Divide and Conquer Algorithms
10 Questions
1 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

Which of the following is a drawback of Merge Sort?

  • Uses less memory than necessary
  • Fully sorts the array in one pass
  • It is an in-place sorting algorithm
  • Requires additional space for another array (correct)

    • Quicksort combines the sorted subarrays by merging them back together.

    False

    What is the primary goal of the Partition procedure in Quicksort?

    To rearrange elements so that all elements less than or equal to a pivot are on one side and those greater are on the other.

    The elements in the array are partitioned such that A[p...j-1] ≤ A[j] ≤ A[j+1...q]. This is the property of ________ sorting algorithm.

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

    Match the following sorting algorithms with their characteristics:

    <p>Merge Sort = Not in place, requires additional space Quicksort = In-place, divides into subarrays Insertion Sort = Efficient for small datasets Selection Sort = Simple but generally slow</p> Signup and view all the answers

    What does the 'Conquer' step in the Quicksort process involve?

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

    In Quicksort, additional work is required to combine the sorted subarrays.

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

    What is the purpose of defining A(n+1) as being greater than or equal to all elements in A(p:q) in the Quicksort algorithm?

    <p>To ensure that the partitioning process can definitively place the pivot element in the correct position during sorting.</p> Signup and view all the answers

    In the Quicksort algorithm, the procedure that identifies the pivot and partitions the array is called ________.

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

    What is the first step undertaken by the Quicksort procedure?

    <p>Divide the array into two subarrays</p> Signup and view all the answers

    Study Notes

    Divide and Conquer Algorithm Design Strategy

    • Divide the problem into smaller subproblems
    • Solve the smaller subproblems recursively
    • Combine the solutions of subproblems to obtain the solution to the original problem

    Divide and Conquer Technique

    • A problem of size n is divided into smaller subproblems of size n/2
    • A solution to each subproblem is obtained recursively
    • The solutions are combined to obtain the solution to the original problem

    Control Abstraction for Divide and Conquer

    • Algorithm DAndC(P)
      • If Small(P), return S(P)
      • Else:
        • Divide P into smaller instances P₁, P₂, ..., Pₖ (k ≥ 1)
        • Apply DAndC to each sub-problem
        • Return Combine(DAndC(P₁), DAndC(P₂), ..., DAndC(Pₖ))

    General Divide-and-Conquer Recurrence

    • For divide-and-conquer algorithms producing sub-problems of the same type:
      • A problem of size n can be divided into b instances of size n/b, with a of them needing to be solved.
      • a and b are constants, a ≥ 1 and b > 1.
      • The running time T(n) follows the recurrence:
        • T(n) = aT(n/b) + f(n) for n > 1
        • T(1) = c for n = 1, where c is a constant
      • n is a power of b, i.e., n = bᵏ
      • f(n) represents time spent on dividing the original problem into smaller ones and combining their solutions

    Solving Recurrence Using Substitution Method

    • Example using T(n) = 2T(n/2) + n with T(1) = 2 and f(n) = n
      • After k substitutions: T(n) = 2ᵏT(n/2ᵏ) + n(1 + 2 + 4 + … + (2^(k-1)))
      • For k = log₂ n, T(n) = nT(1) + n log₂ n
      • T(n) = 2n + n log₂ n (assuming T(1) = 2)
      • Therefore, T(n) ∈ Θ(n log n)
    • A very efficient algorithm for searching in a sorted array A[0…n-1]
      • Compares the target K with the middle element A[m]
      • If they are equal, stop.
      • If K < A[m], search in A[0…m-1].
      • If K > A[m], search in A[m+1…n-1].
    • Procedure BinSrch(a, l, r, x)
      • If l == r, return 1 if x == a[l], else 0
      • Calculate mid = (l + r)/2
      • If x == a[mid], return mid
      • If x < a[mid], recursively call BinSrch on a, l, mid-1, x
      • If x > a[mid], recursively call BinSrch on a, mid+1, r, x
    • Procedure BinSearch(a, n, x)
      • Initialize low = 1, high = n
      • While low <= high:
        • Calculate mid = (low + high)/2
        • If x < a[mid], high = mid - 1
        • Else if x > a[mid], low = mid + 1
        • Else, return mid
      • Return 0 if not found

