Divide and Conquer Strategy in Algorithms
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What is the purpose of the Merge function in the provided code snippet?

  • To reverse the order of elements in an array.
  • To calculate the difference between two arrays.
  • To sort an array recursively by dividing it into sub-arrays and merging them. (correct)
  • To search for a specific element in an array.

The Merge function assumes that the input arrays L and R are already sorted.

True (A)

What initial values are assigned to the variables i and j in the Merge function?

1

The loop invariant states that A[ℓ..k-1] contains the ______ smallest elements of L ∪ R sorted.

<p>k - ℓ</p> Signup and view all the answers

Match the following variables with their corresponding descriptions:

<p>n1 = The size of the right sub-array <code>R</code> ℓ = The starting index of the sub-array <code>A</code> m = The index of the middle element of the array <code>A</code> r = The ending index of the sub-array <code>A</code> n2 = The size of the left sub-array <code>L</code> k = The loop counter in the <code>Merge</code> function</p> Signup and view all the answers

The "Divide and Conquer" strategy involves dividing a problem into smaller instances of the same ______.

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

The Divide and Conquer technique is primarily used for sorting algorithms.

<p>False (B)</p> Signup and view all the answers

What is the primary goal of the "Combine" step in the Divide and Conquer strategy?

<p>To integrate the sorted sub-problems into a complete, sorted solution.</p> Signup and view all the answers

Which of the following is NOT a step involved in the Divide and Conquer strategy?

<p>Filter (D)</p> Signup and view all the answers

Match each step of the Divide and Conquer strategy with its corresponding description.

<p>Divide = Break the problem into smaller subproblems. Conquer = Solve the smaller subproblems. Combine = Integrate the solutions of the subproblems into a final solution.</p> Signup and view all the answers

Give an example of a problem that can be effectively solved using the Divide and Conquer approach.

<p>Merge Sort, Quick Sort, Binary Search</p> Signup and view all the answers

Which advantage does the Divide and Conquer strategy offer?

<p>Improved efficiency for certain problems (D)</p> Signup and view all the answers

The "Combine" step in Divide and Conquer always requires merging sorted subproblems.

<p>False (B)</p> Signup and view all the answers

The Merge function takes four arguments: A, ℓ, m, and r. A represents the array to be merged, ℓ represents the ____ index of the sub-array to be merged, m represents the ____ index, and r represents the ____ index.

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What is the purpose of the two sentinels added to the arrays L and R in the Merge function?

<p>To mark the end of the arrays L and R (C)</p> Signup and view all the answers

In the Merge function, if L[i] is greater than or equal to R[j], the value of R[j] is copied to A[k].

<p>False (B)</p> Signup and view all the answers

What does the loop invariant statement in the Merge function guarantee?

<p>The loop invariant ensures that the subarray <code>A[ℓ..k-1]</code> contains the <code>k-ℓ</code> smallest elements from <code>L∪R</code>, sorted in ascending order.</p> Signup and view all the answers

The Merge function aims to merge two sorted subarrays, A[ℓ..m] and A[m+1..r], into a single sorted subarray A[ℓ..r], which is a core operation for the ______ algorithm.

<p>merge sort</p> Signup and view all the answers

Match the following terms to their corresponding descriptions:

<p>L[n1+1] = Sentinel value in array L R[n2+1] = Sentinel value in array R n1 = Size of the first subarray (A[ℓ..m]) k = Index of the element being copied to A j = Index of the element being considered in array R</p> Signup and view all the answers

The loop invariant in MergeSort algorithm states that the array segment A[ℓ..k-1] contains the ______ smallest elements of L ∪ R sorted, where L and R are subarrays and L[i] and R[j] are the smallest elements in L and R respectively, which haven't been copied to A yet.

<p>k-ℓ</p> Signup and view all the answers

In the Merge Sort algorithm, what is the purpose of the loop invariant?

<p>To ensure that elements from L and R are copied to A in a non-decreasing order. (E)</p> Signup and view all the answers

The Merge Sort algorithm uses recursion to divide the input array into smaller subarrays until each subarray contains only one element, which is considered sorted.

<p>True (A)</p> Signup and view all the answers

What are the two cases considered in the maintenance of the loop invariant during the Merge Sort algorithm?

