INTRODUCTION TO RECURSIVEFORMULATIONS FOR ALGORITHM DESIGN: II

Partha P ChakrabartiIndian Institute of Technology Kharagpur

Overview Algorithms and Programs Pseudo-Code Algorithms + Data Structures = Programs Initial Solutions + Analysis + Solution

Refinement + Data Structures = Final Algorithm

Use of Recursive Definitions as Initial Solutions

Recurrence Equations for Proofs and Analysis

Solution Refinement through Recursion Transformation and Traversal

Data Structures for saving past computation for future use

Sample Problems:1. Finding the Largest2. Largest and Smallest3. Largest and Second Largest4. Fibonacci Numbers5. Searching for an element in an ordered

/ unordered List6. Sorting7. Pattern Matching 8. Permutations and Combinations9. Layout and Routing10. Shortest Paths

3rd Problem: Largest and Second LargestSequential Comparison

Largest and 2nd Largest: Recursive Formulation

Largest and Next: Tournament

Tournaments & Sorting

Final Algorithms & Data Structuring

Algorithms + Data Structures = Progams

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 = Final Algorithm

Use of Recursive Definitions as Initial Solutions

Recurrence Equations for Proofs and Analysis

Solution Refinement through Recursion Transformation and Traversal

Data Structures for saving past computation for future use

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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