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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