The document presents an overview of dynamic programming, outlining its definition, method of operation, and applications in solving complex problems efficiently. Key applications discussed include the shortest path problem, knapsack problem, and the Floyd-Warshall algorithm, which is specifically highlighted for its simplicity and effectiveness in finding shortest paths in weighted graphs. The advantages of dynamic programming include its efficiency, ability to provide optimal solutions, and suitability for complex problem-solving.

1. DYNAMIC PROGRAMMING
PRESENTED BY
GUNDLAPALLY SATHWIKA
22D21A0525
CSE3A

2. TOPICS
• Introduction to Dynamic Programming
• Applications of Dynamic Programming
• Example of DP
• All Pair Shortest path algorithm
• Floyd-warshall algorithm example
• Advantages of Dynamic Programming

3. INTRODUCTION
• Dynamic Programming is an algorithmic technique used in computer science and mathematics to
solve complex problems by breaking them down into smaller overlapping subproblems.
• Dynamic Programming is mainly an optimization over plain recursion.
• How DP Works?
• Identify Subproblems
• Store Solutions
• Build up Solutions
• Avoid Redundancy

4. APPLICATIONS OF DP
1.Finding the shortest path problem
2.Knapsack problem
3.Travelling Salesman problem
4.Optimal Binary Search Tree
Example :
Let N=4

5. ALL PAIR SHORTEST PATHS(FLOYD-WARSHALL
ALGORITHM)
• The Floyd-Warshall algorithm is a dynamic programming algorithm used to
discover the shortest paths in a weighted graph, which includes negative weight
cycles.
• Advantages:
• It is extremely simple.
• It is easy to implement.

6. Example:
Consider the following graph
Steps:
Write the initial distance matrix.
It represents the distance between every pair of vertices in the form of given weights.
For diagonal elements (representing self-loops), distance value = 0.
For vertices having a direct edge between them, distance value = weight of that edge.
For vertices having no direct edge between them, distance value = ∞.

11. ADVANTAGES OF DYNAMIC PROGRAMMING
• Efficiency
• Optimal solutions
• Versatility
• Handles complex problems
• Memory efficiency
• Time Complexity :O(n^3)
• Space Complexity :O(n^2)