    Binary Search- Three Examples

    • Demonstrates how binary search works with different target values in a sorted array

    Binary Decision Tree

    • A tree representing the comparisons made in binary search.
    • The average number of comparisons for successful searches is approximately 3.21, and for unsuccessful searches is approximately 3.93.
    • Successful searches: Best, Average, and Worst case complexity is Θ(log₂n)
    • Unsuccessful searches: Best, Average, and Worst case complexity is Θ(log₂n)

    Mergesort

    • Splits the array into two approximately equal halves.
    • Sorts the halves recursively.
    • Merges the sorted halves into a single sorted array (using an auxiliary array)

    Merge Sort and Merge Algorithm

    • Procedure MergeSort(low, high)

      • If low < high:
        • mid = (low + high)/2
        • MergeSort(low, mid)
        • MergeSort(mid+1, high)
        • Merge(low, mid, high)

    • Procedure Merge(low, mid, high)

    • Merges two sorted sub-arrays of a into the auxiliary array b

    Time Complexity of Merge Sort

    • T(n) = 2T(n/2) + cn for n > 1, T(1) = c, where c is a constant
    • The complexity is Θ(n log n) in all cases

    Analysis of Mergesort

    • All cases have the same efficiency: Θ(n log n)
    • Space requirement: Θ(n) (not in-place)
    • Can be implemented without recursion (bottom-up approach).

    Two Inefficiencies of Merge Sort

    • Not in-place (uses another array)
    • Copy operation between arrays is required
    • Space and time for stack due to recursion
    • For small set sizes, most of the time consumed by recursion instead of sorting

    Quicksort

    • Sorts an array in place using a divide-and-conquer approach
    • Divides the array based on a pivot element
    • Recursive calls to sort the partitions
    • No extra array needed for merging
    • Can be inefficient if partitions are unbalanced

    Quicksort Procedure

    • Procedure Quicksort(p, q)
      • If p<q:
        • j = Partition(a, p, q+1)
        • Quicksort(p, j – 1)
        • Quicksort(j+1, q)

    Quicksort Partition Procedure

    • Procedure Partition(a, m, p) -Partition elements of a such that part of the elements are smaller/equal to a[m] and other part are greater.

    Analysis of Quicksort

    • Best, Average and Worst case analysis:
      • Best Case: Θ(nlogn)
      • Average Case: Θ(nlogn)
      • Worst Case: Θ(n²) (if input is sorted or nearly sorted)

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Related Documents

    Chapter 5 Divide and Conquer PDF

    Description

    This quiz explores the principles and techniques of the Divide and Conquer algorithm design strategy. It covers the process of breaking down problems into smaller subproblems, solving them recursively, and then combining their solutions. Test your understanding of both the methodology and implementation details of this essential algorithmic approach.

    More Like This

    Advanced Divide and Conquer Algorithms
    5 questions

    Advanced Divide and Conquer Algorithms

    WellPositionedMossAgate7535 avatar
    WellPositionedMossAgate7535
    Divide and Conquer Algorithms Quiz
    32 questions

    Divide and Conquer Algorithms Quiz

    HappierSage avatar
    HappierSage
    Divide and Conquer Algorithm - PD Method Quiz
    3 questions

    Divide and Conquer Algorithm - PD Method Quiz

    EnchantedChicago avatar
    EnchantedChicago
    Python Divide and Conquer Algorithm
    10 questions

    Python Divide and Conquer Algorithm

    LivelyCoralReef avatar
    LivelyCoralReef
    Use Quizgecko on...
    Browser
    Open
    Browser
    Browser