10 Questions
What is the goal of analysis of algorithms?
To compare algorithms mainly in terms of running time but also in terms of other factors
What does running time analysis determine?
How running time increases as the size of the problem increases
What does worst case analysis provide?
An upper bound on running time
What does best case analysis provide?
A lower bound on running time
How do we compare algorithms?
By defining a number of objective measures, including comparing execution times
What is a recurrence in the context of algorithms?
An equation or inequality that describes a function in terms of its value on smaller inputs
What is the running time of an algorithm with the recurrence $T(n) = T(n-1) + n$?
$\Theta(n^2)$
What is the running time of an algorithm with the recurrence $T(n) = T(n/2) + c$?
$\Theta(\log n)$
What is the running time of an algorithm with the recurrence $T(n) = T(n/2) + n$?
$\Theta(n)$
What is the running time of an algorithm with the recurrence $T(n) = 2T(n/2) + 1$?
$\Theta(n)$
Study Notes
Analysis of Algorithms
- The goal of analysis of algorithms is to determine their efficiency and scalability.
Running Time Analysis
- Running time analysis determines how long an algorithm takes to complete, usually measured in terms of the number of basic operations performed.
Worst Case Analysis
- Worst case analysis provides an upper bound on the running time of an algorithm, which is the maximum amount of time it takes to complete in the worst possible scenario.
Best Case Analysis
- Best case analysis provides a lower bound on the running time of an algorithm, which is the minimum amount of time it takes to complete in the best possible scenario.
Comparing Algorithms
- Algorithms are compared based on their time and space complexities, which are measured in terms of the size of the input.
Recurrence in Algorithms
- A recurrence in the context of algorithms is a mathematical equation that describes the running time of an algorithm, usually involving the running time of smaller instances of the same problem.
Running Time of Algorithms
- An algorithm with the recurrence $T(n) = T(n-1) + n$ has a running time of O(n^2).
- An algorithm with the recurrence $T(n) = T(n/2) + c$ has a running time of O(log n).
- An algorithm with the recurrence $T(n) = T(n/2) + n$ has a running time of O(n log n).
- An algorithm with the recurrence $T(n) = 2T(n/2) + 1$ has a running time of O(n).
Test your understanding of algorithm analysis with this quiz on asymptotic analysis. Explore the goal of analyzing algorithms and comparing them based on factors like running time, memory requirements, and programmer effort.
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