Recurrence Equations 1. (5 Pts) Stooge Sort Has The Followin ✓ Solved

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Recurrence Equations 1. (5 pts) Stooge sort has the following

1. Stooge sort has the following algorithm. Assume you are given an input array of n integers. Recursively sort the lower two-thirds of an array, then recursively sort the upper two-thirds, then recursively sort the lower two-thirds again. The recursion stops when the array consists of two or fewer elements. If the array size is two, the elements are swapped if necessary. Which of the following recurrence equations describe Stooge sort?

a) T(n) = 3T(n/2) + Θ(n)

b) T(n) = 3T(n/2) + Θ(log n)

c) T(n) = 3T(2n/3) + Θ(1)

d) T(n) = 2T(3n/2) + Θ(1)

e) T(n) = 2T(n/3) + Θ(n)

2. Solve the recurrence you chose in the previous question. How would you compare Stooge sort to other standard comparison sorting algorithms?

3. Assume that the recurrence for merge sort is described by: T(20) = 1 T(2n) = 2T(2n-1) + 2n. Show how to solve this recurrence for T(2n).

4. Analyze the time complexity of the sorting algorithm provided in the implementation.

5. What is the time complexity of computing an inner product?

6. What is the time complexity of computing the cosine between two vectors?

7. What is the time complexity of the matrix product of an n × m matrix and an m × k matrix?

8. Explain why T(n) = 8T(n/2) + cn^2 if n > 1 and T(2) = 8 models the time complexity of matrix multiplication and solve this recurrence.

9. What is the big-O time complexity of the recurrence resulting from Strassen’s method of matrix multiplication?

10. How does this time complexity compare to the standard O(n^3) time cost of matrix multiplication?

11. Describe a fast exponentiation algorithm that computes a^b using binary representation and squaring.

12. Prove the sum of an arithmetic sequence by induction.

13. Provide a mathematical induction proof for arithmetic-geometric sums.

14. Decide whether statements regarding asymptotic order notation are true or false with justifications.

15. Discuss multiprocessor scheduling as an NP-complete problem and analyze a greedy pseudo-code algorithm for it.

16. Describe a dynamic programming algorithm for optimal matrix product parenthesizing.

Paper For Above Instructions

Stooge sort is an unusual sorting algorithm characterized by its recursive approach to sorting elements. The algorithm recursively sorts the lower two-thirds of an array, then the upper two-thirds, and lastly the lower two-thirds again, making it an interesting case for analysis through recurrence relations.

Initially, we need to analyze the provided recurrence equations to identify which correctly describes Stooge sort. The relevant analysis leads to option (c) T(n) = 3T(2n/3) + Θ(1). To understand this, we observe that the sorting's recursive nature necessitates dividing the elements into two-thirds, which aligns with the (2n/3) term in the equation. Therefore, we affirm that option (c) accurately reflects the behavior of the Stooge sort algorithm.

The next step is to solve the recurrence T(n) = 3T(2n/3) + Θ(1). We can apply the Master Theorem for a structured approach. This recurrence indicates that we have a = 3 (the number of recursive calls), b = 3/2 (the factor by which the input size is reduced), and f(n) = Θ(1). According to the Master Theorem, we compute n^(log_b(a)), where log_b(a) = log_(3/2)(3). Using the change of base formula, we find that log_(3/2)(3) = log(3)/log(3/2). This gives us a solution of T(n) = Θ(n^(log_3(3))) = Θ(n) since the logarithm precisely corresponds to the polynomial growth in the parallel branches of the recursion. Therefore, Stooge sort has a time complexity of Θ(n^2.709), which is significantly worse than other sorting methods.

To compare Stooge sort with other standard sorting algorithms, it's essential to recognize its inefficiency. For example, algorithms like Merge sort and Quick sort run in O(n log n) time under average conditions, which vastly outperforms Stooge sort's O(n^2.709). Additionally, Bubble sort, often criticized for its lack of efficiency, operates at an O(n^2) time complexity, yet still outperforms Stooge sort in practical implementations. In essence, Stooge sort is a fascinating example of recursion but stands as an impractical option for sorting due to its exceedingly high time complexity.

Next, considering the recurrence for Merge sort, T(20) = 1 and T(2n) = 2T(2n-1) + 2n, we can apply mathematical techniques to solve it. The standard approach involves unrolling the recurrence step by step or leveraging the Master theorem. We recognize that the recursive calls will form multiple layers contributing increasingly towards the log(n) depth, leading us to establish a relationship that can be resolved into the familiar O(n log n) complexity characteristic of Merge sort's efficient design.

Moving onto linear algebra's critical operations such as the inner product and matrix multiplications, these fundamental computations hold significant implications in computer science and mathematics. For instance, the inner product of vectors results in a time complexity of O(n), dictated by the n operations required for multiplication and summation across the vector dimensions. On the other hand, multiplying matrices A (n×m) and B (m×k), requires O(nmk) time, arising from the need to calculate n rows through m column pairings against k inputs.

The complexity of multiplying matrices via the naive approach is notably O(n^3). However, using Strassen’s algorithm, we can reduce this complex calculation to T(n) = 7T(n/2) + O(n^2), thus upwards approximating with a big-O of O(n^log_2(7)), yielding a faster multiplication when scaling matrix sizes exponentially.

In conclusion, sorting and matrix multiplication highlights significant attributes in algorithmic design and applications. Addressing their complexities provides a framework from which algorithm efficiency can be measured and optimized in computed tasks, leading to better time management in larger data arrays. Combining recursive approaches like Stooge sort or Strassen's more efficient technique reflects fundamentally different strategies important in computational theory.

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