Longest Increasing Subsequence Problem

Oct 13, 2025 by miodeya.com 44 views

Problem Description

A new Amazon intern encountered a challenging task. Currently, the intern has n integers, where the value of the ith element is represented by the array element values [i]. The intern is curious to play with arrays and subsequences. The task is to find the maximum sum of a subsequence in the given array.

Understanding the Problem

The problem can be broken down into smaller sub-problems. The intern needs to find the maximum sum of a subsequence in the given array. A subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements.

Key Concepts

Array: A collection of elements of the same data type stored in contiguous memory locations.
Subsequence: A sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements.
Maximum Sum: The largest possible sum of a subsequence.

Solution Approach

To solve this problem, we can use dynamic programming. Dynamic programming is a method for solving complex problems by breaking them down into smaller sub-problems, solving each sub-problem only once, and storing the solutions to sub-problems to avoid redundant computation.

Code Implementation

Here is a Python code implementation of the solution:

def max_subsequence_sum(arr):
    n = len(arr)
    dp = [0] * n
    dp[0] = arr[0]
    for i in range(1, n):
        dp[i] = max(arr[i], dp[i-1] + arr[i])
    return max(dp)
arr = [-2, -3, 4, -1, -2, 1, 5, -3]
print(max_subsequence_sum(arr))

Explanation

The code implementation uses a dynamic programming approach to solve the problem. The max_subsequence_sum function takes an array as input and returns the maximum sum of a subsequence.

The code initializes a dynamic programming table dp of size n, where n is the length of the input array. The first element of the dp table is set to the first element of the input array.

The code then iterates over the input array from the second element to the last element. For each element, it calculates the maximum sum of a subsequence ending at that element by taking the maximum of the current element and the sum of the current element and the maximum sum of a subsequence ending at the previous element.

Finally, the code returns the maximum sum of a subsequence from the dp table.

Time Complexity

The time complexity of the code implementation is O(n), where n is the length of the input array. This is because the code iterates over the input array once.

Space Complexity

The space complexity of the code implementation is O(n), where n is the length of the input array. This is because the code uses a dynamic programming table of size n.

Conclusion

In this article, we discussed a challenging task faced by a new Amazon intern. The task was to find the maximum sum of a subsequence in a given array. We used a dynamic programming approach to solve the problem and provided a Python code implementation of the solution. The code implementation has a time complexity of O(n) and a space complexity of O(n), where n is the length of the input array.

Future Work

In the future, we can explore other approaches to solve this problem, such as using a greedy algorithm or a divide-and-conquer approach. We can also experiment with different input arrays and analyze the performance of the code implementation.

References

Dynamic Programming: A method for solving complex problems by breaking them down into smaller sub-problems, solving each sub-problem only once, and storing the solutions to sub-problems to avoid redundant computation.
Maximum Sum Subsequence Problem: A problem of finding the maximum sum of a subsequence in a given array.
Python Programming Language: A high-level, interpreted programming language that is widely used for various purposes, including web development, scientific computing, and data analysis.
Q&A: Maximum Sum Subsequence Problem =====================================

Frequently Asked Questions

Q1: What is the Maximum Sum Subsequence Problem?

A1: The Maximum Sum Subsequence Problem is a problem of finding the maximum sum of a subsequence in a given array. A subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements.

Q2: What is a subsequence?

A2: A subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements.

Q3: How do I solve the Maximum Sum Subsequence Problem?

A3: You can solve the Maximum Sum Subsequence Problem using a dynamic programming approach. The dynamic programming approach involves breaking down the problem into smaller sub-problems, solving each sub-problem only once, and storing the solutions to sub-problems to avoid redundant computation.

Q4: What is the time complexity of the dynamic programming approach?

A4: The time complexity of the dynamic programming approach is O(n), where n is the length of the input array.

Q5: What is the space complexity of the dynamic programming approach?

A5: The space complexity of the dynamic programming approach is O(n), where n is the length of the input array.

Q6: Can I use a greedy algorithm to solve the Maximum Sum Subsequence Problem?

A6: No, you cannot use a greedy algorithm to solve the Maximum Sum Subsequence Problem. The greedy algorithm approach is not suitable for this problem because it does not consider all possible subsequences.

Q7: Can I use a divide-and-conquer approach to solve the Maximum Sum Subsequence Problem?

A7: No, you cannot use a divide-and-conquer approach to solve the Maximum Sum Subsequence Problem. The divide-and-conquer approach is not suitable for this problem because it does not consider all possible subsequences.

Q8: How do I implement the dynamic programming approach in Python?

A8: You can implement the dynamic programming approach in Python using the following code:

def max_subsequence_sum(arr):
    n = len(arr)
    dp = [0] * n
    dp[0] = arr[0]
    for i in range(1, n):
        dp[i] = max(arr[i], dp[i-1] + arr[i])
    return max(dp)
arr = [-2, -3, 4, -1, -2, 1, 5, -3]
print(max_subsequence_sum(arr))

Q9: What is the output of the code?

A9: The output of the code is the maximum sum of a subsequence in the given array.

Q10: Can I use the code to solve other problems?

A10: Yes, you can use the code to solve other problems that involve finding the maximum sum of a subsequence.

Conclusion

In this article, we discussed the Maximum Sum Subsequence Problem and provided a dynamic programming approach to solve the problem. We also provided a Python code implementation of the solution and answered frequently asked questions about the problem.

Future Work

References

Dynamic Programming: A method for solving complex problems by breaking them down into smaller sub-problems, solving each sub-problem only once, and storing the solutions to sub-problems to avoid redundant computation.
Maximum Sum Subsequence Problem: A problem of finding the maximum sum of a subsequence in a given array.
Python Programming Language: A high-level, interpreted programming language that is widely used for various purposes, including web development, scientific computing, and data analysis.