Data Structures are a way of organizing data so that it can be accessed more efficiently depending upon the situation. Data Structures are fundamentals of any programming language around which a program is built. Python helps to learn the fundamental of these data structures in a simpler way as compared to other programming languages.
In this article, we will discuss the Data Structures in the Python Programming Language and how they are related to some specific Python Data Types. We will discuss all the in-built data structures like list tuples, dictionaries, etc. as well as some advanced data structures like trees, graphs, etc.
Lists
Python Lists are just like the arrays, declared in other languages which is an ordered collection of data. It is very flexible as the items in a list do not need to be of the same type.
The implementation of Python List is similar to Vectors 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.
We can create a list in python as shown below.
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, sequence is 0 and the ending index is (if N elements are there) N-1.
Example: Python List Operations
Python3
List = ["Geeks", "For", "Geeks"]
print("\nList containing multiple values: ")
print(List)
List2 = [['Geeks', 'For'], ['Geeks']]
print("\nMulti-Dimensional List: ")
print(List2)
print("Accessing element from the list")
print(List[0])
print(List[2])
print("Accessing element using negative indexing")
print(List[-1])
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
Dictionary
Python dictionary is like hash tables in any other language with the time complexity of O(1). It is an unordered collection of data values, used to store data values like a map, which, unlike other Data Types that hold only a single value as an element, Dictionary holds the key:value pair. Key-value is provided in the dictionary to make it more optimized.
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
Dict = {'Name': 'Geeks', 1: [1, 2, 3, 4]}
print("Creating Dictionary: ")
print(Dict)
print("Accessing a element using key:")
print(Dict['Name'])
print("Accessing a element using get:")
print(Dict.get(1))
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}
Tuple
Python Tuple is a collection of Python objects much like a list but Tuples are immutable in nature i.e. the elements in the tuple cannot be added or removed once created. 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: Tuples can also be created with a single element, but it is a bit tricky. Having one element in the parentheses is not sufficient, there must be a trailing ‘comma’ to make it a tuple.
Example: Python Tuple Operations
Python3
Tuple = ('Geeks', 'For')
print("\nTuple with the use of String: ")
print(Tuple)
list1 = [1, 2, 4, 5, 6]
print("\nTuple using List: ")
Tuple = tuple(list1)
print("First element of tuple")
print(Tuple[0])
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
4
Set
Python Set is an unordered collection of data that is mutable and does not allow any duplicate element. 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
Set = set([1, 2, 'Geeks', 4, 'For', 6, 'Geeks'])
print("\nSet with the use of Mixed Values")
print(Set)
print("\nElements of set: ")
for i in Set:
print(i, end =" ")
print()
print("Geeks" in Set)
|
Output
Set with the use of Mixed Values
{1, 2, 'Geeks', 4, 6, 'For'}
Elements of set:
1 2 Geeks 4 6 For
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.
If no parameters are passed, it returns an empty frozenset.
Python3
normal_set = set(["a", "b","c"])
print("Normal Set")
print(normal_set)
frozen_set = frozenset(["e", "f", "g"])
print("\nFrozen Set")
print(frozen_set)
|
Output
Normal Set
{'a', 'c', 'b'}
Frozen Set
frozenset({'g', 'e', 'f'})
String
Python Strings are arrays of bytes representing Unicode characters. In simpler terms, a string is an immutable array of 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)
print("\nFirst character of String is: ")
print(String[0])
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:
s
Bytearray
Python Bytearray gives a mutable sequence of integers in the range 0 <= x < 256.
Example: Python Bytearray Operations
Python3
a = bytearray((12, 8, 25, 2))
print("Creating Bytearray:")
print(a)
print("\nAccessing Elements:", a[1])
a[1] = 3
print("\nAfter Modifying:")
print(a)
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')
Till now we have studied all the data structures that come built-in into core Python. Now let dive more deep into Python and see the collections module that provides some containers that are useful in many cases and provide more features than the above-defined functions.
Collections Module
Python collection module was introduced to improve the functionality of the built-in datatypes. It provides various containers let’s see each one of them in detail.
Counters
A counter is a sub-class of the dictionary. It is used to keep the count of the elements in an iterable in the form of an unordered dictionary where the key represents the element in the iterable and value represents the count of that element in the iterable. This is equivalent to a bag or multiset of other languages.
Example: Python Counter Operations
Python3
from collections import Counter
print(Counter(['B','B','A','B','C','A','B','B','A','C']))
count = Counter({'A':3, 'B':5, 'C':2})
print(count)
count.update(['A', 1])
print(count)
|
Output
Counter({'B': 5, 'A': 3, 'C': 2})
Counter({'B': 5, 'A': 3, 'C': 2})
Counter({'B': 5, 'A': 4, 'C': 2, 1: 1})
OrderedDict
An OrderedDict is also a sub-class of dictionary but unlike a dictionary, it remembers the order in which the keys were inserted.