<p>The two cases considered are <strong>Case (a)</strong> where L[i] ≤ R[j] and <strong>Case (b)</strong> where R[j] &lt; L[i].</p> Signup and view all the answers

Match the following terms with their corresponding explanations in the context of the Merge Sort algorithm:

<p>L[n1+1] = R[n2+1] = ∞ = Represents the sentinel values appended to the subarrays to simplify the comparison process during merging. i = j = 1 = Initializes the indices of the subarrays L and R, pointing to the first elements. A[ℓ .. k-1] = Represents the portion of the final array A that has been filled with sorted elements up to the current iteration. for k = ℓ to r do = Iterates over the indices of the final array A, copying elements from L and R to the appropriate positions.</p> Signup and view all the answers

The initialisation step in the loop invariant sets ______ = ______ and ______ = ______ = ______ , where L[n1+1] and R[n2+1] are the sentinel values appended to the subarrays, i and j are the indices of the subarrays, and n1 and n2 are the sizes of the subarrays L and R respectively.

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

When handling Case (a) (L[i] ≤ R[j]) during the maintenance of the loop invariant, the element L[i] is copied to the current position A[k] in the final array A.

<p>True (A)</p> Signup and view all the answers

In the context of the Merge Sort algorithm, what does the phrase 'L[i] and R[j] are the smallest elements in L and R, which have not been copied into A yet.' mean?

<p>It means that L[i] and R[j] are the next smallest available elements in subarrays L and R, respectively, that are yet to be placed into the final array A.</p> Signup and view all the answers

What is the purpose of the MergeSort function?

<p>The <code>MergeSort</code> function recursively sorts the array <code>A</code> in ascending order by dividing it into sub-arrays, sorting each sub-array, and then merging the sorted sub-arrays back together.</p> Signup and view all the answers

The Merge function is called after the recursive MergeSort calls have completed.

<p>True (A)</p> Signup and view all the answers

The Merge function assumes that the input arrays L and R are already ______.

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

What is the value of m in the MergeSort function?

<p>The floor of the average of <code>ℓ</code> and <code>r</code> (D)</p> Signup and view all the answers

What is the loop invariant for the Merge function?

<p>The loop invariant states that <code>A[ℓ..k-1]</code> contains the <code>k-ℓ</code> smallest elements of <code>L ∪ R</code> sorted, and <code>L[i]</code> and <code>R[j]</code> are the smallest elements in <code>L</code> and <code>R</code> that have not been copied into <code>A</code> yet.</p> Signup and view all the answers

The MergeSort algorithm has a time complexity of O(n log n).

<p>True (A)</p> Signup and view all the answers

How many comparisons does the Merge function make in the worst case?

<p>The <code>Merge</code> function makes exactly <code>r - ℓ + 1</code> comparisons in the worst case.</p> Signup and view all the answers

The MergeSort algorithm is considered a stable sorting algorithm.

<p>True (A)</p> Signup and view all the answers

The MergeSort algorithm uses a ______ approach to sorting.

<p>divide-and-conquer</p> Signup and view all the answers

What must be true for elements in arrays L and R before invoking the Merge function?

<p>They must be sorted. (B)</p> Signup and view all the answers

In the Merge function, the sentinel values L[n1 + 1] and R[n2 + 1] are used to represent the largest possible elements.

<p>True (A)</p> Signup and view all the answers

What is the purpose of having two indices, i and j, in the Merge function?

<p>To track the current position in arrays L and R respectively.</p> Signup and view all the answers

During the Merge function, the condition for copying an element from L to A is that L[i] must be ______ R[j].

<p>less than or equal to</p> Signup and view all the answers

Match the following variables used in the Merge function with their descriptions:

<p>n1 = Number of elements in array L n2 = Number of elements in array R i = Current index for array L j = Current index for array R</p> Signup and view all the answers

How does the loop invariant contribute to the Merge process?

<p>It ensures that the elements of A are always sorted. (A)</p> Signup and view all the answers

The Merge function requires that both L and R are empty prior to beginning the merge process.

<p>False (B)</p> Signup and view all the answers

What happens at the end of the merging process in the Merge function?

<p>All elements from both L and R are merged into A in sorted order.</p> Signup and view all the answers

Flashcards

Divide and Conquer

A problem-solving strategy that divides a problem into smaller subproblems, solves them independently, and combines the results.

Sorting

The process of arranging items in a specified order, often using algorithms.

Subproblem

A smaller instance of a problem that is part of a larger problem.

Combining results

The step in the divide and conquer approach where solutions to subproblems are merged to form a final solution.

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Equal-sized parts

Dividing a problem into parts that are approximately the same size to optimize processing.

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Efficiency

The effectiveness of an algorithm in terms of time and space used to solve a problem.

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Sorting algorithms

Procedures or formulas to arrange data in a specific order, such as quicksort or mergesort.

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Multiple people sorting

Using collaboration or parallel processes to sort different parts of data simultaneously.

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Merge Function

A function combining two sorted arrays into one sorted array.

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Loop Invariant

Condition that remains true before and after each iteration of a loop.

