In-Place Merge Sort Quiz and Flashcards

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Questions and Answers

What is the time complexity of the Mergesort algorithm according to the analysis provided?

$2 N (\log N - 1)$
$2 N \log N$ (correct)
$2 N \log (2N)$
$2 N$

Which of the following statements about the memory usage of Mergesort is correct?

Mergesort does not use any extra memory besides the input array.
Mergesort uses extra space proportional to $2N$.
Mergesort is an in-place sorting algorithm that uses $\leq c \log N$ extra memory.
Mergesort uses extra space proportional to $N$. (correct)

Consider the Mergesort implementation where insertion sort is used for small subarrays. What is the purpose of this optimization?

To improve the time complexity of Mergesort.
To improve the stability of the Mergesort algorithm.
To reduce the overhead of Mergesort for tiny subarrays. (correct)
To reduce the memory usage of Mergesort.

What is the key property of Mergesort that ensures its stability?

Equal items never move past each other during the merge step. (C) Signup and view all the answers

Which of the following sorting algorithms is not stable?

Selection sort (A) Signup and view all the answers

What is the purpose of the `less()` method in the Java code for the Selection sort implementation?

To compare the elements in the array and determine the minimum element. (C) Signup and view all the answers

What is the goal of the in-place merge demo mentioned in the text?

To replace two sorted subarrays with a single sorted subarray (D) Signup and view all the answers

In the Java implementation of the merge function, what does the 'assert isSorted(a, lo, mid);' statement signify?

It ensures the subarray a[lo..mid] is in sorted order. (B) Signup and view all the answers

What does an assertion in Java do when the boolean condition is false?

Throws an exception (A) Signup and view all the answers

For mergesort, which of the following is a best practice when using assertions?

Checking internal invariants (C) Signup and view all the answers

In the context of sorting algorithms, what does an assertion help developers detect?

Logic bugs (C) Signup and view all the answers

When implementing the merge function in Java, what does the 'k' variable represent?

Index for merging elements back into array 'a' (A) Signup and view all the answers

What is the time complexity of the Mergesort algorithm in the worst case?

$O(n \log n)$ (D) Signup and view all the answers

In the merge function implementation, which of the following assertions should be true for the subarrays being merged?

Both the left and right subarrays are sorted in ascending order. (D) Signup and view all the answers

What is the recurrence relation that describes the time complexity of the Mergesort algorithm?

$T(n) = 2T(n/2) + O(n)$ (B) Signup and view all the answers

Which of the following statements is true about the Mergesort algorithm?

It uses the divide-and-conquer approach. (D) Signup and view all the answers

Consider the following assertion in Java: `assert isSorted(a, lo, mid);`. What does this assertion check?

It checks if the subarray <code>a[lo]</code> to <code>a[mid]</code> is sorted in ascending order. (B) Signup and view all the answers

What is the maximum number of auxiliary arrays required by the Mergesort algorithm in the worst case?

$O(n)$ (B) Signup and view all the answers

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Study Notes

Mergesort Time Complexity

Mergesort has a time complexity of O(n log n) in the worst case.
The recurrence relation for Mergesort is T(n) = 2T(n/2) + O(n).

Memory Usage

Mergesort typically requires O(n) auxiliary space for merging.
With an optimized implementation using insertion sort for small subarrays, memory usage may improve for small inputs.

Optimization Purpose

The incorporation of insertion sort for small subarrays enhances efficiency, as insertion sort is faster for small datasets.

Stability of Mergesort

Key property that ensures stability: Mergesort merges equal elements in their original order, preserving relative positions.

Non-Stable Sorting Algorithm

Selection sort is an example of an unstable sorting algorithm since it can change the relative order of equal elements.

`less()` Method in Selection Sort

The less() method is used to compare elements to determine their order during the selection process.

In-Place Merge Demo

The in-place merge demo aims to demonstrate how to merge subarrays without needing additional auxiliary arrays, optimizing space usage.

Assertion in Merge Function

The statement 'assert isSorted(a, lo, mid);' checks that the left half of the array is sorted before merging.

Assertions in Java

An assertion provides a way to test assumptions in the code; if the condition is false, an AssertionError is thrown.

Best Practices for Assertions

Use assertions to enforce pre-conditions in the algorithm, ensuring that input assumptions are valid and improving debugging.

Assertions in Sorting Algorithms

Assertions help developers detect logical errors and incorrect input states during sorting operations.

Role of Variable 'k' in Merge Function

In the merge function, the 'k' variable represents the current index in the array where the next merged element should be placed.

Mergesort Worst-Case Time Complexity

The worst-case time complexity for Mergesort remains at O(n log n), consistent across all cases.

Assertions for Merged Subarrays

Assertions in the merge function should confirm both subarrays are sorted before merging, ensuring correctness.

Maximum Auxiliary Arrays in Mergesort

In the worst case, Mergesort may require at most one auxiliary array for merging operations, especially in its basic form.

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