Example: Python OrderedDict Operations
Python3
from collections import OrderedDict
print("Before deleting:\n")
od = OrderedDict()
od['a'] = 1
od['b'] = 2
od['c'] = 3
od['d'] = 4
for key, value in od.items():
print(key, value)
print("\nAfter deleting:\n")
od.pop('c')
for key, value in od.items():
print(key, value)
print("\nAfter re-inserting:\n")
od['c'] = 3
for key, value in od.items():
print(key, value)
|
Output
Before deleting:
a 1
b 2
c 3
d 4
After deleting:
a 1
b 2
d 4
After re-inserting:
a 1
b 2
d 4
c 3
DefaultDict
DefaultDict is used to provide some default values for the key that does not exist and never raises a KeyError. Its objects can be initialized using DefaultDict() method by passing the data type as an argument.
Note: default_factory is a function that provides the default value for the dictionary created. If this parameter is absent then the KeyError is raised.
Example: Python DefaultDict Operations
Python3
from collections import defaultdict
d = defaultdict(int)
L = [1, 2, 3, 4, 2, 4, 1, 2]
for i in L:
d[i] += 1
print(d)
|
Output
defaultdict(<class 'int'>, {1: 2, 2: 3, 3: 1, 4: 2})
ChainMap
A ChainMap encapsulates many dictionaries into a single unit and returns a list of dictionaries. When a key is needed to be found then all the dictionaries are searched one by one until the key is found.
Example: Python ChainMap Operations
Python3
from collections import ChainMap
d1 = {'a': 1, 'b': 2}
d2 = {'c': 3, 'd': 4}
d3 = {'e': 5, 'f': 6}
c = ChainMap(d1, d2, d3)
print(c)
print(c['a'])
print(c['g'])
|
Output
ChainMap({'a': 1, 'b': 2}, {'c': 3, 'd': 4}, {'e': 5, 'f': 6})
1
KeyError: 'g'
NamedTuple
A NamedTuple returns a tuple object with names for each position which the ordinary tuples lack. For example, consider a tuple names student where the first element represents fname, second represents lname and the third element represents the DOB. Suppose for calling fname instead of remembering the index position you can actually call the element by using the fname argument, then it will be really easy for accessing tuples element. This functionality is provided by the NamedTuple.
Example: Python NamedTuple Operations
Python3
from collections import namedtuple
Student = namedtuple('Student',['name','age','DOB'])
S = Student('Nandini','19','2541997')
print ("The Student age using index is : ",end ="")
print (S[1])
print ("The Student name using keyname is : ",end ="")
print (S.name)
|
Output
The Student age using index is : 19
The Student name using keyname is : Nandini
Deque
Deque (Doubly Ended Queue) is the optimized list for quicker append and pop operations from both sides of the container. It provides O(1) time complexity for append and pop operations as compared to the list with O(n) time complexity.
Python Deque is implemented using doubly linked lists therefore the performance for randomly accessing the elements is O(n).
Example: Python Deque Operations
Python3
import collections
de = collections.deque([1,2,3])
de.append(4)
print("The deque after appending at right is : ")
print(de)
de.appendleft(6)
print("The deque after appending at left is : ")
print(de)
de.pop()
print("The deque after deleting from right is : ")
print(de)
de.popleft()
print("The deque after deleting from left is : ")
print(de)
|
Output
The deque after appending at right is :
deque([1, 2, 3, 4])
The deque after appending at left is :
deque([6, 1, 2, 3, 4])
The deque after deleting from right is :
deque([6, 1, 2, 3])
The deque after deleting from left is :
deque([1, 2, 3])
UserDict
UserDict is a dictionary-like container that acts as a wrapper around the dictionary objects. This container is used when someone wants to create their own dictionary with some modified or new functionality.
Example: Python UserDict
Python3
from collections import UserDict
class MyDict(UserDict):
def __del__(self):
raise RuntimeError("Deletion not allowed")
def pop(self, s = None):
raise RuntimeError("Deletion not allowed")
def popitem(self, s = None):
raise RuntimeError("Deletion not allowed")
d = MyDict({'a':1,
'b': 2,
'c': 3})
print("Original Dictionary")
print(d)
d.pop(1)
|
Output
Original Dictionary
{'a': 1, 'b': 2, 'c': 3}
RuntimeError: Deletion not allowed
UserList
UserList is a list-like container that acts as a wrapper around the list objects. This is useful when someone wants to create their own list with some modified or additional functionality.