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Array L and R

Temporary arrays to hold halves of the original array for merging.

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Case (a) in Merge

Situation when the current element of L is less than or equal to R.

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Initialization in Merge

Setting up initial conditions before merging starts.

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Maintenance in Merge

Keeping the sorted order as elements are added to the merged array.

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n1 and n2

Sizes of temporary arrays used in the merge process: n1 = left size, n2 = right size.

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A[k] assignment

Assigning the smallest element to the merged array in each step.

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Initialization

The setup stage where initial conditions for the algorithm are established.

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Maintenance

Process ensuring the loop invariant remains valid throughout the loop's execution.

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Smallest Element

The current least value from the arrays that is not yet copied into the merged array.

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Element Copying

The action of taking the smallest element from L or R and placing it in the merged array A.

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Termination

The end step when merging is completed and all elements are in place.

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Sentinels

Special values that mark the end of arrays to simplify boundary conditions.

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Sorted Order

The arrangement of elements in a specific sequence (ascending or descending).

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Array Indices

Positions in an array used to access elements.

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Subarray Sizes

n1 = m - ℓ + 1 and n2 = r - m represent the sizes of left and right subarrays.

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Left and Right Arrays

L and R are temporary arrays holding elements for merging.

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Element Comparison

The process of comparing the smallest elements of L and R during merging.

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Copying Elements

Moving the smallest element from L or R to the main array A.

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Final Element Signal

L[n1 + 1] = R[n2 + 1] = ∞ indicates the end of elements to compare.

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Initialisation in MergeSort

Setting initial values of indices before the merging begins.

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Maintenance in MergeSort

Keeping the loop invariant true during each iteration.

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Termination in MergeSort

Condition when the merge process is complete.

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Time Complexity of Merge

Merge operation runs in linear time, O(n).

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Divide in MergeSort

Splitting the array into two halves recursively.

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Conquer in MergeSort

Solving the smaller subproblems created by division.

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Combine in MergeSort

Merging the sorted subarrays back into one sorted array.

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MergeSort Algorithm

A recursive algorithm that uses the divide and conquer approach to sort an array.

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Study Notes

Algorithms and Data Structures - 2nd Lecture Notes

  • Topic: Sorting by other means
  • Focus: Divide and Conquer
  • Idea: Breaking down a large sorting problem into smaller, more manageable subproblems that are solved individually and then combined to produce a solution. Emphasized using a card-stack analogy.
  • General approach: Divide...an instance into smaller instances of the same problem. Conquer...by recursively solving sub-instances (a function calling itself) - only solve very small instances directly. Combine...the partial solutions into a solution to the original instance.
  • Key function: MergeSort
  • Example function: MergeSort(int[] A, int l = 1, int r = A.length)

Divide and Conquer Algorithm for Sorting

  • Idea: Divide the card stack into roughly equal-sized parts. Sort the parts. Combine the sorted parts into a sorted stack.
  • General Description (Divide and Conquer):
    • Divide a problem into smaller subproblems.
    • Conquer the subproblems by solving them recursively.
    • Combine the subproblem solutions to form a solution to the original problem.
  • Example (MergeSort Function):
    • If l < r (the list has more than one element):
      • Find the midpoint m = ⌊(l + r) / 2⌋
      • Recursively call MergeSort(A, l, m) (left half) and MergeSort(A, m + 1, r) (right half). These recursively divide and conquer the subproblems
      • Call Merge(A, l, m, r) to combine the sorted sub-problems into a single sorted list.

Combining Sub Problems (Merge function)

  • Purpose: Merging the sorted sub-problems to produce a sorted overall result. Usually done by performing a comparison-based merging operation.
  • Example function call: Merge(int[] A, int l, int m, int r)
  • Variables: n₁ = m − l + 1, n₂ = r − m.
  • Auxiliary arrays: L = new int [1..n₁ + 1], R = new int [1..n₂ + 1].
  • Data Copy: L[1..n₁] = A[l..m], R[1..n₂] = A[m+1..r].
  • Sentinel Values: L[n₁ + 1] = R[n₂ + 1] = ∞ (a special value ensuring loop termination)
  • Iteration:
    • for k = l to r do - if L[i] ≤ R[j] then - A[k] = L[i] - i = i + 1 - else - A[k] = R[j] - j = j + 1

Summary of Correctness Proofs

  • Iterative algorithms: Use loop invariants (template).
  • Recursive algorithms: Use complete induction.

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This quiz explores the key concepts and components of the Divide and Conquer strategy in algorithm design. It covers the functions, principles, and applications of this technique, particularly in sorting algorithms. Test your understanding of this essential algorithmic approach and its advantages.

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