Lecture 6: Binary Search Formalization & Order Statistics Intro [28/01/2026]

Course: Advanced Data Structures and Algorithms

1. Decrymental Design: Binary Search Logic [00:00:12]

The instructor reviews Binary Search as the canonical example of Decrymental Design (pruning).

Requirement: The array must be sorted. Searching an unsorted array requires Linear Search ($O(n)$), while a sorted array allows $O(\log n)$ efficiency.
Pruning: By checking the middle value, 50% of the search space is eliminated in each iteration.

Translating high-level intuition into concrete code requires defining clear variables and boundaries.

Search Interval: Represented by L (Left) and R (Right).
Subtlety of Boundary [00:17:17]:
- A non-empty sub-array satisfies $L \le R$, containing $R-L+1$ elements.
- If $L > R$, the sub-array is empty (the element is not found).
Iteration Logic:
- $M = \lfloor (L+R)/2 \rfloor$.
- If $A[M] = X$: Success (Return Index).
- If $A[M] > X$: Shrink from right ($R = M-1$).
- If $A[M] < X$: Shrink from left ($L = M+1$).
Termination Convention [00:26:42]: Return -1 to represent "Not Found." This is a design decision to maintain a uniform integer output type.

The instructor provides the Recursive version of Binary Search, which calls itself with updated L or R values.
Programming Fundamentals [00:42:33]:
- Formal Parameters: The template variables (A, X, L, R) used in the function definition/specification.
- Actual Parameters: The specific values (B, Y, 0, 1500) passed during the invocation or call.

The Power of Logarithms: Dividing by 2 repeatedly reaches 1 in $\log n$ steps. For $10^6$ items, this takes only ~20 iterations.
Sorted Array Limitations: While great for searching ($O(\log n)$), sorted arrays are inefficient for dynamic insertions and deletions ($O(n)$ work to shift elements).
Red-Black Trees: Briefly introduced as a structure that handles search, insert, and delete all in $O(\log n)$ time.

A new problem is presented: finding specific positional elements in a set $S$.

Definition of Rank [00:55:13]: The number of elements in the set strictly smaller than $X$.
Kth Smallest Element:
- 1st Smallest = Minimum.
- $n$th Smallest = Maximum.
- The goal is to find the $k$th smallest element for any $k$ (e.g., the 473rd smallest in a set of 7,948).
Relevance: Crucial for competitive exams (percentiles, ranking).
Teaser [01:06:38]: There is a "stunningly amazing" $O(n)$ solution for this selection problem that uses pruning, to be discussed next class.

Class Interruptions: Technical issues (power/net) caused some frustration.
Assignment Access: Several students (August 2025 and new batches) reported missing SML accounts and Google Drive access.
Action Plan: The instructor will personally contact admin (Ravi Chandran) to ensure all students have proper IDs and Drive access.
Deadline Extension: Because of these technical delays, the first assignment deadline is extended by an extra week.
OneNote Difficulty: The instructor mentions that the current iPad version of OneNote has made it difficult to export/mail lecture notes, contributing to the delay in posting PDFs.

Key Terms Corrected: