INTRODUCTION

What is an algorithm?

What is a Problem?

Problem Instance Determining GCD is a problem – How many actual problems? GCD of 1 and 2 ? GCD of 234 and 42? More? Obviously yes. Problem instance – A problem with a specific input – E.g., find a GCD of 42 and 14

Purpose of this class

Calculating Fibonacci Sequence Fibonacci sequence – 1, 1, 2, 3, 5, 7, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584

The Problem Input: – a positive number N Output: – F n (the n th Fibonacci Number) Example instances – Ex. 1: N = 10 – Ex. 2: N = 15 – Ex. 3: N = 0 N = -4 is not an instances of this problem!!!

Approach 1 Array based method: dynamic programming on linear structure (linear time)

Approach 2 method: Recursive (exponential time) F(9) F(8)F(7) F(6) F(5) … …

Approach 3 Method: Divide and Conquer (logarithmic)

Approach 3 Find exponential Method: Divide and Conquer (logarithmic)

Approach 4 Method: Closed form solution Golden Ratio

Conclusion Difference Design  Difference Performance This class emphasizes on designing “efficient algorithm”

Algorithm Again It is the essence of the computation

Side Note on Algorithm Named after a Persian mathematician “Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī” Wrote book on linear equation Introduce number 0

TOPICS OVERVIEW Analysis part

Asymptotic Notation Measurement of “efficiency” of algorithms – How to compare two algorithms – How to predict behavior of the algorithm

Big O analysis How to determine Big O of some code Recurrent Relation

NP-Complete What computer could solve – Efficiently – Inefficiently The difference between “Efficiency” and “Inefficiency”

TOPICS OVERVIEW Synthesis part

Divide and Conquer Solve an instance of the problem by dividing it into smaller instances – Based on induction Example Problems: – Sorting (Quick Sort & Merge Sort) – Maximum Contiguous Sum of Subsequence – Modulo Exponential – Closest Pair

Dynamic Programming Aim to reduce redundancy in computation For the case when there are several overlapping sub-problems Example Problems: – Fibonacci Number – Binomial Coefficient – Matrix Chain Multiplication – Longest Common Subsequence – Largest Square in Binary Picture (Island problem)

Greedy Algorithm Solve the problem by doing the best for the current step Proof of correctness Example Problems: – Minimum Spanning Tree Prim’s Algorithm Kruskal’s Algorithm

Graph Algorithm Algorithm related to graph structure – Breadth First Search – Depth First Search – Shortest Path

Search Solve the problem by enumeration – Systematical enumeration – Performance improvement Branch and Bound Backtracking