
    / \     / \
4 None None None
/ \
None None'''

Tree Traversal

Trees can be traversed in different ways. Following are the generally used ways for traversing trees. Let us consider the below tree –

     tree
    ----
     1    <-- root
   /   \
  2     3
 / \
4   5

Depth First Traversals:

• Inorder (Left, Root, Right) : 4 2 5 1 3
• Preorder (Root, Left, Right) : 1 2 4 5 3
• Postorder (Left, Right, Root) : 4 5 2 3 1

Algorithm Inorder(tree)

• Traverse the left subtree, i.e., call Inorder(left-subtree)
• Visit the root.
• Traverse the right subtree, i.e., call Inorder(right-subtree)

Algorithm Preorder(tree)

• Visit the root.
• Traverse the left subtree, i.e., call Preorder(left-subtree)
• Traverse the right subtree, i.e., call Preorder(right-subtree)

Algorithm Postorder(tree)

• Traverse the left subtree, i.e., call Postorder(left-subtree)
• Traverse the right subtree, i.e., call Postorder(right-subtree)
• Visit the root.
• Python3

# Python program to for tree traversals
  
# A class that represents an individual node in a
# Binary Tree
class Node:
    def __init__(self, key):
        self.left = None
        self.right = None
        self.val = key
  
  
# A function to do inorder tree traversal
def printInorder(root):
  
    if root:
  
        # First recur on left child
        printInorder(root.left)
  
        # then print the data of node
        print(root.val),
  
        # now recur on right child
        printInorder(root.right)
  
  
# A function to do postorder tree traversal
def printPostorder(root):
  
    if root:
  
        # First recur on left child
        printPostorder(root.left)
  
        # the recur on right child
        printPostorder(root.right)
  
        # now print the data of node
        print(root.val),
  
  
# A function to do preorder tree traversal
def printPreorder(root):
  
    if root:
  
        # First print the data of node
        print(root.val),
  
        # Then recur on left child
        printPreorder(root.left)
  
        # Finally recur on right child
        printPreorder(root.right)
  
  
# Driver code
root = Node(1)
root.left = Node(2)
root.right = Node(3)
root.left.left = Node(4)
root.left.right = Node(5)
print("Preorder traversal of binary tree is")
printPreorder(root)
  
print("\nInorder traversal of binary tree is")
printInorder(root)
  
print("\nPostorder traversal of binary tree is")
printPostorder(root)

Output

Preorder traversal of binary tree is
1
2
4
5
3


Inorder traversal of binary tree is
4
2
5
1
3


Postorder traversal of binary tree is
4
5
2
3

Time Complexity –  O(n)

Breadth-First or Level Order Traversal

Level order traversal of a tree is breadth-first traversal for the tree. The level order traversal of the above tree is 1 2 3 4 5.

For each node, first, the node is visited and then its child nodes are put in a FIFO queue. Below is the algorithm for the same –

• Create an empty queue q
• temp_node = root /*start from root*/
• Loop while temp_node is not NULL
  • print temp_node->data.
  • Enqueue temp_node’s children (first left then right children) to q
  • Dequeue a node from q
• Python3

# Python program to print level
# order traversal using Queue
  
# A node structure
class Node:
  
    # A utility function to create a new node
    def __init__(self ,key):
        self.data = key
        self.left = None
        self.right = None
  
# Iterative Method to print the
# height of a binary tree
def printLevelOrder(root):
  
    # Base Case
    if root is None:
        return
      
    # Create an empty queue
    # for level order traversal
    queue = []
  
    # Enqueue Root and initialize height
    queue.append(root)
  
    while(len(queue) > 0):
      
        # Print front of queue and
        # remove it from queue
        print (queue[0].data)
        node = queue.pop(0)
  
        # Enqueue left child
        if node.left is not None:
            queue.append(node.left)
  
        # Enqueue right child
        if node.right is not None:
            queue.append(node.right)
  
# Driver Program to test above function
root = Node(1)
root.left = Node(2)
root.right = Node(3)
root.left.left = Node(4)
root.left.right = Node(5)
  
print ("Level Order Traversal of binary tree is -")
printLevelOrder(root)

Output

Level Order Traversal of binary tree is -
1
2
3
4
5

Time Complexity: O(n) 

Binary Search Tree

Binary Search Tree is a node-based binary tree data structure that has the following properties:

• The left subtree of a node contains only nodes with keys lesser than the node’s key.
• The right subtree of a node contains only nodes with keys greater than the node’s key.
• The left and right subtree each must also be a binary search tree.

