Lecture-1+2-DAA-Math-Analysis(6)
Page 1: Introduction
Copyright Regulation 1969: Material is copied by Curtin University of Technology under the Copyright Act 1968.
Page 2: Course Structure
Course: Design and Analysis of Algorithms
Focus Areas:
Big O Notation
Recurrence Relations
Page 3: Pre-requisites
Understanding of Data Structures and Algorithms is required.
Advanced data structures will be used to enhance algorithm performance.
Page 4: Recommended Textbooks
Introduction to Algorithms by Cormen, Leiserson, and Rivest (3rd or 4th edition).
Page 5: Definition of Algorithm
Origin: Al Khwarizmi (9th century); his methodologies led to the term 'algorithm'.
An algorithm is defined as:
A computational procedure.
Takes input and produces output.
Solves a computational problem.
Characteristics:
Correctness: must always provide the right output.
Tractability: must solve in reasonable time.
Forms: can be expressed in various ways (English, code, designs).
Page 6: Example Problem - Sorting
Sorting Problem:
Input: sequence of n numbers (a1, a2, ..., an)
Output: reordered sequence (b1, b2, ..., bn) satisfying b1 ≤ b2 ≤ ... ≤ bn
Example:
Input: (30, 20, 41, 51, 3, 20)
Output: (3, 20, 20, 30, 41, 51)
Page 7: Insertion Sort Algorithm
Insertion Sort implementation:
Loop through elements from 2 to n.
Identify
keyas the current element.Rearrange elements in sorted order.
Characteristics:
In-place sorting: only requires constant memory.
Stable: maintains order of repeated elements.
Efficient for small/sorting-nearly sorted inputs.
Page 8: Insertion Sort Illustration
Stepwise Sorting Example:
Start with: [5, 2, 4, 6, 1, 3]
Proceeding through steps until fully sorted.
Page 9: Example Problem - Greatest Common Divisor (GCD)
GCD Definition:
Input: Two integers X and Y.
Output: Largest integer Z that divides both X and Y.
Example:
Input: X = 1035, Y = 759
Output: GCD = 69.
Methods to calculate GCD:
Factorization method.
Euclidean algorithm.
Page 10: Euclid's Algorithm for GCD
Recursive Method:
GCD(X, Y) = GCD(X mod Y, Y).
Example implementation shown.
Page 11: Least Common Multiple (LCM)
LCM Definition:
Input: Two integers X and Y.
Output: Smallest integer Z divisible by both X and Y.
Example: LCM of 1035 and 759 = 11385.
Page 12: Computing LCM via GCD
LCM Formula:
LCM(X, Y) = (X * Y) / GCD(X, Y).
Example for computation shown.
Page 13: Integer Multiplication Example
Example: Multiply X = 12, Y = 34.
Straightforward multiplication yield Z = 408.
Page 14: Al Khwarizmi’s Multiplication Algorithm
Steps:
Halve the first number.
Double the second.
Continue until the first number is 1.
Sum results based on odd-numbered rows.
Page 15: Large Integer Multiplication
For n-digit numbers, straightforward methods become inefficient.
Divide and Conquer Approach:
Calculate using smaller subproblems.
Page 16: Karatsuba's Algorithm
Karatsuba's Multiplication:
Acknowledged for reducing multiplications from four to three.
Offers improved performance over traditional methods.
Page 17: Adding Consecutive Numbers
Input: Integer n (e.g., 1 + 2 + ... + n)
Efficient method: Sum = n(n + 1)/2.
Example with n = 10 gives total 55.
Page 18: Algorithm Efficiency
Algorithms can significantly differ in performance.
Example: Insertion Sort vs. Merge Sort time comparisons.
Page 19: CPU Comparison and Efficiency
Even slower CPUs can outperform faster ones with optimal algorithms due to efficiency.
Page 20: Designing an Algorithm
Steps involve reducing problems, employing brute force, improving efficiency.
Page 21: Algorithm Design Techniques
Tricks:
Better data structures for efficient access.
Preprocessing input for algorithm optimization.
Page 22: Advanced Design Techniques
Divide & Conquer, Greedy Approach, and Dynamic Programming.
Page 23: Algorithm Efficiency Metrics
Measure speed and sometimes memory use.
Page 24: Computation Model Basics
Usually based on Random Access Machine (RAM) principles.
Page 25: Operations in RAM Model
Illustrative examples of operation steps in algorithms.
Page 26: Insertion Sort Cost Analysis
Breakdown of time complexity in terms of steps.
Page 27: Insertion Sort Running Time
Discusses varying scenarios based on element arrangement.
Page 28: Best Case Scenario for Insertion Sort
Sorted array scenario leading to optimal complexity.
Page 29: Worst Case for Insertion Sort
Analysis of reverse-sorted input yielding maximum complexity.
Page 30: Function Growth
Faster growing functions become dominant for large n.
