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Efficient algorithms: Your guide to mastering complexity

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Nov 18, 2024

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Exploring algorithm complexity Techniques for improving algorithm efficiency 1. Optimizing the algorithms Example: The Two Sum problem Naive solution Improved solution Optimized solution 2. Choosing the right data structure 3. Applying algorithmic paradigms Conclusion

In software development, algorithms are the building blocks for creating efficient and scalable applications. Well-designed algorithms can minimize resource usage and optimize performance, which is crucial for modern software solutions. Good algorithms improve the software performance and user experience. Therefore, companies are always looking for developers who can write efficient code. As a result, knowledge of efficient algorithms is a primary element of coding interviews because they help demonstrate how well you can solve problems and think logically.

In this blog, we'll discuss what algorithm efficiency means, strategies to improve it, and practical coding examples to understand these principles.

Exploring algorithm complexity#

Before diving into how to optimize algorithms, it’s important to understand how to assess their efficiency using algorithm complexity metrics. Algorithm complexity refers to measuring an algorithm’s efficiency in terms of time and space. It provides insight into how an algorithm’s running time or space requirement grows relative to the input size. It lets us compare different algorithms and choose the most efficient one for a given problem.

Different notations are used to express algorithm complexity, which are as follows:

Big O (O): Indicates the upper bound of an algorithm complexity, representing the worst-case scenario.
Big Omega (Ω): Indicates the lower bound of an algorithm complexity, representing the best-case scenario.
Big Theta (Θ): Indicates upper and lower bounds, representing the average time complexity.

Regarding algorithm complexity, two metrics used to evaluate its efficiency are time and space complexity.

Time complexity evaluates the time an algorithm takes to complete as a function of the input size. It helps us understand how an algorithm’s running time grows relative to the input size.
Space complexity measures the amount of memory an algorithm occupies as a function of the input size. It includes the memory allocated for data structures and the call stack during recursive calls.

Common complexity classes include constant $O(1)$ , logarithmic $O(\log n)$ , linear $O(n)$ , quadratic $O(n^2)$ , and exponential $O(2^n)$ .

Techniques for improving algorithm efficiency#

Developing key skills that optimize solutions is important when preparing for coding interviews. An efficient algorithm uses less time and space to solve the problem, which is important for handling large datasets and complex architectures. Because of this, interviewers strongly emphasize algorithmic efficiency to see if you can devise solutions that work better within the time and space constraints.

There are many ways to solve a particular coding problem, but not all solutions are efficient enough to be practical in real-world applications. We can apply the following strategies to improve the algorithm efficiency:

Optimizing the algorithms.
Choosing the right data structure.
Applying algorithmic paradigms.

1. Optimizing the algorithms#

Given a coding problem, its naive algorithm is the straightforward solution to solve it without taking care of the efficiency. On the other hand, the main focus of an optimized algorithm is to provide a correct solution while ensuring efficiency. An optimized algorithm gives similar correct results as the naive algorithm but with minimal time and space complexity.

Let’s examine an example in which we discuss three different strategies for solving the same problem and see how applying optimized approaches improves an algorithm’s efficiency.

Example: The Two Sum problem#

Given an array of integers, nums, and an integer, target, return indexes of the two numbers such that they add up to target.

What would the time and space complexities of an optimized solution for the Two Sum problem be?

def two_sum(arr, target):
    n = len(arr)
    for i in range(n):
        for j in range(i + 1, n):
            if arr[i] + arr[j] == target:
                return [i, j]
    
# Driver code
def main():
    arrs = [[3, 7, 1, 2, 8, 4, 5],
            [-1, 2, 1, -4, 5, -3],
            [2, 3, 4, 1, 7, 9],
            [1, -1, 0],
            [5, 4, 2, 7, 6, 0, -8]]
    targets = [3, -1, 12, 0, 8]
    for i in range(len(arrs)):
        print(i + 1, ".\tInput array: ", arrs[i], sep="")
        print("\tTarget: ", targets[i], sep="")
        print("\tSolution: ", two_sum(arrs[i], targets[i]), sep="")
        print("-"*100)
if __name__ == '__main__':
    main()

Two Sum: Naive solution

def binary_search(arr, target, start):
    left, right = start, len(arr) - 1
    while left <= right:
        mid = (left + right) // 2
        if arr[mid] == target:
            return mid
        elif arr[mid] < target:
            left = mid + 1
        else:
            right = mid - 1
    return -1
def two_sum(arr, target):
    arr_with_indices = [(num, i) for i, num in enumerate(arr)]
    arr_with_indices.sort()
    
    for i in range(len(arr_with_indices)):
        complement = target - arr_with_indices[i][0]
        temp_arr = [num for num, _ in arr_with_indices]
        j = binary_search(temp_arr, complement, i + 1)
        if j != -1:
            return [arr_with_indices[i][1], arr_with_indices[j][1]]
    
# Driver code
def main():
    arrs = [[3, 7, 1, 2, 8, 4, 5],
            [-1, 2, 1, -4, 5, -3],
            [2, 3, 4, 1, 7, 9],
            [1, -1, 0],
            [5, 4, 2, 7, 6, 0, -8]]
    targets = [3, -1, 12, 0, 8]
    for i in range(len(arrs)):
        print(i + 1, ".\tInput array: ", arrs[i], sep="")
        print("\tTarget: ", targets[i], sep="")
        print("\tSolution: ", two_sum(arrs[i], targets[i]), sep="")
        print("-"*100)
if __name__ == '__main__':
    main()

Two Sum: Improved solution

Optimized solution#

Imagine your interviewer has asked you to further optimize your solution. Now, you need to think out of the box and develop an even more optimized solution. One such approach involves iterating and inserting elements into a hash table. While iterating, we also check if the complement of the current element already exists in the hash table. If yes, we return the indexes of the two elements.

The time and space complexities of this solution are $O(n)$ and $O(n)$ , respectively.

def two_sum(arr, target):
    hashmap = {}
    for i in range(len(arr)):
        complement = target - arr[i]
        if complement in hashmap:
            return [i, hashmap[complement]]
        hashmap[arr[i]] = i
    
# Driver code
def main():
    arrs = [[3, 7, 1, 2, 8, 4, 5],
            [-1, 2, 1, -4, 5, -3],
            [2, 3, 4, 1, 7, 9],
            [1, -1, 0],
            [5, 4, 2, 7, 6, 0, -8]]
    targets = [3, -1, 12, 0, 8]
    for i in range(len(arrs)):
        print(i + 1, ".\tInput array: ", arrs[i], sep="")
        print("\tTarget: ", targets[i], sep="")
        print("\tSolution: ", two_sum(arrs[i], targets[i]), sep="")
        print("-"*100)
if __name__ == '__main__':
    main()

Two Sum: Optimized solution

Therefore, one of the effective strategies for optimizing the algorithms is implementing suitable coding patterns. Coding patterns are reusable templates or approaches to solve similar programming problems. The Grokking the Coding Interview Patterns course series is a valuable resource for learning and practicing coding patterns.

2. Choosing the right data structure#

Almost every problem asked in the interview involves the use of data structures. Choosing the right data structure can significantly improve the performance of an algorithm, optimizing its time and space complexity. For example:

Arrays provide random access in constant time but tend to be inefficient for dynamic data where frequent insertions or deletions are needed.
Linked lists are introduced to handle dynamic data at the expense of access times.
Trees offer good search time by keeping the data in an order based on certain criteria. For example, binary search trees maintain the data in sorted order.
Hash maps offer constant-time search operations by keeping the data in key-value pairs using a hash function.

It can be observed in the different solutions to the Two Sum problem discussed above that choosing the right data structure improved the computation time from $O(n\log n)$ to $O(n)$ . Let’s look into another example where the choice of the data structure significantly improves the algorithm complexity.

Consider there’s a continuous stream of integers, and the task is to find the median of the numbers at any point in time in $O(1)$ . The naive approach to solving this problem is to sort the list every time a new number is inserted. This would result in $O(n \log n)$ time complexity for each insertion. Its optimized solution is to maintain two heaps: a max-heap to store the lower half of the numbers and a min-heap for the upper half. For each new number, add it to the max-heap if it’s smaller than or equal to the top of the max-heap. Otherwise, add it to the min-heap. The median can be calculated using the top elements of the two heaps in constant time.

Therefore, choosing the right data structure is key to implementing and designing optimal algorithms. Resources like Data Structures for Coding Interviews in Java, available in various programming languages, are helpful for revising key data structure concepts. These courses provide practical examples and exercises that reinforce learning. Knowing the strengths and limitations of different data structures enables us to choose the right one for our problems, ensuring that our algorithms perform optimally and handle different edge cases effectively.

3. Applying algorithmic paradigms#

Algorithmic paradigms are fundamental approaches used to solve a wide range of complex problems in a structured and effective manner. They empower algorithms to handle large test cases that would otherwise fail using naive algorithms. One of the fundamental principles of algorithmic paradigms is to divide a complex problem into smaller subproblems.

Imagine you’re in a coding interview, and the interviewer poses a complex coding problem. You’re struggling to find an optimal solution. A good starting point is to check if any of the following algorithmic paradigms can be applied to the problem:

Divide and conquer: Break a problem into smaller subproblems, solve them independently, and combine their solutions for the final result.
Greedy algorithms: Find a locally optimal solution at each step to find a global optimal solution.
Dynamic programming: Solve a problem by combining the solutions of overlapping subproblems to avoid redundant calculations. A dynamic programming interview prep course, like this one, can be very helpful when preparing to apply this algorithmic paradigm.

Conclusion#

Efficient algorithms are necessary for any developer to create high-performance software solutions, whether they’re just starting their career or have been in the industry for a long time. Consistency is the key—practicing different data structures and algorithm problems to master these skills. To serve that purpose, Educative-99 is a special list of curated coding problems to help learners ace their coding interview preparation in a reasonable amount of time. Whether you’re preparing for a coding interview or working on real-world applications, algorithm efficiency will always be the backbone of your success.

To help you effectively prepare for coding interviews, Educative-99 is available in multiple languages, including Python, JavaScript, Java, C++, and Go.

Frequently Asked Questions

How do you calculate the time complexity of an algorithm?

Time complexity is typically calculated by counting the number of basic operations an algorithm performs as a function of input size, usually expressed using Big-O notation.

Which time complexity is best?

O(1), representing constant time complexity, is considered the most optimal. This means the algorithm executes a single operation without needing any iteration. Following that is O(log n), which is also favorable, along with other complexities ranked accordingly.

Why is Big-O notation important in analyzing algorithms?

Big-O notation provides a way to express the worst-case scenario of an algorithm’s running time, helping developers understand the algorithm’s scalability and efficiency as the input size grows.

What is the most efficient algorithm ever?

Quicksort is widely recognized as a highly efficient sorting algorithm with numerous applications. It is particularly well-suited for large datasets, as its average-case time complexity of $O(n \log n)$ allows it to sort substantial amounts of data rapidly.

Can an algorithm be efficient for time but not space, or vice versa?

Yes, an algorithm can be time-efficient but use a large amount of memory (space-inefficient), or it can be space-efficient but require a significant amount of time to execute (time-inefficient).

What are the features of an efficient algorithm with an example?

An efficient algorithm typically has five key properties: well-defined inputs, well-defined outputs, unambiguity, finiteness, language independence, and feasibility. These characteristics enable algorithms to serve as powerful tools for innovation and effective problem-solving across various domains.

Written By:

Muhammad Hamza

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Data Structure	Solution	Time Complexity	Space Complexity
None	Sort the list every time a new number is inserted.	O(n^2)	O(n)
Heaps	Maintain two heaps: a max-heap to store the lower half of the numbers and a min-heap for the upper half. Add the new number to the max-heap if it’s smaller than or equal to its top. Otherwise, add it to the min-heap. The median can be calculated using the top elements of the two heaps.	O(logn)	O(n)