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Questions and Answers
When analyzing the time efficiency of recursive algorithms, the initial step involves solving the recurrence for the order of growth of its solution.
When analyzing the time efficiency of recursive algorithms, the initial step involves solving the recurrence for the order of growth of its solution.
False (B)
A recurrence relation of $T(n) = T(n-1) + 1$ corresponds to an algorithm with $O(log n)$ time complexity.
A recurrence relation of $T(n) = T(n-1) + 1$ corresponds to an algorithm with $O(log n)$ time complexity.
False (B)
The recurrence relation $T(n) = 2T(n/2) + 1$ is representative of the time complexity of a tree traversal algorithm.
The recurrence relation $T(n) = 2T(n/2) + 1$ is representative of the time complexity of a tree traversal algorithm.
True (A)
For a recurrence relation $T(n) = T(n-k) + k$, if $k = n + 1$, then $T(n) = T(1) + n + 1$.
For a recurrence relation $T(n) = T(n-k) + k$, if $k = n + 1$, then $T(n) = T(1) + n + 1$.
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If an algorithm has a recurrence relation of $T(n) = 2T(n/2) + n$, its time complexity is $O(n^2)$.
If an algorithm has a recurrence relation of $T(n) = 2T(n/2) + n$, its time complexity is $O(n^2)$.
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In analyzing the time efficiency of an iterative algorithm, the first step is to pinpoint the algorithm's least significant operation.
In analyzing the time efficiency of an iterative algorithm, the first step is to pinpoint the algorithm's least significant operation.
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When analyzing algorithms, the best-case scenario provides an upper bound on the algorithm's performance.
When analyzing algorithms, the best-case scenario provides an upper bound on the algorithm's performance.
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The summation $\sum_{i=0}^{n} i$ simplifies to $\frac{n(n+1)}{2}$.
The summation $\sum_{i=0}^{n} i$ simplifies to $\frac{n(n+1)}{2}$.
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If an algorithm's time complexity is expressed as $\frac{n^2 - 7n + 4}{2}$, it belongs to $\Theta(n^3)$.
If an algorithm's time complexity is expressed as $\frac{n^2 - 7n + 4}{2}$, it belongs to $\Theta(n^3)$.
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In Example 1, the outer loop iterates from $i = 0$ to $n-1$, and the inner loop iterates from $j = i+1$ to $n-1$, with a constant time operation inside. The time complexity is $\Theta(n)$.
In Example 1, the outer loop iterates from $i = 0$ to $n-1$, and the inner loop iterates from $j = i+1$ to $n-1$, with a constant time operation inside. The time complexity is $\Theta(n)$.
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In the binary representation formula $b = (\log_2 n) + 1$, 'b' represents the base of the logarithm.
In the binary representation formula $b = (\log_2 n) + 1$, 'b' represents the base of the logarithm.
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When analyzing iterative algorithms that repeatedly halve the input size, a logarithmic time complexity, $O(\log n)$, is often observed.
When analyzing iterative algorithms that repeatedly halve the input size, a logarithmic time complexity, $O(\log n)$, is often observed.
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The time complexity of an algorithm where the value of n is halved on each repetition of a loop is $O(n\log n)$.
The time complexity of an algorithm where the value of n is halved on each repetition of a loop is $O(n\log n)$.
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Flashcards
Empirical Analysis
Empirical Analysis
A method of algorithm analysis based on practical experiments and measurements.
Iterative Algorithms
Iterative Algorithms
Algorithms that repeat a sequence of operations until a condition is met.
Recursive Algorithms
Recursive Algorithms
Algorithms that solve a problem by breaking it down into smaller subproblems of the same type.
Time Efficiency
Time Efficiency
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Basic Operation
Basic Operation
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Summation Formula
Summation Formula
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O(log n)
O(log n)
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Algorithm Analysis Steps
Algorithm Analysis Steps
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Recurrence Relation
Recurrence Relation
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Time Complexity
Time Complexity
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Binary Search
Binary Search
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Merge Sort
Merge Sort
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Selection Sort
Selection Sort
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Study Notes
Algorithm Analysis & Design
- Course: CSC645
- Chapter 3: Methods of Algorithm Analysis
- Author: Nur Azmina Mohamad Zamani
Chapter Overview
- Analysis of iterative algorithms
- Analysis of recursive algorithms
- Includes empirical analysis
Time Efficiency of Iterative Algorithms
- General Plan for Analysis
- Decide on parameter n, indicating input size
- Identify algorithm's basic operation
- Determine worst, average and best cases for input of size n
- Set up a sum for the number of times the basic operation is executed
- Simplify the sum using standard formulas and rules
Summation Formulas & Rules
- Σcaᵢ = cΣaᵢ (R1)
- Σ(aᵢ + bᵢ) = Σaᵢ + Σbᵢ (R2)
- Σi = 1 = n(n+1) /2 ≈ n² (S2)
- Where i ∈ {1, 2... n}
Summation Formula of i
- Multiple summation formulas for different powers of i given
Example 1: Unique Elements
- Algorithm: Identifies whether all elements in an array are distinct
- Input: Array A[0..n-1]
- Output: True if distinct, False otherwise
- Nested loops examine all pairs of elements for equality
- Time complexity using summation notation is shown
Solution for Example 1
- Calculation of the summation involved is shown leading to time complexity of θ(n²)
Exercises
- Multiple exercises are present
Example 2: Binary Algorithm
- Algorithm calculates the number of binary digits in a decimal integer.
- Input: Positive decimal integer n
- Output: Number of binary digits
- The algorithm repeatedly divides by 2, incrementing a count with each division
Solution for Example 2
- Algorithm steps involved
- Time complexity is θ(log n)
Time Efficiency of Recursive Algorithms
- General Plan for Analysis
- Decide on parameter n, indicating input size
- Identify algorithm's basic operation
- Determine worst, average and best cases for input of size n
- Set up recurrence relation for the number of times the basic operation is executed
- Solve the recurrence for the order of growth of the solution
Recurrence Relation for Some Algorithms
- Provides examples of recurrence relations (e.g. Binary search, Sequential search, Tree Traversal, Selection sort, Merge sort, Quick sort) and their corresponding Big-Oh solutions
Example 1 (Recurrence Relation)
- Describes an algorithm with recurrence relation T(n) = T(n - 1) + 1, T(1) = 0
- Solves the recurrence relation to find algorithm complexity θ(n)
Example 2 (Recurrence Relation)
- Algorithm with recurrence relation T(n) = 2T(n/2) + n, T(1) = 1
- Solves the recurrence showing time complexity θ(n log n)
Reference
- Reference provided on algorithm analysis
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Description
Explore the methods of algorithm analysis in Chapter 3 of CSC645. This quiz covers the analysis of iterative and recursive algorithms, including techniques for empirical analysis and summation formulas. Test your understanding of time efficiency and algorithm performance.
