Dynamic programming is an optimization technique invented by Richard Bellman in 1950, designed to solve problems with overlapping subproblems by storing solutions to avoid recomputation. It involves characterizing the structure of optimal solutions, defining those solutions recursively, and computing the solutions either through top-down caching or a bottom-up approach. Applications of dynamic programming include problems like the knapsack problem, shortest path calculations, and matrix chain multiplication.

1. Prepared by-
▪ Bhavin Darji
Guided by –
SUBJECT-ADA (2150703)
Introduction to Dynamic Programming, Principle of Optimality

2. Introduction
▪ Typically applied to optimization problem.
▪ Invented by a U.S. Mathematician Richard Bellman in 1950.
▪ In the word Dynamic Programming the word programming stands for
planning.

3. ▪ Dynamic Programming is technique for solving problems with
overlapping subproblems.
▪ In this method each subproblem is solved only once.
▪ The result of each subproblem is recorded in a table from which we
can obtain a solution to the original problem.
▪ In Dynamic computing duplications in solution is avoided totally.
▪ Efficient then Divide and conquer strategy.
▪ Uses bottom up approach of problem solving.

4. Dynamic Programming
▪ Dynamic Programming is an algorithm design technique for
optimization problems: often minimizing or maximizing.
▪ Like divide and conquer, DP solves problems by combining solutions
to subproblems.
▪ Unlike divide and conquer, subproblems are not independent.
▪ Subproblems may share subproblems
▪ However, solution to one subproblem may not affect the solutions
to other subproblems of the same problem.

5. DP reduces computation by
▪ Solving subproblems in a bottom-up fashion.
▪ Storing solution to a subproblem the first time it is solved.
▪ Looking up the solution when subproblem is encountered again.

6. Key Idea
 The key idea is to save answers of overlapping smaller sub-problems
to avoid recomputation.
▪ determine structure of optimal solutions.

7. Steps in Dynamic Programming
1. Characterize structure of an optimal solution.
2. Define value of optimal solution recursively.
3. Compute optimal solution values either top-down with
caching or bottom-up in a table.
4. Construct an optimal solution from computed values.

8. Dynamic Programming Properties
• An instance is solved using the solutions for smaller instances.
• The solutions for a smaller instance might be needed multiple times, so
store their results in a table.
• Thus each smaller instance is solved only once.
• Additional space is used to save time.

9. Principle of Optimality
▪ Obtains the solution using principle of optimality.
▪ In an optimal sequence of decisions or choices, each subsequence must
also be optimal.
▪ When it is not possible to apply the principle of optimality it is almost
impossible to obtain the solution using the dynamic programming
approach.
▪ Example: Finding of shortest path in a given graph uses the principle
of optimality.

10. Problem solving using Dynamic Programming
▪ Various problem that can be solved using dynamic
programming are-
▪ Calculating the Binomial coefficient
▪ Making change problem
▪ Assembly line scheduling
▪ Knapsack problem
▪ Shortest path
▪ Matrix chain Multiplication

11. ThankYou 