Divide and Conquer Algorithms Quiz
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Questions and Answers

What is the main strategy of the divide and conquer algorithm design?

  • Divide the problem into sub-problems and conquer them separately (correct)
  • Incrementally increase the complexity of the problem and solve it
  • Combine the problem with other problems and solve them together
  • Randomly select multiple sub-problems and solve them concurrently

    • Which algorithm is NOT listed as a notable example of the divide and conquer strategy?

  • Cooley-Tukey fast Fourier transform algorithm
  • Euclid's algorithm for computing greatest common divisor
  • Merge sort and quicksort algorithms
  • Insertion sort algorithm (correct)

    • What does the concept of 'Divide and Conquer' refer to outside of algorithm design?

  • A strategy to establish large power structures
  • A strategy to prevent smaller power groups from linking up (correct)
  • A strategy to unify rivalries and discord among people
  • A strategy to randomly distribute power among different groups

    • Which algorithm is an example of a recursive implementation of the divide and conquer strategy?

    <p>Karatsuba's multiplication algorithm</p> Signup and view all the answers

    What is a common characteristic of the sub-problems in the divide and conquer strategy?

    <p>They are trivial for small sizes or use the same strategy</p> Signup and view all the answers

    According to the text, what is John von Neumann thinking about that is much more important than bombs?

    <p>The impact of computers on society</p> Signup and view all the answers

    What is the primary strategy used in the merge sort algorithm?

    <p>Divide and conquer</p> Signup and view all the answers

    Which auxiliary function helps in merging two ordered vectors in the merge sort algorithm?

    <p>Merge function</p> Signup and view all the answers

    In the merge procedure, what indices does the input array A have?

    <p>p, q, r such that p≤q1</p> Signup and view all the answers

    Who developed a fast algorithm for multiplying n-digit numbers with complexity O(nlg3)?

    <p>Anatolii Karatsuba</p> Signup and view all the answers

    What was disproved by Karatsuba's algorithm regarding the number of operations required for multiplying n-digit numbers?

    <p>Andrey Kolmogorov's 1956 conjecture that Ω(n2 ) operations would be required</p> Signup and view all the answers

    In Karatsuba's algorithm, what is the base case for the recursive multiplication of binary numbers?

    <p>n=1</p> Signup and view all the answers

    What is the running time complexity of Karatsuba's algorithm for n > 2?

    <p>$T (n) ≤ 3T (n/2) + c n lg 3$</p> Signup and view all the answers

    What is the number of bits in x and y in Karatsuba's recursive multiplication algorithm?

    <p>$n$</p> Signup and view all the answers

    How many recursive calls are made in Karatsuba's algorithm to obtain quadratic running time?

    <p>$3$</p> Signup and view all the answers

    What is the value of p in the Recursive-Multiply(x,y) algorithm if n=1?

    <p>$x * y$</p> Signup and view all the answers

    What is the primary strategy used in the divide and conquer algorithm design?

    <p>Divide a problem into sub-problems and combine the solutions</p> Signup and view all the answers

    Which algorithm is an example of a recursive implementation of the divide and conquer strategy?

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

    What is the number of bits in x and y in Karatsuba's recursive multiplication algorithm?

    <p>4</p> Signup and view all the answers

    Which auxiliary function helps in merging two ordered vectors in the merge sort algorithm?

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

    What does the concept of 'Divide and Conquer' refer to outside of algorithm design?

    <p>Breaking up existing power structures to prevent smaller power groups from linking up</p> Signup and view all the answers

    What is the value of p in the Recursive-Multiply(x,y) algorithm if n=1?

    <p>$y$</p> Signup and view all the answers

    What is the time complexity of the merge sort algorithm?

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

    What did Anatolii Karatsuba disprove with his algorithm?

    <p>Andrey Kolmogorov's 1956 conjecture that Ω(n^2) operations would be required to multiply n-digit numbers</p> Signup and view all the answers

    What is the value of p in the Recursive-Multiply(x,y) algorithm if n=1?

    <p>x * y</p> Signup and view all the answers

    What is the base case for the recursive multiplication of binary numbers in Karatsuba's algorithm?

    <p>$n = 1$</p> Signup and view all the answers

    Which auxiliary function helps in merging two ordered vectors in the merge sort algorithm?

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

    Who developed a fast algorithm for multiplying n-digit numbers with complexity O(nlg3)?

    <p>Anatolii Karatsuba</p> Signup and view all the answers

    How many recursive calls are made in Karatsuba's algorithm to obtain quadratic running time?

    <p>$4$</p> Signup and view all the answers

    What is the main strategy of the divide and conquer algorithm design?

    <p>Combine and Conquer</p> Signup and view all the answers

    Which algorithm is NOT listed as a notable example of the divide and conquer strategy?

    <p><code>Insertion sort</code></p> Signup and view all the answers

    What is a common characteristic of the sub-problems in the divide and conquer strategy?

    <p><code>They are homogeneous in content</code></p> Signup and view all the answers

    Study Notes

    Divide and Conquer Algorithm Design

    • The main strategy is to break down complex problems into smaller sub-problems, solve them, and then combine the solutions to solve the original problem.

    Notable Examples

    • Merge sort is an example of a recursive implementation of the divide and conquer strategy.
    • Karatsuba's algorithm is another example of a recursive implementation of the divide and conquer strategy.

    Concept of Divide and Conquer

    • Outside of algorithm design, the concept refers to breaking down complex tasks or problems into smaller, manageable parts to solve them.

    Sub-problems

    • A common characteristic of the sub-problems in the divide and conquer strategy is that they are smaller instances of the same problem.

    Merge Sort Algorithm

    • The primary strategy used in the merge sort algorithm is the divide and conquer strategy.
    • The auxiliary function Merge helps in merging two ordered vectors.

    Merge Procedure

    • The input array A has indices ranging from 1 to n.

    Karatsuba's Algorithm

    • Karatsuba developed a fast algorithm for multiplying n-digit numbers with a complexity of O(nlg3).
    • The base case for the recursive multiplication of binary numbers is when n = 1.
    • The running time complexity of Karatsuba's algorithm for n > 2 is O(n^log2(3)).
    • The number of bits in x and y in Karatsuba's recursive multiplication algorithm is n.
    • Three recursive calls are made in Karatsuba's algorithm to obtain quadratic running time.
    • If n = 1, the value of p in the Recursive-Multiply(x,y) algorithm is 1.
    • Anatolii Karatsuba disproved the idea that the number of operations required for multiplying n-digit numbers is O(n^2).

    John von Neumann

    • John von Neumann was thinking about the human cost of war, which he believed was much more important than bombs.

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    Description

    Test your knowledge of divide and conquer algorithms with this quiz focused on algorithm design strategies, including insertion sort, Dijkstra's algorithm, merge sort, quicksort, and all-pairs-shortest-path. This quiz covers concepts from Chapter 3 of the book 'Algorithms Illuminated'.

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