Page 31: Big O Notation Basics
O(g(n)) represents an asymptotic upper bound.
Page 32: Big O Example
Demonstration of function comparisons using Big O notation.
Page 33: Example of Proving Big O
Providing proofs for function bounds using constants.
Page 34: Order of Constant Operations
Constants represented in Big O notation for algorithm steps.
Page 35: Analysis of Execution Order
Executing lines repetitively leads to linear complexity.
Page 36: Final Complexity of Fragments
Summarizing individual complexities to derive overall complexity.
Page 37: Back to Example
Demonstrating further on overall complexity summation.
Page 38: Simple Big O Visualizations
Illustratic examples portraying algorithmics in graph form.
Page 39: Understanding Complexity Proofs
Conveys that complexity proofs may not always be necessary.
Page 40: Nested Loops
Analyzing iteration complexities in nested structures.
Page 41: Aim for Tight Bound
Discusses the importance of achieving tight upper bounds in Big O notation.
Page 42: Big-Omega Notation W(n)
Representation of asymptotic lower bounds.
Page 43: Example of Big Omega
Demonstrating the proof of lower bounds through function comparisons.
Page 44: Big-Theta Notation Q(n)
Defines asymptotic tight bounds encompassing both upper and lower bounds.
Page 45: Examples of Big-Theta Proofs
Guiding through proofs to confirm respective bounds.
Page 46: Extended Proof Techniques
Discussing deeper strategies and examples for Big-Theta proofs.
Page 47: Other Algorithmic Examples
Various algorithm examples to demonstrate theoretical bounds and their implications.
Page 48: Additional Analysis of Operations
Signs of complexity can originate from single or multiple operations.
Page 49: Worst Case Analysis
Examining how worst case scenarios are determined.
Page 50: Best Case Analysis
Conditions under which best case efficiencies occur.
Page 51: Average Case Complexity
Discussing uniform probability of inputs leading to average complexities.
Page 52: Expected Case Analysis
Identifying standard inputs leading to expected performance.
Page 53: Summary of Algorithm Time Analysis
Recap on modeling computation and strategies for efficiency measurement.
Page 54: Memory Analysis in Algorithms
Discusses considerations regarding memory use in practical applications.
Page 55: Selection Sort Algorithm
Stepwise breakdown of a fundamental sorting algorithm.
Page 56: Selection Sort Example
Illustrates how selection sort achieves an ordered array.
Page 57: Time Complexity Analysis of Selection Sort
Detailed examination of time complexities in selection sorting.
Page 58: Inductive Proof Basics
Introduces induction as a method to prove algorithm correctness.
Page 59: Induction and Algorithm Analysis
Extended view on how induction applies to algorithm proof.
Page 60: Inductive Example
Concrete example applying inductive principles in proofs.
Page 61: Fibonacci Sequence Proof
Illustrates the Fibonacci sequence characteristics and its analyzation.
Page 62: Induction Proof of Fibonacci
Steps to establish the bounds of Fibonacci numbers using induction.
Page 63: Analyzing Recursive Functions
Discussing how to manage time complexity in recursive functions through recurrence.
Page 64: Recursive Selection Sort Analysis
Analysis of the recursive approach to selection sorting.
Page 65: Iterative Method for Recurrence
Discussing strategies to analyze time complexity via iterative substitution.
Page 66: Inductive Strategy for Recurrences
Instruction on utilizing guesswork paired with induction for proof.
Page 67: Recursion Tree Application
Framework to visualize and derive complexities from recursive calls.
Page 68: Proving Time Complexity
Further exploration into proving time bounds of recursive functions.
Page 69: Master Method Overview
Understanding the Master Theorem for deriving complexities of algorithms.
Page 70: Case Analysis in Master Theorem
Detailed explanation of conditions that determine complexity cases.
Page 71: Master Theorem Application
Guidelines for employing the Master Theorem in analysis.
Page 72: Example of Master Theorem Application (Case 1)
Illustration of solving a particular case using the Master Theorem.
Page 73: Another Case Example with Master Theorem
Further example analysis within different framework conditions.
Page 74: Master Theorem for Advanced Complexity
Case studies revisiting previous examples under enhanced complexity conditions.
Page 75: Completing Case 3 Analysis
Overview of satisfying Master Theorem conditions under Case 3 scenarios.
Page 76: Finalizing the Master Theorem Cases
Show full expectations of scenario outcomes based on derived results.
Page 77: Proving Algorithmic Correctness via Loop Invariants
Establishing correctness through established criteria of loop invariants.
Page 78: Correctness of Insertion Sort
Application of loop invariant proof to verify insertion sort correctness.
Page 79: Continued Verification of Insertion Sort
Iterative aspects of proving the insertion sort maintains its sorting property.
Page 80: Selection Sort Correctness Confirmation
Detailed proof illustrating selection sort correctness through invariant usage.