Dynamic Programming: Optimizing Solutions
- 1. DYNAMIC PROGRAMMING: OPTIMIZING SOLUTIONS IN DATA STRUCTURES AND ALGORITHMS LARANA, INC.
- 2. INTRODUCTI ON Dynamic Programming (DP) Importance Link to Data Structures and Algorithms
- 3. Steps in Dynamic Programming UNDERSTANDING DYNAMIC PROGRAMMING - Optimization technique for problems with overlapping subproblems and optimal substructure Definition and Key Concept - Characterizing the problem (subproblems) - Defining the state and state transition - Constructing the solution
- 4. 1D DP CORE TECHNIQUES IN DP 2D DP Memoization vs Tabulation
- 5. APPLICATION OF DATA STRUCTURES IN DP Arrays and Matrices: Store answers to subproblems for fast access (e.g., knapsack problem). Graphs: Find shortest paths or solve problems involving nodes and edges. Strings: Solve problems like finding the longest matching sequence or editing one string to become another.
- 6. 1. DEFINE SUBPROBLEMS: BREAK THE BIG PROBLEM INTO SMALLER, SIMILAR PROBLEMS. PROBLEM-SOLVING APPROACH How to Recognize a DP Problem: Overlapping Subproblems: The problem can be broken into smaller parts that repeat. Example: Climbing stairs — the number of ways to reach each step depends on the steps before it. Optimal Substructure: Solving smaller parts leads to the best solution for the whole problem. Example: Finding the shortest path in a maze — solving smaller sections helps find the overall shortest path. Steps to Solve a DP Problem: 2. RECURRENCE RELATION: FIND A RULE THAT CONNECTS EACH SMALL PROBLEM TO THE NEXT. 3. BASE CASE(S): SET STARTING POINTS THAT DON’T DEPEND ON SMALLER PROBLEMS. 4. BUILD THE SOLUTION ITERATIVELY: SOLVE EACH SUBPROBLEM STEP-BY-STEP, STARTING FROM THE BASE CASE AND WORKING UPWARDS.
- 7. COMPLEXITY ANALYSIS IN DP Time Complexity: How long it takes to solve depends on how many smaller problems there are. Space Complexity: How much memory it uses depends on how many answers it needs to store. Trick to Save Space: Only keep what you need right now, like storing results for the last two steps in the stairs example.
- 9. EXAMPLE PROBLEMS AND SOLUTIONS COIN CHANGE
- 10. EXAMPLE PROBLEMS AND SOLUTIONS LONGEST INCREASING SUBSEQUENCE (LIS):
- 11. EXAMPLE PROBLEMS AND SOLUTIONS KNAPSACK PROBLEM
- 12. EXAMPLE PROBLEMS AND SOLUTIONS TRAVELING SALESMAN PROBLEM (TSP)
- 13. EXAMPLE PROBLEMS AND SOLUTIONS DP WITH BIT MASKING
- 14. CHALLENGES IN DYNAMIC PROGRAMMING High memory consumption Difficulty in identifying subproblems Optimization pitfalls
- 15. DYNAMIC PROGRAMMING IN REAL-WORLD APPLICATIONS AI AND MACHINE LEARNING (MARKOV DECISION PROCESSES) GAME THEORY AND DECISION-MAKING ALGORITHMS BIOINFORMATICS: SEQUENCE ALIGNMENT (E.G., DNA SEQUENCING)
- 16. ADVANTAGES AND DISADVANTAGES Advantages • Saves Time: By reusing results, DP avoids solving the same problem multiple times. • Efficient: Can handle complex problems that would take too long otherwise. • Organized: Provides a clear, step-by-step process to reach the solution. Disadvantages • High Memory Use: Storing many solutions can take up a lot of space. • Hard to Identify: It can be tricky to figure out if a problem can be solved with DP. • Setup Time: Setting up subproblems and rules can be time- consuming.
- 17. CONCLUSI ON Dynamic Programming (DP) is a powerful method for solving complex problems efficiently by breaking them into smaller, manageable parts and reusing solutions to save time. It works best when problems have repeating subparts and can be solved step-by-step. By recognizing patterns, defining subproblems, and following a structured approach, DP helps find optimal solutions to challenges in areas like optimization, pathfinding, and sequence alignment. With practice, understanding and applying DP can significantly improve problem-solving skills, especially when combined with the right data structures.
- 18. THANK YOU! MEMBERS: Bonto, Rhyle Abrio, Ayer Khali Arevalo, Gabriel Montalan, Mark Galanga, Jenelle