Dynamic programming is a technique used in computer science and mathematics to solve complex optimization problems by breaking them down into smaller overlapping subproblems. It is often used in competitive programming to efficiently solve problems that would otherwise take a long time to compute.
Dynamic programming is a method used to solve problems by breaking them down into smaller subproblems and storing the solutions to those subproblems. By storing the solutions to subproblems, dynamic programming avoids redundant computations, saving time and making the solution more efficient.
In other words, dynamic programming solves a problem by solving smaller versions of the same problem and combining their solutions to find the final solution. It is based on the principle of "optimal substructure," which states that an optimal solution can be constructed from optimal solutions of its subproblems.
Dynamic programming problems often possess two important characteristics, namely overlapping subproblems and optimal substructure.
Overlapping Subproblems: Dynamic programming problems can be divided into smaller subproblems that are solved independently. These subproblems often overlap, meaning that the same subproblem is solved multiple times. Dynamic programming takes advantage of this overlapping property by storing the solutions to subproblems and reusing them when needed.
Optimal Substructure: Dynamic programming problems exhibit optimal substructure, meaning that a globally optimal solution can be obtained by combining optimal solutions to its subproblems. This allows dynamic programming to efficiently compute the globally optimal solution by solving smaller subproblems and building up to the final solution.
To solve a dynamic programming problem, you can follow these general steps:
Identify if the problem can be divided into smaller subproblems. Look for patterns that indicate overlapping subproblems.
Formulate a recurrence relation that defines the solution to the problem in terms of solutions to its subproblems. This recurrence relation serves as a mathematical equation that expresses how the problem can be broken down into smaller subproblems.
Determine the base case(s) - the smallest subproblems that can be solved directly. These base cases usually represent the simplest or smallest instance of the problem.
Design an algorithm to compute the solution to the problem by solving the subproblems and combining their solutions efficiently. Dynamic programming algorithms often use memoization or tabulation techniques to store and retrieve solutions to subproblems.
Analyze the time and space complexity of the algorithm to ensure it meets the desired efficiency requirements. Dynamic programming algorithms can have various time and space complexity depending on the specific problem and the chosen approach.
The Fibonacci series is a classic example of a problem that can be efficiently solved using dynamic programming. The Fibonacci series is defined as a sequence of numbers where the next number is the sum of the previous two numbers in the sequence.
Using dynamic programming, the Fibonacci series can be computed efficiently by breaking it down into smaller subproblems. We can use either a bottom-up or top-down approach to solve this problem.
In the bottom-up approach, we start from the base cases (F(0) = 0 and F(1) = 1) and iteratively compute the Fibonacci numbers up to the desired number. Each step involves solving smaller subproblems and storing their solutions to avoid redundant computations.
In the top-down approach (also known as memoization), we start from the desired number and recursively compute the Fibonacci numbers by memoizing the solutions to subproblems. This saves time by avoiding redundant computations and using stored solutions when available.
Both approaches demonstrate the power of dynamic programming in efficiently solving a well-known mathematical problem.
Dynamic programming is a powerful technique used in computer science to solve complex optimization problems efficiently. By breaking down a problem into smaller subproblems and storing their solutions, dynamic programming avoids redundant computations and provides an optimal solution. Understanding the concept of dynamic programming can greatly enhance your problem-solving skills, especially in competitive programming using C++.
noob to master © copyleft
noob to master