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?

    <p>If the index of S1 becomes equal to the length of S1</p> 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?

    <p>dlog n e</p> Signup and view all the answers

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

    <p>When the size of the array is 1 or less</p> Signup and view all the answers

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

    <p>Heap Sort</p> Signup and view all the answers

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

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

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

    <p>O(n)</p> Signup and view all the answers

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

    <p>To create a copy of a specified range of elements</p> Signup and view all the answers

    In which case would the mergeSort function return without sorting?

    <p>When the input size is less than 2</p> Signup and view all the answers

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

    <p>O(n log n)</p> Signup and view all the answers

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

    <p>Side-by-side merging of sorted halves</p> Signup and view all the answers

    During quick sort, how is the pivot typically chosen?

    <p>The last element</p> Signup and view all the answers

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

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

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

    <p>Bubble Sort</p> Signup and view all the answers

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

    <p>The algorithm runs in O(n^2)</p> 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?

    <p>O(n log n)</p> Signup and view all the answers

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

    <p>Binary tree</p> Signup and view all the answers

    What is the height of the merge sort tree?

    <p>O(log n)</p> Signup and view all the answers

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

    <p>Merging L, E, and G</p> Signup and view all the answers

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

    <p>Recursively sort the subarrays</p> Signup and view all the answers

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

    <p>The partitioned elements</p> Signup and view all the answers

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

    <p>The sequence is sorted</p> Signup and view all the answers

    What does the running time of quick sort depend on?

    <p>All of the above</p> Signup and view all the answers

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

    <p>Three</p> Signup and view all the answers

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

    <p>To track the size of L</p> Signup and view all the answers

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

    <p>Randomized quick sort</p> 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.

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