The above properties of the Binary Search Tree provide an ordering among keys so that the operations like search, minimum and maximum can be done fast. If there is no order, then we may have to compare every key to search for a given key.

Searching Element

• Start from the root.
• Compare the searching element with root, if less than root, then recurse for left, else recurse for right.
• If the element to search is found anywhere, return true, else return false.
• Python3

# A utility function to search a given key in BST
def search(root,key):
      
    # Base Cases: root is null or key is present at root
    if root is None or root.val == key:
        return root
  
    # Key is greater than root's key
    if root.val < key:
        return search(root.right,key)
  
    # Key is smaller than root's key
    return search(root.left,key)

Insertion of a key 

• Start from the root.
• Compare the inserting element with root, if less than root, then recurse for left, else recurse for right.
• After reaching the end, just insert that node at left(if less than current) else right.
• Python3

# Python program to demonstrate
# insert operation in binary search tree
  
# A utility class that represents
# an individual node in a BST
class Node:
    def __init__(self, key):
        self.left = None
        self.right = None
        self.val = key
  
# A utility function to insert
# a new node with the given key
def insert(root, key):
    if root is None:
        return Node(key)
    else:
        if root.val == key:
            return root
        elif root.val < key:
            root.right = insert(root.right, key)
        else:
            root.left = insert(root.left, key)
    return root
  
# A utility function to do inorder tree traversal
def inorder(root):
    if root:
        inorder(root.left)
        print(root.val)
        inorder(root.right)
  
  
# Driver program to test the above functions
# Let us create the following BST
# 50
# /     \
# 30     70
# / \ / \
# 20 40 60 80
  
r = Node(50)
r = insert(r, 30)
r = insert(r, 20)
r = insert(r, 40)
r = insert(r, 70)
r = insert(r, 60)
r = insert(r, 80)
  
# Print inorder traversal of the BST
inorder(r)

Output

20
30
40
50
60
70
80

Graphs

A graph is a nonlinear data structure consisting of nodes and edges. The nodes are sometimes also referred to as vertices and the edges are lines or arcs that connect any two nodes in the graph. More formally a Graph can be defined as a Graph consisting of a finite set of vertices(or nodes) and a set of edges that connect a pair of nodes.

In the above Graph, the set of vertices V = {0,1,2,3,4} and the set of edges E = {01, 12, 23, 34, 04, 14, 13}. The following two are the most commonly used representations of a graph.

• Adjacency Matrix
• Adjacency List

Adjacency Matrix

Adjacency Matrix is a 2D array of size V x V where V is the number of vertices in a graph. Let the 2D array be adj[][], a slot adj[i][j] = 1 indicates that there is an edge from vertex i to vertex j. The adjacency matrix for an undirected graph is always symmetric. Adjacency Matrix is also used to represent weighted graphs. If adj[i][j] = w, then there is an edge from vertex i to vertex j with weight w. 

• Python3

# A simple representation of graph using Adjacency Matrix
class Graph:
    def __init__(self,numvertex):
        self.adjMatrix = [[-1]*numvertex for x in range(numvertex)]
        self.numvertex = numvertex
        self.vertices = {}
        self.verticeslist =[0]*numvertex
  
    def set_vertex(self,vtx,id):
        if 0<=vtx<=self.numvertex:
            self.vertices[id] = vtx
            self.verticeslist[vtx] = id
  
    def set_edge(self,frm,to,cost=0):
        frm = self.vertices[frm]
        to = self.vertices[to]
        self.adjMatrix[frm][to] = cost
          
