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Questions and Answers
What is the computational complexity of insertion sort in the best case?
What is the divide and conquer approach composed of?
What happens when the input data size of subproblems is small enough in the Conquer step?
What is the purpose of the Combine step in the Divide and Conquer approach?
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What is the result of the Divide step in the Merge sort algorithm?
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What is the purpose of the merge function in the Merge sort algorithm?
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What is the computational cost necessary to solve, divide and combine subproblems having dimension 1?
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How are the subproblems solved in the Conquer step of the Divide and Conquer approach?
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What is the relationship between the number of elements in sequences S1 and S2 in the Divide step of the Merge sort algorithm?
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What is the main advantage of using a recursive approach in algorithms?
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In the context of merge sort, what is the purpose of the divide step?
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What is the relationship between the number of elements in the original sequence and the number of elements in the subproblems in merge sort?
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What is the main difference between the best case and worst case scenarios for the computational complexity of insertion sort?
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What is the purpose of the merge function in the merge sort algorithm?
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What is the main advantage of using the divide and conquer approach in algorithms?
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In the context of merge sort, what is the purpose of the conquer step?
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What is the relationship between the number of levels of recursion and the size of the subproblems in merge sort?
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What is the main difference between the divide and conquer approach and other algorithmic approaches?
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What is the purpose of the combine step in the divide and conquer approach?
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Study Notes
Algorithm Analysis
- The computational cost of an algorithm depends on the input size.
- Different computational costs may depend on different degrees of sorting of the input arrays.
- Input size refers to the number of elements composing the input data.
- Computational cost refers to the number of elementary instructions to be executed.
Insertion Sort
- Insertion sort is a well-defined procedure that takes a certain value as input and produces a certain value as output.
- It is a sequence of computational steps that transforms an input into an output.
- An algorithm is an instrument to solve a well-defined computational problem.
- The specific problem defines the relation existing between input and output.
- Insertion sort describes a specific computational procedure to obtain this relation between input and output.
- Example: sorting a sequence of n numbers a1, a2, ..., an to produce a permutation a1′, a2′, ..., an′ of the input sequence such that a1′ ≤ a2′ ≤ ... ≤ an′.
Insertion Sort Algorithm
- The algorithm starts with an unsorted array and iteratively inserts elements into their correct position.
- At the beginning of each iteration, the first j-1 elements are sorted, while the rest are unsorted.
- Elements previously occupying positions 1 to j-1 are now sorted.
- The computational complexity of insertion sort is linear in the best case and quadratic in the worst case.
Divide and Conquer
- Divide and conquer is an approach used by recursive algorithms to solve problems.
- The approach consists of three steps: divide, conquer, and combine.
- Divide: divide the problem into subproblems of the same kind but with smaller input data size.
- Conquer: recursively solve the subproblems until their input data size is small enough.
- Combine: combine the solutions of the subproblems to obtain the solution of the original problem.
Merge Sort
- Merge sort is an example of a divide and conquer algorithm.
- Goal: sort a sequence S composed of n elements using the divide and conquer approach.
- Divide: divide the sequence S into two sequences S1 and S2 of approximately equal size.
- Conquer: recursively sort sequences S1 and S2.
- Combine: merge the two sorted sequences S1 and S2 into a sorted sequence S.
Algorithm Analysis
- The computational cost of an algorithm depends on the input size.
- Different computational costs may depend on different degrees of sorting of the input arrays.
- Input size refers to the number of elements composing the input data.
- Computational cost refers to the number of elementary instructions to be executed.
Insertion Sort
- Insertion sort is a well-defined procedure that takes a certain value as input and produces a certain value as output.
- It is a sequence of computational steps that transforms an input into an output.
- An algorithm is an instrument to solve a well-defined computational problem.
- The specific problem defines the relation existing between input and output.
- Insertion sort describes a specific computational procedure to obtain this relation between input and output.
- Example: sorting a sequence of n numbers a1, a2, ..., an to produce a permutation a1′, a2′, ..., an′ of the input sequence such that a1′ ≤ a2′ ≤ ... ≤ an′.
Insertion Sort Algorithm
- The algorithm starts with an unsorted array and iteratively inserts elements into their correct position.
- At the beginning of each iteration, the first j-1 elements are sorted, while the rest are unsorted.
- Elements previously occupying positions 1 to j-1 are now sorted.
- The computational complexity of insertion sort is linear in the best case and quadratic in the worst case.
Divide and Conquer
- Divide and conquer is an approach used by recursive algorithms to solve problems.
- The approach consists of three steps: divide, conquer, and combine.
- Divide: divide the problem into subproblems of the same kind but with smaller input data size.
- Conquer: recursively solve the subproblems until their input data size is small enough.
- Combine: combine the solutions of the subproblems to obtain the solution of the original problem.
Merge Sort
- Merge sort is an example of a divide and conquer algorithm.
- Goal: sort a sequence S composed of n elements using the divide and conquer approach.
- Divide: divide the sequence S into two sequences S1 and S2 of approximately equal size.
- Conquer: recursively sort sequences S1 and S2.
- Combine: merge the two sorted sequences S1 and S2 into a sorted sequence S.
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Description
This quiz covers the analysis of the insertion sort algorithm, including its computational cost and how it depends on the input size and degree of sorting.
