Lecture 7: Partitioning & Linear-Time Selection [02/02/2026]

Course: Advanced Data Structures and Algorithms

1. The Partition Problem [00:01:46]

Goal: Rearrange an array $A[p..r]$ around a pivot $x$ such that all elements $\le x$ are on the left, $x$ is in the middle, and all elements $> x$ are on the right.
Lomuto Partition (Incremental):
1. Choose the first element $A[p]$ as the pivot $x$.
2. Maintain an index $q$ (end of smaller region) and $j$ (scanning index).
3. If $A[j] \le x$, swap $A[q+1]$ with $A[j]$ and increment $q$.
4. At the end, swap the pivot $A[p]$ with $A[q]$ to place it at the boundary.
Complexity: $O(n)$ linear time.

Definition: Find the $k$th smallest element in an array of $n$ distinct integers.
Naive Recursive Solution:
- Partition the array and check the pivot's rank $q$.
- Case 1: $q=k$. Return $A[q]$.
- Case 2: $q>k$. Recurse on the left portion $A[1..q-1]$.
- Case 3: $q<k$. Recurse on the right portion $A[q+1..n]$ with new rank $k-q$.
The Bottleneck [01:12:46]: If the pivot choice $A[1]$ is consistently the minimum or maximum, the pruning is ineffective (only 1 element removed). This leads to a Worst-Case Complexity of $O(n^2)$.

To force a linear $O(n)$ complexity, the pivot must be chosen "cleverly."

The Strategy:
1. Divide $n$ elements into groups of 5.
2. Sort each group of 5 (takes $O(1)$ per group, $O(n)$ total).
3. Extract the median of each group.
4. Median of Medians: Recursively call the Selection algorithm on the $\lceil n/5 \rceil$ medians to find the median of the medians ($x$).
5. Use $x$ as the pivot for the main Partition.

Pruning Guarantee: The Median of Medians $x$ is guaranteed to be greater than at least 30% of the elements and smaller than at least 30% of the elements.
The Recurrence Relation: $$G(n) \le 3n + G(n/5) + G(7n/10)$$
- $3n$: Cost of sorting groups and partitioning.
- $G(n/5)$: Cost of finding the median of medians.
- $G(7n/10)$: Cost of the recursive call on the remaining $\approx 70%$ of the data.
Close Form Solution [01:53:47]: Using Strong Induction, it is proven that $G(n) \le 30n$.
Key Criterion [02:08:33]: For a recurrence $G(n) \le cn + G(pn) + G(qn)$ to be linear, the sum of fractions $p+q$ must be strictly less than 1.
- Here: $1/5 + 7/10 = 9/10 < 1$.
- Note: Grouping by 3 ($1/3 + 2/3 = 1$) fails to be linear.

Communication Gap: Many students missed the official mail due to asynchronous registration and technical hiccups with SML accounts.
The Fix:
- Deadline: Extended to February 10th.
- Protocol: TAs will post to a new group email (CS5800-2026) and the Google Drive simultaneously.
Philosophical Encouragement [02:20:01]: The instructor urges students to value "Natural Intelligence"—critical thinking, spotting weaknesses, and fine-tuning designs—over automated solutions.

Key Terms Corrected: