Merge Sort Algorithm Overview

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Questions and Answers

What is the time complexity of Merge Sort in terms of its running time?

O(n^2)

O(n log n) (correct)

O(log n)

O(n)

Which of the following steps is NOT part of the divide and conquer pattern used in Merge Sort?

Merge the solutions into a final solution

Divide the input into smaller subsets

Conquer the subproblems recursively

Generate random numbers for sorting (correct)

What type of tree represents the execution of the Merge Sort algorithm?

Red-black tree

Binary tree (correct)

B-tree

AVL tree

In the merge step of Merge Sort, what condition is checked to determine which element to merge next?

If the index of S1 becomes equal to the length of S1 Signup and view all the answers

What is the height of the Merge Sort tree associated with an execution on a sequence of size n?

dlog n e Signup and view all the answers

What is the base case condition for the Merge Sort recursive function?

When the size of the array is 1 or less Signup and view all the answers

Which sorting algorithm will be introduced later in the course after discussing heaps?

Heap Sort Signup and view all the answers

When merging two sorted arrays, what does line 19 of the merge function check?

If the current element of S1 is less than or equal to that of S2 Signup and view all the answers

What type of time complexity does the merge function have in Merge Sort?

O(n) Signup and view all the answers

What is the purpose of the `copyOfRange` method in Merge Sort implementation?

To create a copy of a specified range of elements Signup and view all the answers

In which case would the `mergeSort` function return without sorting?

When the input size is less than 2 Signup and view all the answers

What is the overall time complexity of the merge sort algorithm?

O(n log n) Signup and view all the answers

What kind of sorting is performed by the combination step of Merge Sort?

Side-by-side merging of sorted halves Signup and view all the answers

During quick sort, how is the pivot typically chosen?

The last element Signup and view all the answers

What does the notation O(n1 + n2) indicate in the context of the Merge Sort algorithm?

The merge function's running time for any two arrays Signup and view all the answers

Which of the following sorting algorithms is NOT classified under divide and conquer?

Bubble Sort Signup and view all the answers

What occurs if the pivot in quick sort is the unique maximum or minimum element?

The algorithm runs in O(n^2) Signup and view all the answers

What is the expected running time of quick sort when subsequences L and G are roughly the same size?

O(n log n) Signup and view all the answers

What type of tree structure represents the execution of quick sort?

Binary tree Signup and view all the answers

What is the height of the merge sort tree?

O(log n) Signup and view all the answers

What does the 'Combine' step in quick sort involve?

Merging L, E, and G Signup and view all the answers

Which of the following best describes the purpose of the 'Conquer' step in quick sort?

Recursively sort the subarrays Signup and view all the answers

In the implementation of quick sort, what does the variable 'temp' store?

The partitioned elements Signup and view all the answers

What happens when the size of the sequence is less than or equal to 1 in quick sort?

The sequence is sorted Signup and view all the answers

What does the running time of quick sort depend on?

All of the above Signup and view all the answers

In the quick sort algorithm, how many subsequences are produced after partitioning around the pivot?

Three Signup and view all the answers

What is the role of the variable 'm' in the implementation of quick sort?

To track the size of L Signup and view all the answers

What is a variation of quick sort that selects pivots at random?

Randomized quick sort Signup and view all the answers

Study Notes

Sorting in N-Log-N And Linear Time

Sorting problem: Given an unordered data set, arrange elements in non-decreasing order.

Merge Sort

Divide and Conquer Pattern: break down a problem into a smaller problem, solve the smaller problems recursively, and combine the solutions.
Algorithm:
- Divide: if the input has 1 or fewer elements, it is already sorted. Otherwise, split the input into two halves.
- Conquer: recursively sort the two halves.
- Combine: merge the two sorted halves into a single sorted sequence.
Performance:
- Time Complexity: O(n log n)
- Space Complexity: O(n) due to the extra space needed for the merging process.

Merge Sort Illustration

The merge sort process can be visualized as a binary tree.
- Each node represents a recursive call, storing both the unsorted sequence before processing and the sorted sequence after.
- The root node represents the initial call to the sorting algorithm.
- Leaf nodes represent calls on subsequences of size 0 or 1, meaning they are already sorted.
The height of the merge sort tree is O(log n), as each recursive call divides the sequence in half.

Quick Sort

Algorithm:
- Divide:
  - If input has 1 or fewer elements, it is already sorted. Otherwise, select a pivot element (often the last element in the sequence).
  - Partition the input into:
    - L (elements smaller than the pivot)
    - E (elements equal to the pivot)
    - G (elements greater than the pivot)
- Conquer: Recursively sort L and G.
- Combine: Join L, E, and G together.
Tree Visualization:Similar to merge sort, quick sort can be visualized as a binary tree with a similar interpretation of nodes and their meaning.
Performance:
- Worst-Case Time Complexity: O(n^2)
- Best-Case Time Complexity: O(n log n)
- Average Time Complexity: O(n log n)
- Space Complexity: O(log n) on average due to the recursive call stack.

Quick Sort Illustration

The pivot element is chosen (often the last element), and the sequence is partitioned around this pivot.
The process recurses through the partitions until base cases are reached, and the sorted sub-sequences are joined back together.

Bucket Sort

A linear time sorting algorithm, which implies its performance is directly proportional to the number of elements to be sorted.
Suitable for data that is evenly distributed within a known range.
Algorithm:
- Create a number of buckets equal to the range of the input data.
- For each element in the input:
  - Calculate its bucket index based on its value.
  - Insert the element into the corresponding bucket.
- Sort the elements within each bucket (using other sorting algorithms).
- Concatenate the sorted buckets to produce the final sorted sequence.

Radix Sort

Also a non-comparison based sorting algorithm, which can achieve linear time complexity in certain cases.
Algorithm:
- Sort the input sequence digit by digit, starting from the least significant digit.
- For each digit position:
  - Create buckets for each possible digit value (0-9).
  - Distribute input elements into buckets based on their digit value in that position.
  - Concatenate the buckets to form the sorted sequence for this digit position.
  - Use the output of the previous stage as input for the next digit position.

Heap Sort

Will be covered in future lectures on heaps.

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Description

This quiz covers the Merge Sort algorithm, a popular divide and conquer sorting method. Learn about its processes, including dividing the input, conquering through recursion, and combining sorted halves. It also discusses the time and space complexities involved.