What are Dynamic Programming Problems?

Last updated on December 24th, 2024 at 11:44 pm

This article explores Dynamic Programming Problems, the top-down and bottom-up approaches, their differences with recursion, and some classic examples like Fibonacci and Longest Increasing Subsequence.

1. What are Dynamic Programming Problems?

Dynamic Programming is a powerful technique to solve a complicated problem by breaking it down into simpler subproblems. It can be solved by breaking it down into smaller sub-problems, solving each sub-problem only once, and storing the solutions to sub-problems to avoid redundant computation. This approach is particularly useful for problems that have overlapping sub-problems or that can be decomposed into smaller sub-problems.

The idea of dynamic programming is to avoid making redundant method calls. In Divide and Conquer, sub-problems are independent. While In Dynamic Programming, sub-problems are dependent.

Majority of the Dynamic Programming problems can be divided into two types:

1. Optimization problems
2. Combinatorial problems

Some famous Dynamic Programming algorithms are:

• Unix diff – for comparing two files
• Bellman-Ford – for shortest path routing in networks
• TeX – the ancestor of LaTeX
• WASP – Winning and Score predictor

2. Dynamic Programming Strategy

The major components of Dynamic Programming are:

1. Recursion: Solves subproblems recursively.
2. Memoization: Stores already computed values in the table (Memoization means caching)

Dynamic Programming ≈ Recursion + Memoization (i.e. reuse)

Dynamic Programming ≈ controlled brute force

By using memoization, dynamic programming reduces the exponential complexity to polynomial complexity (O(n²), O(n³), etc) for many problems.

3. Dynamic Programming vs Recursion

Dynamic Programming and Recursion are two related but distinct concepts in programming.

1. Recursion: Recursion is a programming technique where a function calls itself repeatedly until it reaches a base case. Recursion can be used to solve problems that have a recursive structure, but it can be inefficient if the problem has overlapping sub-problems.
2. Dynamic Programming: Dynamic Programming is a technique that solves 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. Dynamic Programming can be used to solve problems that have overlapping sub-problems or that can be decomposed into smaller sub-problems.

4. Top Down vs Bottom Up Dynamic Programming

There are two primary approaches to solving Dynamic Programming Problems: Top-Down and Bottom-Up Approach

1. Top-Down Dynamic Programming: In this approach, we start with the original problem and recursively break it down into smaller sub-problems until we reach the base case. The solutions to sub-problems are stored in a memory-based data structure, such as a hash table or an array, to avoid redundant computation.
2. Bottom-Up Dynamic Programming: In this approach, we start with the smallest sub-problems and iteratively build up the solutions to larger sub-problems until we reach the original problem. This approach typically uses a tabular data structure to store the solutions to sub-problems.

5. Fibonacci Dynamic Programming

The Fibonacci sequence is a classic example of a Dynamic Programming Problem. The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding numbers: 0, 1, 1, 2, 3, 5, 8, 13, and so on.

1. Naive Recursive Solution: A naive recursive solution to the Fibonacci problem would involve recursively calculating the nth Fibonacci number by summing the (n-1)th and (n-2)th Fibonacci numbers. However, this approach is inefficient because it involves redundant computation.
2. Dynamic Programming Solution: A Dynamic Programming solution to the Fibonacci problem would involve storing the solutions to sub-problems in a memory-based data structure, such as an array or a hash table. This approach avoids redundant computation and has a time complexity of O(n).

6. Longest Increasing Subsequence Dynamic Programming

The Longest Increasing Subsequence (LIS) problem is another classic example of a Dynamic Programming Problem. The LIS problem involves finding the longest increasing subsequence in a given sequence of numbers.

• Naive Recursive Solution: A naive recursive solution to the LIS problem would involve recursively finding the longest increasing subsequence by considering all possible subsequences. However, this approach is inefficient because it involves redundant computation.
• Dynamic Programming Solution: A Dynamic Programming solution to the LIS problem would involve storing the solutions to sub-problems in a memory-based data structure, such as an array or a hash table. This approach avoids redundant computation and has a time complexity of O(n^2).

7. Common Dynamic Programming Problems and Solutions

Some common Dynamic Programming Problems include:

1. Knapsack Problem: Optimizing the selection of items to maximize value within a weight limit.
2. Edit Distance: Calculating the minimum number of operations to convert one string into another.
3. Coin Change Problem: Finding the minimum number of coins needed to make a specific amount.
4. Matrix Chain Multiplication: Determining the most efficient way to multiply a series of matrices.

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