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How can you determine the optimal substructure of a problem?

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Many problems in computer science can be solved more efficiently by breaking them down into smaller and simpler subproblems, and then reusing the solutions of these subproblems to build the solution of the original problem. This technique is called dynamic programming, and it relies on a key property of the problem: optimal substructure. In this article, you will learn what optimal substructure means, how to identify it, and how to use it to design efficient algorithms.

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Saurabh Singh Gautam

SDE-2 @Adobe | Previous: Microsoft | IIT Roorkee (CSE-21)
Soumojit Ghosh

Engineering @ Lexmark International (R&D) | ex Streebo, Odessa | Transforming logic into innovation

1 What is optimal substructure?

Optimal substructure is the property of a problem that says that an optimal solution of the problem can be constructed from optimal solutions of its subproblems. For example, consider the problem of finding the shortest path between two nodes in a weighted graph. If you know the shortest path from node A to node B, and from node B to node C, then you can easily find the shortest path from node A to node C by concatenating the two paths. Therefore, the shortest path problem has optimal substructure.

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Soumojit Ghosh

Engineering @ Lexmark International (R&D) | ex Streebo, Odessa | Transforming logic into innovation
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In a layman’s term , an optimal substructure is like the small piece of a puzzle which when solved constitutes to solving the bigger puzzle. This forms the basis of any recurrence relation.

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Gourav Verma

Software Developer | @Mentor at Unstop, Ex - Campus Ambassador at IIM–Bangalore, Ex- Campus Ambassador at eDC IIT Delhi and Quantum Robotics | GDSC Member
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Optimal substructure is a key concept in dynamic programming and algorithm design. It refers to the property of a problem where an optimal solution to the problem can be constructed efficiently from optimal solutions of its subproblems. In other words, a problem exhibits optimal substructure if an optimal solution to the overall problem can be composed by efficiently combining optimal solutions to its smaller subproblems. This property is crucial for the application of dynamic programming techniques. The optimal substructure property simplifies problem-solving by breaking down a complex problem into smaller, more manageable subproblems.

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2 How to identify optimal substructure?

One way to identify optimal substructure is to look for recursive or iterative patterns in the problem. If you can express the problem as a function of smaller instances of the same problem, then you may have optimal substructure. For example, consider the problem of computing the nth Fibonacci number, which is defined as F(n) = F(n-1) + F(n-2) for n > 1. You can see that the problem is recursive, and that the solution of F(n) depends on the solutions of F(n-1) and F(n-2), which are smaller instances of the same problem.

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3 How to use optimal substructure?

Once you have identified optimal substructure, you can use it to design a dynamic programming algorithm. The basic idea is to store the solutions of the subproblems in a data structure, such as an array or a table, and then use them to compute the solution of the original problem. This way, you avoid recomputing the same subproblems multiple times, and save time and space. For example, to compute the nth Fibonacci number, you can use an array to store the solutions of F(0), F(1), ..., F(n-1), and then use them to calculate F(n).

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SONU RAJ

Engineering at Zepto | Ex-Flipkart Search | Mentor at PrePlaced | Topmate 1:1
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To determine the optimal substructure of a problem, analyze its characteristics to identify if the problem can be broken down into smaller, overlapping subproblems. Look for patterns where the solution to the overall problem relies on the solutions to its subproblems. If a problem exhibits the optimal substructure property, dynamic programming or recursive approaches are often suitable for solving it efficiently. Consider exploring the problem's recursive nature and how solutions to smaller instances contribute to solving the larger problem.

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4 What are the challenges of optimal substructure?

Optimal substructure is not always easy to find or exploit. Sometimes, the problem may not have optimal substructure at all, or it may have multiple optimal substructures that are not compatible with each other. For example, consider the problem of finding the longest common subsequence of two strings. A subsequence is a sequence of characters that appears in the same order in both strings, but not necessarily consecutively. For example, "ABC" is a subsequence of "ABCD" and "AECB". The problem of finding the longest common subsequence does have optimal substructure, but it is not unique. There may be more than one longest common subsequence for a given pair of strings, and they may not share any common characters. Therefore, you cannot simply concatenate the solutions of smaller subproblems to get the solution of the original problem.

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5 How to deal with multiple optimal substructures?

One way to deal with multiple optimal substructures is to use a more sophisticated data structure that can store multiple solutions for each subproblem, and then choose the best one according to some criterion. For example, to find the longest common subsequence of two strings, you can use a matrix to store the lengths of the longest common subsequences for each pair of prefixes of the strings, and then use a backtracking algorithm to reconstruct the actual subsequences. Alternatively, you can use a heuristic or a randomized algorithm that can approximate the optimal solution with high probability, but without guaranteeing its optimality.

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6 How to practice optimal substructure?

The best way to practice optimal substructure is to solve problems that require dynamic programming. You can find many examples of such problems on online platforms, such as LeetCode, HackerRank, or Codeforces. You can also read books or articles that explain the theory and applications of dynamic programming, such as Introduction to Algorithms by Cormen et al., or Dynamic Programming for Coding Interviews by Mehta and Gupta. By practicing optimal substructure, you will improve your algorithmic thinking and problem-solving skills, and become a better programmer.

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7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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Saurabh Singh Gautam

SDE-2 @Adobe | Previous: Microsoft | IIT Roorkee (CSE-21)
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It's very simple yet very hard for untrained eyes to spot the pattern. CP becomes a lot easier once you start spotting the patterns. And to achieve that 1. You have to actually know the pattern i.e., learn from somewhere or someone 2. Practice, Practice, Practice. I can't stress enough how important it is to practice solving problems and train yourself to spot the pattern in problems, to solve other problems based on the same pattern or with little variation easily. Thanks for reading! Happy Coding!

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How can you determine the optimal substructure of a problem?

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3

4

5

6

7

1 What is optimal substructure?

2 How to identify optimal substructure?

3 How to use optimal substructure?

4 What are the challenges of optimal substructure?

5 How to deal with multiple optimal substructures?

6 How to practice optimal substructure?

7 Here’s what else to consider

Algorithms

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How can you determine the optimal substructure of a problem?

1

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3

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5

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1 What is optimal substructure?

2 How to identify optimal substructure?

3 How to use optimal substructure?

4 What are the challenges of optimal substructure?

5 How to deal with multiple optimal substructures?

6 How to practice optimal substructure?

7 Here’s what else to consider

Algorithms

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