        # for directed graph do not add this
        self.adjMatrix[to][frm] = cost
  
    def get_vertex(self):
        return self.verticeslist
  
    def get_edges(self):
        edges=[]
        for i in range (self.numvertex):
            for j in range (self.numvertex):
                if (self.adjMatrix[i][j]!=-1):
                    edges.append((self.verticeslist[i],self.verticeslist[j],self.adjMatrix[i][j]))
        return edges
          
    def get_matrix(self):
        return self.adjMatrix
  
G =Graph(6)
G.set_vertex(0,'a')
G.set_vertex(1,'b')
G.set_vertex(2,'c')
G.set_vertex(3,'d')
G.set_vertex(4,'e')
G.set_vertex(5,'f')
G.set_edge('a','e',10)
G.set_edge('a','c',20)
G.set_edge('c','b',30)
G.set_edge('b','e',40)
G.set_edge('e','d',50)
G.set_edge('f','e',60)
  
print("Vertices of Graph")
print(G.get_vertex())
  
print("Edges of Graph")
print(G.get_edges())
  
print("Adjacency Matrix of Graph")
print(G.get_matrix())

Output

Vertices of Graph

[‘a’, ‘b’, ‘c’, ‘d’, ‘e’, ‘f’]

Edges of Graph

[(‘a’, ‘c’, 20), (‘a’, ‘e’, 10), (‘b’, ‘c’, 30), (‘b’, ‘e’, 40), (‘c’, ‘a’, 20), (‘c’, ‘b’, 30), (‘d’, ‘e’, 50), (‘e’, ‘a’, 10), (‘e’, ‘b’, 40), (‘e’, ‘d’, 50), (‘e’, ‘f’, 60), (‘f’, ‘e’, 60)]

Adjacency Matrix of Graph

[[-1, -1, 20, -1, 10, -1], [-1, -1, 30, -1, 40, -1], [20, 30, -1, -1, -1, -1], [-1, -1, -1, -1, 50, -1], [10, 40, -1, 50, -1, 60], [-1, -1, -1, -1, 60, -1]]

Adjacency List

An array of lists is used. The size of the array is equal to the number of vertices. Let the array be an array[]. An entry array[i] represents the list of vertices adjacent to the ith vertex. This representation can also be used to represent a weighted graph. The weights of edges can be represented as lists of pairs. Following is the adjacency list representation of the above graph. 

• Python3

# A class to represent the adjacency list of the node
class AdjNode:
    def __init__(self, data):
        self.vertex = data
        self.next = None
  
  
# A class to represent a graph. A graph
# is the list of the adjacency lists.
# Size of the array will be the no. of the
# vertices "V"
class Graph:
    def __init__(self, vertices):
        self.V = vertices
        self.graph = [None] * self.V
  
    # Function to add an edge in an undirected graph
    def add_edge(self, src, dest):
      
        # Adding the node to the source node
        node = AdjNode(dest)
        node.next = self.graph[src]
        self.graph[src] = node
  
        # Adding the source node to the destination as
        # it is the undirected graph
        node = AdjNode(src)
        node.next = self.graph[dest]
        self.graph[dest] = node
  
    # Function to print the graph
    def print_graph(self):
        for i in range(self.V):
            print("Adjacency list of vertex {}\n head".format(i), end="")
            temp = self.graph[i]
            while temp:
                print(" -> {}".format(temp.vertex), end="")
                temp = temp.next
            print(" \n")
  
  
# Driver program to the above graph class
if __name__ == "__main__":
    V = 5
    graph = Graph(V)
    graph.add_edge(0, 1)
    graph.add_edge(0, 4)
    graph.add_edge(1, 2)
    graph.add_edge(1, 3)
    graph.add_edge(1, 4)
    graph.add_edge(2, 3)
    graph.add_edge(3, 4)
  
    graph.print_graph()

Output

Adjacency list of vertex 0
 head -> 4 -> 1


Adjacency list of vertex 1
 head -> 4 -> 3 -> 2 -> 0


Adjacency list of vertex 2
 head -> 3 -> 1


Adjacency list of vertex 3
 head -> 4 -> 2 -> 1


Adjacency list of vertex 4
 head -> 3 -> 1 ->

Graph Traversal

Breadth-First Search or BFS

Breadth-First Traversal for a graph is similar to Breadth-First Traversal of a tree. The only catch here is, unlike trees, graphs may contain cycles, so we may come to the same node again. To avoid processing a node more than once, we use a boolean visited array. For simplicity, it is assumed that all vertices are reachable from the starting vertex.

For example, in the following graph, we start traversal from vertex 2. When we come to vertex 0, we look for all adjacent vertices of it. 2 is also an adjacent vertex of 0. If we don’t mark visited vertices, then 2 will be processed again and it will become a non-terminating process. A Breadth-First Traversal of the following graph is 2, 0, 3, 1.

• Python3

# Python3 Program to print BFS traversal
# from a given source vertex. BFS(int s)
# traverses vertices reachable from s.
from collections import defaultdict
  
# This class represents a directed graph
# using adjacency list representation
class Graph:
  
    # Constructor
    def __init__(self):
  
        # default dictionary to store graph
        self.graph = defaultdict(list)
  
    # function to add an edge to graph
    def addEdge(self,u,v):
        self.graph[u].append(v)
  
    # Function to print a BFS of graph
    def BFS(self, s):
  
        # Mark all the vertices as not visited
        visited = [False] * (max(self.graph) + 1)
  
        # Create a queue for BFS
        queue = []
  
        # Mark the source node as
        # visited and enqueue it
        queue.append(s)
        visited[s] = True
  
        while queue:
  
            # Dequeue a vertex from
            # queue and print it
            s = queue.pop(0)
            print (s, end = " ")
  
            # Get all adjacent vertices of the
            # dequeued vertex s. If a adjacent
            # has not been visited, then mark it
            # visited and enqueue it
            for i in self.graph[s]:
                if visited[i] == False:
                    queue.append(i)
                    visited[i] = True
  
# Driver code
  
# Create a graph given in
# the above diagram
g = Graph()
g.addEdge(0, 1)
g.addEdge(0, 2)
g.addEdge(1, 2)
g.addEdge(2, 0)
g.addEdge(2, 3)
g.addEdge(3, 3)
  
print ("Following is Breadth First Traversal"
                " (starting from vertex 2)")
g.BFS(2)

Output

Following is Breadth First Traversal (starting from vertex 2)
2 0 3 1 

Time Complexity: O(V+E) where V is the number of vertices in the graph and E is the number of edges in the graph.

Depth First Search or DFS

Depth First Traversal for a graph is similar to Depth First Traversal of a tree. The only catch here is, unlike trees, graphs may contain cycles, a node may be visited twice. To avoid processing a node more than once, use a boolean visited array.

Algorithm:

• Create a recursive function that takes the index of the node and a visited array.
• Mark the current node as visited and print the node.
• Traverse all the adjacent and unmarked nodes and call the recursive function with the index of the adjacent node.
• Python3

# Python3 program to print DFS traversal
# from a given  graph
from collections import defaultdict
  
# This class represents a directed graph using
# adjacency list representation
class Graph:
  
    # Constructor
    def __init__(self):
  
        # default dictionary to store graph
        self.graph = defaultdict(list)
  
    # function to add an edge to graph
    def addEdge(self, u, v):
        self.graph[u].append(v)
  
    # A function used by DFS
    def DFSUtil(self, v, visited):
  
        # Mark the current node as visited
        # and print it
        visited.add(v)
        print(v, end=' ')
  
        # Recur for all the vertices
        # adjacent to this vertex
        for neighbour in self.graph[v]:
            if neighbour not in visited:
                self.DFSUtil(neighbour, visited)
  
    # The function to do DFS traversal. It uses
    # recursive DFSUtil()
    def DFS(self, v):
  
        # Create a set to store visited vertices
        visited = set()
  
        # Call the recursive helper function
        # to print DFS traversal
        self.DFSUtil(v, visited)
  
# Driver code
  
# Create a graph given
# in the above diagram
g = Graph()
g.addEdge(0, 1)
g.addEdge(0, 2)
g.addEdge(1, 2)
g.addEdge(2, 0)
g.addEdge(2, 3)
g.addEdge(3, 3)
  
print("Following is DFS from (starting from vertex 2)")
g.DFS(2)

Output

Following is DFS from (starting from vertex 2)
2 0 1 3 

Recursion

The process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. Using the recursive algorithms, certain problems can be solved quite easily. Examples of such problems are Towers of Hanoi (TOH), Inorder/Preorder/Postorder Tree Traversals, DFS of Graph, etc.

What is the base condition in recursion?

In the recursive program, the solution to the base case is provided and the solution of the bigger problem is expressed in terms of smaller problems. 

def fact(n):


    # base case
    if (n < = 1)
        return 1
    else
        return n*fact(n-1)

In the above example, base case for n < = 1 is defined and larger value of number can be solved by converting to smaller one till base case is reached.

How memory is allocated to different function calls in recursion?

When any function is called from main(), the memory is allocated to it on the stack. A recursive function calls itself, the memory for a called function is allocated on top of memory allocated to the calling function and a different copy of local variables is created for each function call. When the base case is reached, the function returns its value to the function by whom it is called and memory is de-allocated and the process continues.

Let us take the example of how recursion works by taking a simple function. 

• Python3

# A Python 3 program to
# demonstrate working of
# recursion
  
def printFun(test):
  
    if (test < 1):
        return
    else:
  
        print(test, end=" ")
        printFun(test-1)  # statement 2
        print(test, end=" ")
        return
  
# Driver Code
test = 3
printFun(test)

Output

3 2 1 1 2 3 

The memory stack has been shown in below diagram.

Dynamic Programming

Dynamic Programming is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of subproblems, so that we do not have to re-compute them when needed later. This simple optimization reduces time complexities from exponential to polynomial. For example, if we write simple recursive solution for Fibonacci Numbers, we get exponential time complexity and if we optimize it by storing solutions of subproblems, time complexity reduces to linear.

Tabulation vs Memoization

There are two different ways to store the values so that the values of a sub-problem can be reused. Here, will discuss two patterns of solving dynamic programming (DP) problem: 

• Tabulation: Bottom Up
• Memoization: Top Down

Tabulation

As the name itself suggests starting from the bottom and accumulating answers to the top. Let’s discuss in terms of state transition.

Let’s describe a state for our DP problem to be dp[x] with dp[0] as base state and dp[n] as our destination state. So,  we need to find the value of destination state i.e dp[n].

If we start our transition from our base state i.e dp[0] and follow our state transition relation to reach our destination state dp[n], we call it the Bottom-Up approach as it is quite clear that we started our transition from the bottom base state and reached the topmost desired state.

Now, Why do we call it tabulation method?

To know this let’s first write some code to calculate the factorial of a number using bottom up approach. Once, again as our general procedure to solve a DP we first define a state. In this case, we define a state as dp[x], where dp[x] is to find the factorial of x.

Now, it is quite obvious that dp[x+1] = dp[x] * (x+1)

# Tabulated version to find factorial x.
dp = [0]*MAXN


# base case
dp[0] = 1;
for i in range(n+1):
   dp[i] = dp[i-1] * i

Memoization

Once, again let’s describe it in terms of state transition. If we need to find the value for some state say dp[n] and instead of starting from the base state that i.e dp[0] we ask our answer from the states that can reach the destination state dp[n] following the state transition relation, then it is the top-down fashion of DP.

Here, we start our journey from the top most destination state and compute its answer by taking in count the values of states that can reach the destination state, till we reach the bottom-most base state.

