Merge-Sort Algorithm Overview

Merge-Sort Overview

• Running time for merge-sort is Θ(n).
• Merge-sort is a divide-and-conquer algorithm for sorting an array.
• It recursively divides an array into two halves, sorting each half before merging them back together.

Merge-Sort Process

• The initial call is made with MERGE-SORT(A, 1, A.length) to sort the entire array A.
• The division process involves calculating q = ⌊n/2⌋ to split the array into sub-arrays A[p,..., q] and A[q + 1,..., r].

Recurrence Relation

• T(n) defines the running time for an algorithm with n elements.
• For small problems, when n ≤ c (c is a constant), the solution takes Θ(1) time.
• For larger problems, with a = b = 2 in merge-sort, the recurrence relation is:
  • T(n) = aT(n/b) + D(n) + C(n)

Master Theorem

• The Master Theorem provides a method to analyze the running time of divide-and-conquer algorithms.
• If T(n) = aT(n/b) + f(n), then specific conditions determine T(n):
  • T(n) = Θ(n^(log_b a)) if f(n) = O(n^(log_b a - ϵ)) for some ϵ > 0.

Merge Procedure

• The MERGE function combines two sorted sub-arrays A[p,..., q] and A[q + 1,..., r].
• It uses two sentinel values (∞) to simplify the merging process.
• The pseudo-code includes determining lengths of sub-arrays and initializing loops for merging.

Merge Function Analysis

• The merge process is efficient with a time complexity of O(n).
• Essential steps include copying elements and merging in a single pass.

Importance of Sorting Algorithms

• Sorting is fundamental in algorithms, often serving as a preprocessing step for other applications.
• Various sorting techniques exist, including insertion-sort, which is Θ(n²) in the worst case.
• Merge-sort and other algorithms like heapsort and quicksort offer better performance.

Divide-and-Conquer Strategy

• The divide-and-conquer strategy involves three main steps:
  • Divide: Split the problem into smaller sub-problems.
  • Conquer: Solve each sub-problem recursively.
  • Combine: Merge the solutions of the sub-problems to form a complete solution.

Recursive Functions Example

• Recursive functions can be demonstrated through Fibonacci numbers with defined relationships:
  • f(0) = 0, f(1) = 1, and f(n) = f(n-1) + f(n-2) for n ≥ 2.

This quiz covers the merge-sort algorithm, a fundamental concept in computer science for sorting arrays. It details the procedure of dividing arrays into subarrays and the merging process. Perfect for understanding time complexity and algorithm design.