Examples: Python UserList
Python3
from collections import UserList
class MyList(UserList):
def remove(self, s = None):
raise RuntimeError("Deletion not allowed")
def pop(self, s = None):
raise RuntimeError("Deletion not allowed")
L = MyList([1, 2, 3, 4])
print("Original List")
print(L)
L.append(5)
print("After Insertion")
print(L)
L.remove()
|
Output
Original List
[1, 2, 3, 4]
After Insertion
[1, 2, 3, 4, 5]
RuntimeError: Deletion not allowed
UserString
UserString is a string-like container and just like UserDict and UserList, it acts as a wrapper around string objects. It is used when someone wants to create their own strings with some modified or additional functionality.
Example: Python UserString
Python3
from collections import UserString
class Mystring(UserString):
def append(self, s):
self.data += s
def remove(self, s):
self.data = self.data.replace(s, "")
s1 = Mystring("Geeks")
print("Original String:", s1.data)
s1.append("s")
print("String After Appending:", s1.data)
s1.remove("e")
print("String after Removing:", s1.data)
|
Output
Original String: Geeks
String After Appending: Geekss
String after Removing: Gkss
Now after studying all the data structures let’s see some advanced data structures such as stack, queue, graph, linked list, etc. that can be used in Python Language.
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
class Node:
def __init__(self, data):
self.data = data
self.next = None
class LinkedList:
def __init__(self):
self.head = None
|
Let us create a simple linked list with 3 nodes.
Python3
class Node:
def __init__(self, data):
self.data = data
self.next = None
class LinkedList:
def __init__(self):
self.head = None
if __name__=='__main__':
list = LinkedList()
list.head = Node(1)
second = Node(2)
third = Node(3)
list.head.next = second;
second.next = third;
|
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
class Node:
def __init__(self, data):
self.data = data
self.next = None
class LinkedList:
def __init__(self):
self.head = None
def printList(self):
temp = self.head
while (temp):
print (temp.data)
temp = temp.next
if __name__=='__main__':
list = LinkedList()
list.head = Node(1)
second = Node(2)
third = Node(3)
list.head.next = second;
second.next = third;
list.printList()
|
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)
Python Stack Implementation
Stack in Python can be implemented using the following ways:
- list
- Collections.deque
- queue.LifoQueue
Implementation using List
Python’s built-in data structure list can be used as a stack. Instead of push(), append() is used to add elements to the top of the stack while pop() removes the element in LIFO order.
Python3
stack = []
stack.append('g')
stack.append('f')
stack.append('g')
print('Initial stack')
print(stack)
print('\nElements popped from stack:')
print(stack.pop())
print(stack.pop())
print(stack.pop())
print('\nStack after elements are popped:')
print(stack)
|
Output
Initial stack
['g', 'f', 'g']
Elements popped from stack:
g
f
g
Stack after elements are popped:
[]
Implementation using collections.deque:
Python stack can be implemented using the deque class from the collections module. Deque is preferred over the list in the cases where we need quicker append and pop operations from both the ends of the container, as deque provides an O(1) time complexity for append and pop operations as compared to list which provides O(n) time complexity.
Python3
from collections import deque
stack = deque()
stack.append('g')
stack.append('f')
stack.append('g')
print('Initial stack:')
print(stack)
print('\nElements popped from stack:')
print(stack.pop())
print(stack.pop())
print(stack.pop())
print('\nStack after elements are popped:')
print(stack)
|
Output
Initial stack:
deque(['g', 'f', 'g'])
Elements popped from stack:
g
f
g
Stack after elements are popped:
deque([])
Implementation using queue module
The queue module also has a LIFO Queue, which is basically a Stack. Data is inserted into Queue using the put() function and get() takes data out from the Queue.
Python3
from queue import LifoQueue
stack = LifoQueue(maxsize = 3)
print(stack.qsize())
stack.put('g')
stack.put('f')
stack.put('g')
print("Full: ", stack.full())
print("Size: ", stack.qsize())
print('\nElements popped from the stack')
print(stack.get())
print(stack.get())
print(stack.get())
print("\nEmpty: ", stack.empty())
|
Output
0
Full: True
Size: 3
Elements popped from the stack
g
f
g
Empty: True
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)
Python queue Implementation
Queue in Python can be implemented in the following ways:
- list
- collections.deque
- queue.Queue
Implementation using list
Instead of enqueue() and dequeue(), append() and pop() function is used.
