https://file.aiquickdraw.com/mathbot/file/1725575772272qnpu76zu.pdf

The uploaded PDF contains a set of problems related to data structures, algorithm analysis, and recurrence relations. Here is a breakdown of the problems and the provided solutions:

### Problem Breakdown and Solutions:

#### **1. Basics (7 points total)**
(a) Is \( f(n) \in O(n^2) \) also in \( o(n^2) \)?  
- Answer: NO. A function in \(O(n^2)\) is not necessarily in \(o(n^2)\) because \(O(n^2)\) includes functions that grow as fast as or slower than \(n^2\), but \(o(n^2)\) specifically refers to functions that grow strictly slower than \(n^2\).

(b) Is \( f(n) \in \Omega(n \log^2 n) \) also in \( o(n^2) \)?  
- Answer: MAYBE. This depends on the growth rate of \(f(n)\). If \(f(n) = n \log^2 n\), it is indeed in \(o(n^2)\). However, if the growth rate is slightly higher, it may not be.

(c) Is \( f(n) = n \log_2 n \) in \( o(2^n) \)?  
- Answer: YES. \( n \log_2 n \) grows much slower than \(2^n\), so it is in \(o(2^n)\).

#### **2. Bubble Sort and Binary Search (8 points total)**
(a) Provide an input array for which Bubble Sort takes \( \Theta(nk) \) time.  
- Solution: An example array and explanation are given, showing the process and the number of iterations required.

(b) A modified binary search algorithm with a recurrence relation is analyzed.  
- Solution: The recurrence relation \( T(n) = T(n/4) + 3 \) is solved, and it's concluded that the modified algorithm is slower than the standard binary search's \( O(\log n) \) complexity.

#### **3. Tries and Balanced Binary Trees (5 points total)**
Analysis of insertion and querying times for a set of integers treated as strings in base 3, using prefix trees.  
- Solution: Both insertion and query operations are shown to take \( O(\log_3 N) \). The binary search tree comparison is also made, suggesting that while a balanced binary tree is more efficient in terms of comparisons, both are asymptotically similar.

#### **4. Recurrences (17 points total)**
Several recurrence relations are provided, and the task is to solve them using appropriate methods (Master theorem, Akra-Bazzi, etc.).  
- Example (a): \( T(n) = T(n/2) + n + 2 \) with base case \( T(1) = 2 \).  
- Example (b): \( T(n) = n T(\sqrt{n})^2 \) with base cases \( T(1) = 4 \) and \( T(1) = 16 \).  
- Example (c): A divide-and-conquer analysis comparing different split strategies.

#### **5. Computing Intersections (5 points total)**
An alternative algorithm is proposed to compute intersections of inverted index sets with a logarithmic complexity improvement.  
- Solution: The algorithm and its time complexity are explained using binary search to reduce the number of operations needed.

#### **6. Dynamic Arrays: Is Doubling Important? (8 points total)**
Amortized analysis of a new "add" procedure for dynamic arrays where the array size is increased by 50 instead of doubling.  
- Solution: The total cost and amortized cost analysis conclude that this approach results in \( O(N) \) amortized time complexity for \( N \) add operations.

### Next Steps

Would you like a detailed solution for a specific problem from this set? Here are some follow-up questions to consider:

1. Could you explain in detail the Master Theorem application for Problem 4(a)?
2. How does the alternative binary search method in Problem 2(b) compare to other searching algorithms?
3. What would be an optimal way to implement the alternative intersection algorithm in Problem 5?
4. Can you provide a step-by-step breakdown of the amortized analysis for Problem 6?
5. Are there any more efficient algorithms than Bubble Sort for cases like in Problem 2(a)?

**Tip:** Always double-check your recurrence relations and their base cases when applying solving techniques like the Master Theorem.

This solution breaks down problems related to algorithm complexity, including Big-O notation, recurrence relations, and amortized analysis. Learn how to solve recurrence relations using the Master theorem and Akra-Bazzi method, analyze Bubble Sort, and explore dynamic array growth.

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