Once again, let’s write the code for the factorial problem in the top-down fashion

# Memoized version to find factorial x.
#  To speed up we store the values
# of calculated states


# initialized to -1
dp[0]*MAXN


# return fact x!
def solve(x):
   if (x==0)
       return 1
   if (dp[x]!=-1)
       return dp[x]
   return (dp[x] = x * solve(x-1)

Searching Algorithms

Linear Search

• Start from the leftmost element of arr[] and one by one compare x with each element of arr[]
• If x matches with an element, return the index.
• If x doesn’t match with any of the elements, return -1.

• Python3

# Python3 code to linearly search x in arr[].
# If x is present then return its location,
# otherwise return -1
  
  
def search(arr, n, x):
  
    for i in range(0, n):
        if (arr[i] == x):
            return i
    return -1
  
  
# Driver Code
arr = [2, 3, 4, 10, 40]
x = 10
n = len(arr)
  
# Function call
result = search(arr, n, x)
if(result == -1):
    print("Element is not present in array")
else:
    print("Element is present at index", result)

Output

Element is present at index 3

The time complexity of the above algorithm is O(n). 

For more information, refer to Linear Search.

Binary Search

Search a sorted array by repeatedly dividing the search interval in half. Begin with an interval covering the whole array. If the value of the search key is less than the item in the middle of the interval, narrow the interval to the lower half. Otherwise, narrow it to the upper half. Repeatedly check until the value is found or the interval is empty.

• Python3

# Python3 Program for recursive binary search.
  
# Returns index of x in arr if present, else -1
def binarySearch (arr, l, r, x):
  
    # Check base case
    if r >= l:
  
        mid = l + (r - l) // 2
  
        # If element is present at the middle itself
        if arr[mid] == x:
            return mid
          
        # If element is smaller than mid, then it
        # can only be present in left subarray
        elif arr[mid] > x:
            return binarySearch(arr, l, mid-1, x)
  
        # Else the element can only be present
        # in right subarray
        else:
            return binarySearch(arr, mid + 1, r, x)
  
    else:
        # Element is not present in the array
        return -1
  
# Driver Code
arr = [ 2, 3, 4, 10, 40 ]
x = 10
  
# Function call
result = binarySearch(arr, 0, len(arr)-1, x)
  
if result != -1:
    print ("Element is present at index % d" % result)
else:
    print ("Element is not present in array")

Output

Element is present at index  3

The time complexity of the above algorithm is O(log(n)). 

For more information, refer to Binary Search.

Sorting Algorithms

Selection Sort

The selection sort algorithm sorts an array by repeatedly finding the minimum element (considering ascending order) from unsorted part and putting it at the beginning. In every iteration of selection sort, the minimum element (considering ascending order) from the unsorted subarray is picked and moved to the sorted subarray. 

Flowchart of the Selection Sort: 

• Python3

# Python program for implementation of Selection
# Sort
import sys
  
  
A = [64, 25, 12, 22, 11]
  
# Traverse through all array elements
for i in range(len(A)):
      
    # Find the minimum element in remaining
    # unsorted array
    min_idx = i
    for j in range(i+1, len(A)):
        if A[min_idx] > A[j]:
            min_idx = j
              
    # Swap the found minimum element with
    # the first element    
    A[i], A[min_idx] = A[min_idx], A[i]
  
# Driver code to test above
print ("Sorted array")
for i in range(len(A)):
    print("%d" %A[i]),

Output

Sorted array
11
12
22
25
64

Time Complexity: O(n²) as there are two nested loops.

Auxiliary Space: O(1) 

Bubble Sort

Bubble Sort is the simplest sorting algorithm that works by repeatedly swapping the adjacent elements if they are in wrong order.

Illustration : 

• Python3

# Python program for implementation of Bubble Sort
  
def bubbleSort(arr):
    n = len(arr)
  
    # Traverse through all array elements
    for i in range(n):
  
        # Last i elements are already in place
        for j in range(0, n-i-1):
  
            # traverse the array from 0 to n-i-1
            # Swap if the element found is greater
            # than the next element
            if arr[j] > arr[j+1] :
                arr[j], arr[j+1] = arr[j+1], arr[j]
  
# Driver code to test above
arr = [64, 34, 25, 12, 22, 11, 90]
  
bubbleSort(arr)
  
print ("Sorted array is:")
for i in range(len(arr)):
    print ("%d" %arr[i]),

Output

Sorted array is:
11
12
22
25
34
64
90

Time Complexity: O(n²)

Insertion Sort

To sort an array of size n in ascending order using insertion sort:

• Iterate from arr[1] to arr[n] over the array.
• Compare the current element (key) to its predecessor.
• If the key element is smaller than its predecessor, compare it to the elements before. Move the greater elements one position up to make space for the swapped element.

Illustration:

• Python3

# Python program for implementation of Insertion Sort
  
# Function to do insertion sort
def insertionSort(arr):
  
    # Traverse through 1 to len(arr)
    for i in range(1, len(arr)):
  
        key = arr[i]
  
        # Move elements of arr[0..i-1], that are
        # greater than key, to one position ahead
        # of their current position
        j = i-1
        while j >= 0 and key < arr[j] :
                arr[j + 1] = arr[j]
                j -= 1
        arr[j + 1] = key
  
  
# Driver code to test above
arr = [12, 11, 13, 5, 6]
insertionSort(arr)
for i in range(len(arr)):
    print ("% d" % arr[i])

Output

 5
 6
 11
 12
 13

Time Complexity: O(n²))

Merge Sort

Like QuickSort, Merge Sort is a Divide and Conquer algorithm. It divides the input array into two halves, calls itself for the two halves, and then merges the two sorted halves. The merge() function is used for merging two halves. The merge(arr, l, m, r) is a key process that assumes that arr[l..m] and arr[m+1..r] are sorted and merges the two sorted sub-arrays into one. 

MergeSort(arr[], l,  r)
If r > l
     1. Find the middle point to divide the array into two halves:
             middle m = l+ (r-l)/2
     2. Call mergeSort for first half:
             Call mergeSort(arr, l, m)
     3. Call mergeSort for second half:
             Call mergeSort(arr, m+1, r)
     4. Merge the two halves sorted in step 2 and 3:
             Call merge(arr, l, m, r)

• Python3

# Python program for implementation of MergeSort
def mergeSort(arr):
    if len(arr) > 1:
  
        # Finding the mid of the array
        mid = len(arr)//2
  
        # Dividing the array elements
        L = arr[:mid]
  
        # into 2 halves
        R = arr[mid:]
  
        # Sorting the first half
        mergeSort(L)
  
        # Sorting the second half
        mergeSort(R)
  
        i = j = k = 0
  
        # Copy data to temp arrays L[] and R[]
        while i < len(L) and j < len(R):
            if L[i] < R[j]:
                arr[k] = L[i]
                i += 1
            else:
                arr[k] = R[j]
                j += 1
            k += 1
  
        # Checking if any element was left
        while i < len(L):
            arr[k] = L[i]
            i += 1
            k += 1
  
        while j < len(R):
            arr[k] = R[j]
            j += 1
            k += 1
  
# Code to print the list
  
  
def printList(arr):
    for i in range(len(arr)):
        print(arr[i], end=" ")
    print()
  
  
# Driver Code
if __name__ == '__main__':
    arr = [12, 11, 13, 5, 6, 7]
    print("Given array is", end="\n")
    printList(arr)
    mergeSort(arr)
    print("Sorted array is: ", end="\n")
    printList(arr)

Output

Given array is
12 11 13 5 6 7
Sorted array is:
5 6 7 11 12 13 

Time Complexity: O(n(logn))

QuickSort

Like Merge Sort, QuickSort is a Divide and Conquer algorithm. It picks an element as pivot and partitions the given array around the picked pivot. There are many different versions of quickSort that pick pivot in different ways.

Always pick first element as pivot.

• Always pick last element as pivot (implemented below)
• Pick a random element as pivot.
• Pick median as pivot.

The key process in quickSort is partition(). Target of partitions is, given an array and an element x of array as pivot, put x at its correct position in sorted array and put all smaller elements (smaller than x) before x, and put all greater elements (greater than x) after x. All this should be done in linear time.

/* low  --> Starting index,  high  --> Ending index */
quickSort(arr[], low, high)
{
    if (low < high)
    {
        /* pi is partitioning index, arr[pi] is now
           at right place */
        pi = partition(arr, low, high);


        quickSort(arr, low, pi - 1);  // Before pi
        quickSort(arr, pi + 1, high); // After pi
    }
}

Partition Algorithm

There can be many ways to do partition, following pseudo code adopts the method given in CLRS book. The logic is simple, we start from the leftmost element and keep track of index of smaller (or equal to) elements as i. While traversing, if we find a smaller element, we swap current element with arr[i]. Otherwise we ignore current element. 

/* low  --> Starting index,  high  --> Ending index */
quickSort(arr[], low, high)
{
    if (low < high)
    {
        /* pi is partitioning index, arr[pi] is now
           at right place */
        pi = partition(arr, low, high);


        quickSort(arr, low, pi - 1);  // Before pi
        quickSort(arr, pi + 1, high); // After pi
    }
}

• Python3

# Python3 implementation of QuickSort
  
# This Function handles sorting part of quick sort
# start and end points to first and last element of
# an array respectively
def partition(start, end, array):
      
    # Initializing pivot's index to start
    pivot_index = start
    pivot = array[pivot_index]
      
    # This loop runs till start pointer crosses
    # end pointer, and when it does we swap the
    # pivot with element on end pointer
    while start < end:
          
        # Increment the start pointer till it finds an
        # element greater than pivot
        while start < len(array) and array[start] <= pivot:
            start += 1
              
        # Decrement the end pointer till it finds an
        # element less than pivot
        while array[end] > pivot:
            end -= 1
          
        # If start and end have not crossed each other,
        # swap the numbers on start and end
        if(start < end):
            array[start], array[end] = array[end], array[start]
      
    # Swap pivot element with element on end pointer.
    # This puts pivot on its correct sorted place.
    array[end], array[pivot_index] = array[pivot_index], array[end]
      
    # Returning end pointer to divide the array into 2
    return end
      
# The main function that implements QuickSort
def quick_sort(start, end, array):
      
    if (start < end):
          
        # p is partitioning index, array[p]
        # is at right place
        p = partition(start, end, array)
          
        # Sort elements before partition
        # and after partition
        quick_sort(start, p - 1, array)
        quick_sort(p + 1, end, array)
          
# Driver code
array = [ 10, 7, 8, 9, 1, 5 ]
quick_sort(0, len(array) - 1, array)
  
print(f'Sorted array: {array}')

Output

Sorted array: [1, 5, 7, 8, 9, 10]

Time Complexity: O(n(logn))

ShellSort

ShellSort is mainly a variation of Insertion Sort. In insertion sort, we move elements only one position ahead. When an element has to be moved far ahead, many movements are involved. The idea of shellSort is to allow the exchange of far items. In shellSort, we make the array h-sorted for a large value of h. We keep reducing the value of h until it becomes 1. An array is said to be h-sorted if all sublists of every h^th element is sorted.

• Python3

# Python3 program for implementation of Shell Sort
  
def shellSort(arr):
    gap = len(arr) // 2 # initialize the gap
  
    while gap > 0:
        i = 0
        j = gap
          
        # check the array in from left to right
        # till the last possible index of j
        while j < len(arr):
      
            if arr[i] >arr[j]:
                arr[i],arr[j] = arr[j],arr[i]
              
            i += 1
            j += 1
          
            # now, we look back from ith index to the left
            # we swap the values which are not in the right order.
            k = i
            while k - gap > -1:
  
                if arr[k - gap] > arr[k]:
                    arr[k-gap],arr[k] = arr[k],arr[k-gap]
                k -= 1
  
        gap //= 2
  
  
# driver to check the code
arr2 = [12, 34, 54, 2, 3]
print("input array:",arr2)
  
shellSort(arr2)
print("sorted array",arr2)

Output

input array: [12, 34, 54, 2, 3]
sorted array [2, 3, 12, 34, 54]

Time Complexity:  O(n²).