Instructor & TA Team

• Instructor: Tianle Ma
• TA: Fan Li

Student Introductions

• Who are you?
• Name & Major
• Why are you taking this course?
• One interesting fact / Favorite Algorithm?
• Turn to your neighbor and introduce yourself!

Welcome to Design and Analysis of Algorithms

• What is this course about?
• Moving from 'programming' to 'algorithm design'
• Correctness and Efficiency as primary goals

Syllabus Highlights

• 24+ Lectures + Midterm + Final
• 10 Assignments (Written & Programming)
• Java is the required language (StdIn/StdOut)
• No complex IDE dependencies (grading via terminal)

Academic Integrity & AI

• AI tools (ChatGPT, Gemini, Claude, Grok, etc.) ARE allowed for assistance.
• THE CATCH: You must understand everything you submit.
• We WILL conduct random academic audits (oral/video).
• If you can't explain it, you get a zero.

Definition

• An algorithm is a well-defined computational procedure.
• Input: A value or set of values.
• Output: A value or set of values.
• Constraint: Must be finite and effective.

The Sorting Problem

Example: Insertion Sort (Logical)

Input: sequence of n numbers 
Output: permutation  such that a'1 <= a'2 <= ... <= a'n

1. Start with second element.
2. Compare with previous elements.
3. Shift elements greater than key to the right.
4. Insert key in correct position.

Correctness

• Does the algorithm solve the problem for ALL valid inputs?
• Does it terminate?
• Incorrect algorithms might loop forever or return wrong answers for edge cases.

Efficiency

• Time Complexity: How does runtime grow with input size?
• Space Complexity: How much memory is needed?
• We usually care about 'Worst Case' analysis.
• Why? Reliability.

Java Example: Linear Search

public static int linearSearch(int[] arr, int target) {
    for (int i = 0; i < arr.length; i++) {
        if (arr[i] == target) return i;
    }
    return -1;
}

Analyzing Linear Search

• Best Case: Target is at index 0 (1 comparison).
• Worst Case: Target is not present or at end (n comparisons).
• As n grows, time grows linearly.

Logarithms

• We use base 2 ($\lg n$) in CS mostly.
• Definition: $x = \lg n$ means $2^x = n$.
• $\lg(ab) = \lg a + \lg b$
• $\lg(a^b) = b \cdot \lg a$
• Change of base: $\log_b a = \frac{\ln a}{\ln b}$

Series & Summations

• Arithmetic Series: $1 + 2 + ... + n = \frac{n(n+1)}{2} = \Theta(n^2)$
• Geometric Series: $1 + x + x^2 + ... + x^n = \frac{x^{n+1} - 1}{x - 1}$
• Infinite Geometric Series ($|x| < 1$): $\frac{1}{1-x}$

Floors and Ceilings

• $\lfloor x \rfloor$: Greatest integer $\le x$
• $\lceil x \rceil$: Smallest integer $\ge x$
• Useful for splitting odd-sized arrays (e.g., $n/2$).

Watch Out For...

• Confusing 0-based vs 1-based indexing.
• Thinking 'faster computer' fixes 'bad algorithm'.

Activity 1: Insertion Sort Trace

• ❓ Problem:
• Sort A: [5, 2, 4, 6, 1, 3] using Insertion Sort.
• Take 2 minutes to solve this.

Solution 1

• Sort A: [5, 2, 4, 6, 1, 3] using Insertion Sort.
• Pass 1: [2, 5, 4, 6, 1, 3] (insert 2)
• Pass 2: [2, 4, 5, 6, 1, 3] (insert 4)
• Pass 3: [2, 4, 5, 6, 1, 3] (6 in place)
• Pass 4: [1, 2, 4, 5, 6, 3] (insert 1)
• Pass 5: [1, 2, 3, 4, 5, 6] (insert 3)

Activity 2: Pseudocode Translation

• ❓ Problem:
• Write pseudocode to find the minimum in an array A.
• Take 2 minutes to solve this.

Solution 2

min = A[0]
for i = 1 to n:
    if A[i] < min:
        min = A[i]
return min

Interactive: Insertion Sort

Interactive Demo: insertion_sort_demo.html

Interactive: Linear Search

Interactive Demo: linear_search_demo.html

Interactive: Why Efficiency Matters

Interactive Demo: efficiency_demo.html

Lecture Summary

• An algorithm is a well-defined computational procedure.
• We analyze correctness (does it work?) and efficiency (how fast?).
• Insertion Sort: O(n²) worst case, O(n) best case.
• Next lecture: Asymptotic notation (Big-O, Omega, Theta).

Supplementary Resources

Interactive Demo: worked_examples.html
Interactive Demo: reading_practice.html
Interactive Demo: handout.html