introduction to recursive formulations for …
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INTRODUCTION TO RECURSIVEFORMULATIONS FOR ALGORITHM DESIGN: IV
Partha P ChakrabartiIndian Institute of Technology Kharagpur
Algorithm Design by Recursion Transformation1. Initial Solution
a. Recursive Definition – A set of Solutionsb. Inductive Proof of Correctnessc. Analysis Using Recurrence Relations
2. Exploration of Possibilitiesa. Decomposition or Unfolding of the Recursion Treeb. Examination of Structures formedc. Re-composition Properties
3. Choice of Solution & Complexity Analysisa. Balancing the Split, Choosing Pathsb. Identical Sub-problems
4. Data Structures & Complexity Analysisa. Remembering Past Computation for Futureb. Space Complexity
5. Final Algorithm & Complexity Analysisa. Traversal of the Recursion Treeb. Pruning
6. Implementationa. Available Memory, Time, Quality of Solution, etc
Algorithms and Programs Pseudo-Code Algorithms + Data Structures = Programs Initial Solutions + Analysis + Solution
Refinement + Data Structures = FinalAlgorithm
Use of Recursive Definitions as InitialSolutions
Recurrence Equations for Proofs andAnalysis
Solution Refinement through RecursionTransformation and Traversal
Data Structures for saving pastcomputation for future use
Coin Selection Problem
First Recursive Definition
Example
Improved Recursive Definition
Alternative Recursive Definition
Example
Traversal and Potential Pruning
Finalizing the Algorithm
Special Case
Summary
Overview of Algorithm Design 1. Initial Solution
a. Recursive Definition – A set of Solutionsb. Inductive Proof of Correctnessc. Analysis Using Recurrence Relations
2. Exploration of Possibilitiesa. Decomposition or Unfolding of the Recursion Treeb. Examination of Structures formedc. Re-composition Properties
3. Choice of Solution & Complexity Analysisa. Balancing the Split, Choosing Pathsb. Identical Sub-problems
4. Data Structures & Complexity Analysisa. Remembering Past Computation for Futureb. Space Complexity
5. Final Algorithm & Complexity Analysisa. Traversal of the Recursion Treeb. Pruning
6. Implementationa. Available Memory, Time, Quality of Solution, etc
1. Core Methodsa. Divide and Conquerb. Greedy Algorithmsc. Dynamic Programmingd. Branch-and-Bounde. Analysis using Recurrencesf. Advanced Data Structuring
2. Important Problems to be addresseda. Sorting and Searchingb. Strings and Patternsc. Trees and Graphsd. Combinatorial Optimization
3. Complexity & Advanced Topicsa. Time and Space Complexityb. Lower Boundsc. Polynomial Time, NP-Hardd. Parallelizability, Randomization
Thank you
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