Python3
queue = []
queue.append('g')
queue.append('f')
queue.append('g')
print("Initial queue")
print(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)
|
Output
Initial queue
['g', 'f', 'g']
Elements dequeued from queue
g
f
g
Queue after removing elements
[]
Implementation using collections.deque
Deque is preferred over the list in the cases where we need quicker append and pop operations from both the ends of the container, as deque provides an O(1) time complexity for append and pop operations as compared to list which provides O(n) time complexity.
Python3
from collections import deque
q = deque()
q.append('g')
q.append('f')
q.append('g')
print("Initial queue")
print(q)
print("\nElements dequeued from the queue")
print(q.popleft())
print(q.popleft())
print(q.popleft())
print("\nQueue after removing elements")
print(q)
|
Output
Initial queue
deque(['g', 'f', 'g'])
Elements dequeued from the queue
g
f
g
Queue after removing elements
deque([])
Implementation using the queue.Queue
queue.Queue(maxsize) initializes a variable to a maximum size of maxsize. A maxsize of zero ‘0’ means an infinite queue. This Queue follows the FIFO rule.
Python3
from queue import Queue
q = Queue(maxsize = 3)
print(q.qsize())
q.put('g')
q.put('f')
q.put('g')
print("\nFull: ", q.full())
print("\nElements dequeued from the queue")
print(q.get())
print(q.get())
print(q.get())
print("\nEmpty: ", q.empty())
q.put(1)
print("\nEmpty: ", q.empty())
print("Full: ", q.full())
|
Output
0
Full: True
Elements dequeued from the queue
g
f
g
Empty: True
Empty: False
Full: False
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
class PriorityQueue(object):
def __init__(self):
self.queue = []
def __str__(self):
return ' '.join([str(i) for i in self.queue])
def isEmpty(self):
return len(self.queue) == 0
def insert(self, data):
self.queue.append(data)
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 queue (or heapq)
heapq module in Python provides the heap data structure that is mainly used to represent a priority queue. The property of this data structure in Python is that each time the smallest heap element is popped(min-heap). 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.
Python3
import heapq
li = [5, 7, 9, 1, 3]
heapq.heapify(li)
print ("The created heap is : ",end="")
print (list(li))
heapq.heappush(li,4)
print ("The modified heap after push is : ",end="")
print (list(li))
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
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
class Node:
def __init__(self,key):
self.left = None
self.right = None
self.val = key
root = Node(1)
root.left = Node(2);
root.right = Node(3);
root.left.left = Node(4);
|
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
class Node:
def __init__(self, key):
self.left = None
self.right = None
self.val = key
def printInorder(root):
if root:
printInorder(root.left)
print(root.val),
printInorder(root.right)
def printPostorder(root):
if root:
printPostorder(root.left)
printPostorder(root.right)
print(root.val),
def printPreorder(root):
if root:
print(root.val),
printPreorder(root.left)
printPreorder(root.right)
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
1
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
class Node:
def __init__(self ,key):
self.data = key
self.left = None
self.right = None
def printLevelOrder(root):
if root is None:
return
queue = []
queue.append(root)
while(len(queue) > 0):
print (queue[0].data)
node = queue.pop(0)
if node.left is not None:
queue.append(node.left)
if node.right is not None:
queue.append(node.right)
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)
Graph
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
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
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
class AdjNode:
def __init__(self, data):
self.vertex = data
self.next = None
class Graph:
def __init__(self, vertices):
self.V = vertices
self.graph = [None] * self.V
def add_edge(self, src, dest):
node = AdjNode(dest)
node.next = self.graph[src]
self.graph[src] = node
node = AdjNode(src)
node.next = self.graph[dest]
self.graph[dest] = node
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")
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 -> 0
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
from collections import defaultdict
class Graph:
def __init__(self):
self.graph = defaultdict(list)
def addEdge(self,u,v):
self.graph[u].append(v)
def BFS(self, s):
visited = [False] * (max(self.graph) + 1)
queue = []
queue.append(s)
visited[s] = True
while queue:
s = queue.pop(0)
print (s, end = " ")
for i in self.graph[s]:
if visited[i] == False:
queue.append(i)
visited[i] = True
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
from collections import defaultdict
class Graph:
def __init__(self):
self.graph = defaultdict(list)
def addEdge(self, u, v):
self.graph[u].append(v)
def DFSUtil(self, v, visited):
visited.add(v)
print(v, end=' ')
for neighbour in self.graph[v]:
if neighbour not in visited:
self.DFSUtil(neighbour, visited)
def DFS(self, v):
visited = set()
self.DFSUtil(v, visited)
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
Time complexity: O(V + E), where V is the number of vertices and E is the number of edges in the graph.
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