Mergesort and Quicksort Algorithms

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Questions and Answers

What are the three steps in the basic plan for mergesort?

Divide the array into two halves. 2. Recursively sort each half. 3. Merge the two halves.

In the context of the abstract in-place merge demo, what is the goal?

Given two sorted subarrays a[lo] to a[mid] and a[mid+1] to a[hi], replace with sorted subarray a[lo] to a[hi].

What condition must be met to stop the standard mergesort algorithm early (before merging) and what type of arrays benefit most from this optimization?

Stop if the largest item in the first half is less than or equal to the smallest item in the second half. Partially ordered arrays benefit the most from this optimization.

What is the key idea behind bottom-up mergesort, and how does it differ from the standard mergesort approach?

Bottom-up mergesort involves passing through the array, merging subarrays of increasing sizes, starting from size 1, instead of recursively dividing the array. Signup and view all the answers

In terms of computational complexity, what is an 'optimal algorithm'?

An optimal algorithm is one that provides the best possible cost guarantee for solving a particular problem X. Signup and view all the answers

Explain the relevance of a 'decision tree' in the context of sorting algorithms' computational complexity.

A decision tree models all possible comparisons an algorithm makes. The height of the tree represents the worst-case number of comparisons required to sort the elements. Signup and view all the answers

What is the significance of the statement: "Lower bound may not hold if the algorithm can take advantage of the initial order of the input"?

If an algorithm can leverage existing order in the input data, it might perform better than the theoretical lower bound, which assumes a worst-case scenario. Signup and view all the answers

In the context of sorting algorithms, what does it mean for a sort to be 'stable'?

A stable sort preserves the relative order of items with equal keys. Signup and view all the answers

Of insertion sort, selection sort, shellsort and mergesort, which are stable?

Insertion sort and mergesort are stable. Signup and view all the answers

Explain why insertion sort is stable.

Equal items never move past each other. Signup and view all the answers

Explain why selection sort is not stable.

Long-distance exchange can move one equal item past another one. Signup and view all the answers

Explain why shellsort is not stable.

Long-distance exchanges. Signup and view all the answers

Explain why mergesort is stable.

Takes from left subarray if equal keys. Signup and view all the answers

In terms of the 'running time estimates', how many compares per second can a laptop execute?

10^8 compares per second. Signup and view all the answers

In terms of the 'running time estimates', how many compares per second can a supercomputer execute?

10^12 compares per second. Signup and view all the answers

What is the number of compares $C(N)$ to mergesort an array of length $N$?

$C(N) \le C([N/2]) + C([N/2]) + N$ for $N > 1$, with $C(1) = 0$. Signup and view all the answers

Mergesort uses ≤ how many array accesses to sort an array of length $N$?

$6 N lg N$ Signup and view all the answers

Mergesort uses extra space proportional to what?

$N$ Signup and view all the answers

What is 'natural mergesort'?

Exploit pre-existing order by identifying naturally-occurring runs. Signup and view all the answers

Name one algorithm that is now widely used and is 'natural mergesort'.

Timsort Signup and view all the answers

What sorting algorithm is commonly used for small subarrays within mergesort to improve performance, and why is it beneficial?

Insertion sort. It has lower overhead for small sizes. Signup and view all the answers

Explain the purpose of the 'eliminate the copy to the auxiliary array' optimization in mergesort and how it's achieved.

To save time (but not space). It achieves this by switching the roles of the input and auxiliary array in each recursive call. Signup and view all the answers

What is the primary advantage of using bottom-up mergesort over the standard recursive mergesort, and what is its main disadvantage?

Advantage: Simplicity and non-recursive implementation. Disadvantage: It is slower than the typical top-down mergesort on typical systems. Signup and view all the answers

What is Timsort and what are its key features?

Timsort is an adaptive, stable, natural mergesort use binary insertion sort to make initial runs (if needed) and a few more clever optimizations. Signup and view all the answers

Explain what the term 'computational complexity' refers to in the context of algorithms.

Framework to study efficiency of algorithms for solving a particular problem $X$. Signup and view all the answers

Distinguish between the 'Upper bound' and 'Lower bound' in computational complexity.

Upper bound: Cost guarantee provided by some algorithm for X. Lower bound: Proven limit on cost guarantee of all algorithms for X. Signup and view all the answers

Why is mergesort considered an optimal algorithm?

Algorithm with best possible cost guarantee for $X$. Signup and view all the answers

Is mergesort optimal with respect to space usage.

Mergesort is not optimal with respect to space usage. Signup and view all the answers

Java system sort uses what basic algorithm for sorting objects?

Mergesort Signup and view all the answers

Which of the algorithms discussed can access information based only through compares.

Mergesort Signup and view all the answers

What is the 'complexity of sorting'?

Upper bound: ~$N lg N$ from mergesort. Lower bound: ~$N lg N$. Optimal algorithm = mergesort. Signup and view all the answers

If $D(N)$ satisfies $D (N) = 2 D (N / 2) + N$ for $N > 1$, with $D (1) = 0$, then $D (N)$ = ?

$N lg N$ Signup and view all the answers

What is the key advantage of using comparators in sorting algorithms?

Decouples the definition of the data type from the definition of what it means to compare two objects of that type. Signup and view all the answers

How does mergesort's space complexity affect its suitability for sorting very large datasets?

Mergesort's requires extra space proportional to N makes it less suitable for sorting very large datasets due to potential memory constraints. In-place merge algorithms are very hard to create. Signup and view all the answers

In what scenarios would you choose bottom-up mergesort over the standard recursive mergesort?

When simplicity and a non-recursive version of mergesort is needed. Signup and view all the answers

Describe a scenario where the stability of a sorting algorithm is crucial.

First, sort by name; then sort by section. Signup and view all the answers

What are the typical steps involved in implementing a comparator in Java?

Define a (nested) class that implements the Comparator interface. Implement the compare() method. Signup and view all the answers

Explain what optimization is applied to MergeSort to help reduce running time when using Java 6?

Cutoff to insertion sort = 7, Stop-if-already-sorted test, and Eliminate-the-copy-to-the-auxiliary-array trick. Signup and view all the answers

If you need to sort strings of varying case (both lowercase and uppercase), how would you use a Comparator to achieve case-insensitive sorting?

Arrays.sort(a, String.CASE_INSENSITIVE_ORDER); Signup and view all the answers

List the steps of how to create a 'natural order' sort.

public class Date implements Comparable<Date> ... public int compareTo(Date that) { ... } Signup and view all the answers

Flashcards

Mergesort Basic Plan

Divide array in half, recursively sort each half, then merge halves.

In-place Merge Goal

Given two sorted subarrays, replace with single sorted subarray.