Big O notation is one of the most essential concepts in computer science and software engineering. It allows developers to understand the efficiency of algorithms, specifically in terms of time complexity and space complexity. In coding interviews, knowledge of Big O notation is crucial because it helps you assess how well your solution will scale with large inputs.
In this article, we will explore Big O notation, how it is used to describe algorithm efficiency, and discuss various common complexities that every software engineer should know. We’ll also cover tips and tricks to help you master the concept and ace your coding interviews.
Big O notation is a mathematical notation used to describe the upper bound (or worst-case) performance of an algorithm. It provides a way to express how the runtime or space requirements of an algorithm grow as the size of the input increases. In simpler terms, Big O helps you understand how the performance of an algorithm scales.
Big O notation focuses on the relationship between the size of the input (usually denoted as n) and the number of operations an algorithm performs. It ignores constants and less significant terms to give a high-level approximation of the algorithm’s growth rate.
Both time and space complexity can be expressed using Big O notation.
Below are some of the most common Big O complexities that you should know for coding interviews.
An algorithm is said to have O(1) time complexity if its running time does not depend on the size of the input. This means that no matter how large the input is, the algorithm takes a constant amount of time to execute.
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def get_element(arr, index):
return arr[index] # O(1) operation
In this example, accessing an element in an array takes constant time, regardless of the array’s size.
An algorithm is said to have O(log n) time complexity if its running time grows logarithmically as the input size increases. Logarithmic time complexities often arise in search algorithms where the input is halved with each step (such as binary search).
def binary_search(arr, target):
low, high = 0, len(arr) - 1
while low <= high:
mid = (low + high) // 2
if arr[mid] == target:
return mid
elif arr[mid] < target:
low = mid + 1
else:
high = mid - 1
return -1
In binary search, the search space is halved at each step, which results in O(log n) time complexity.
An algorithm is said to have O(n) time complexity if the running time grows linearly with the size of the input. This is typically seen in algorithms that process each element of the input once.
def find_max(arr):
max_value = arr[0]
for num in arr:
if num > max_value:
max_value = num
return max_value
In this case, we iterate over each element in the array exactly once, making this an O(n) operation.
O(n log n) time complexity typically occurs in algorithms that combine both linear and logarithmic operations. These types of complexities are common in efficient sorting algorithms like Merge Sort and QuickSort.
def merge_sort(arr):
if len(arr) <= 1:
return arr
mid = len(arr) // 2
left_half = merge_sort(arr[:mid])
right_half = merge_sort(arr[mid:])
return merge(left_half, right_half)
def merge(left, right):
sorted_arr = []
i = j = 0
while i < len(left) and j < len(right):
if left[i] < right[j]:
sorted_arr.append(left[i])
i += 1
else:
sorted_arr.append(right[j])
j += 1
sorted_arr.extend(left[i:])
sorted_arr.extend(right[j:])
return sorted_arr
Here, the algorithm divides the array into halves recursively (log n) and then merges them (n), resulting in an overall O(n log n) time complexity.
An algorithm is said to have O(n²) time complexity if its running time grows quadratically with the size of the input. This often happens with nested loops, where each iteration of an outer loop triggers another full iteration of an inner loop.
def bubble_sort(arr):
n = len(arr)
for i in range(n):
for j in range(0, n - i - 1):
if arr[j] > arr[j + 1]:
arr[j], arr[j + 1] = arr[j + 1], arr[j]
In bubble sort, there are two nested loops, and the algorithm compares each pair of adjacent elements. As a result, the time complexity is O(n²).
An algorithm is said to have O(2^n) time complexity if its running time doubles with each additional element in the input. This type of complexity is often seen in brute-force recursive algorithms that solve problems by exploring all possible solutions.
def fibonacci(n):
if n <= 1:
return n
return fibonacci(n - 1) + fibonacci(n - 2)
In the above recursive Fibonacci function, the algorithm computes the Fibonacci number by making two recursive calls at each step. The result is an O(2^n) time complexity due to the exponentially growing number of recursive calls.
An algorithm is said to have O(n!) time complexity if its running time grows as a factorial of the input size. This is typically found in algorithms that explore all possible permutations of the input.
The Traveling Salesman Problem (TSP) involves finding the shortest possible route that visits each city exactly once. A brute-force solution explores every possible route, which results in O(n!) time complexity.
Focus on the Worst-Case Scenario:
Simplify Your Expressions:
Look for Dominant Terms:
Use Best, Worst, and Average Case Analysis:
Optimize Space Complexity:
Practice with Real-World Examples:
def quickselect(arr, k):
if len(arr) == 1:
return arr[0]
pivot = arr[len(arr) // 2]
left = [x for x in arr if x < pivot]
right = [x for x in arr if x > pivot]
if k <= len(left):
return quickselect(left, k)
elif k > len(arr) - len(right):
return quickselect(right, k - (len(arr) - len(right)))
else:
return pivot
Big O notation is an essential concept to understand for any software engineer, especially when preparing for coding interviews. It provides a way to evaluate and compare the efficiency of algorithms, helping you make informed decisions about which algorithms to use based on time and space constraints. By mastering Big O notation and practicing with different algorithmic complexities, you will be well-equipped to tackle coding interview questions with confidence.
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