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- Data structure and Algorithm
- Characteristics of Algorithms
- Behaviour of algorithm
- Asymptotic Notations
Data structure and Algorithm
- A data structure is a particular way of storing and organizing data. The purpose is to effectively access and modify data effictively.
- A procedure to solve a specific problem is called Algorithm.
During programming we use data structures and algorithms that work on that data.
Characteristics of Algorithms
An algorithm has follwing characteristics.
- Input : Zero or more quantities are externally supplied to algorithm.
- Output : An algorithm should produce atleast one output.
- Finiteness : The algorithm should terminate after a finite number of steps. It should not run infinitely.
- Definiteness : Algorithm should be clear and unambiguous. All instructions of an algorithm must have a single meaning.
- Effectiveness : Algorithm must be made using very basic and simple operations that a computer can do.
- Language Independance : A algorithm is language independent and can be implemented in any programming language.
Behaviour of algorithm
The behaviour of an algorithm is the analysis of the algorithm on basis of Time and Space.
- Time complexity : Amount of time required to run the algorithm.
- Space complexity : Amount of space (memory) required to execute the algorithm.
The behaviour of algorithm can be used to compare two algorithms which solve the same problem. \\ The preference is traditionally/usually given to better time complexity. But we may need to give preference to better space complexity based on needs.
Best, Worst and Average Cases
The input size tells us the size of the input given to algorithm. Based on the size of input, the time/storage usage of the algorithm changes. Example, an array with larger input size (more elements) will taken more time to sort.
- Best Case : The lowest time/storage usage for the given input size.
- Worst Case : The highest time/storage usage for the given input size.
- Average Case : The average time/storage usage for the given input size.
Bounds of algorithm
Since algorithms are finite, they have bounded time taken and bounded space taken. Bounded is short for boundries, so they have a minimum and maximum time/space taken. These bounds are upper bound and lower bound.
- Upper Bound : The maximum amount of space/time taken by the algorithm is the upper bound. It is shown as a function of worst cases of time/storage usage over all the possible input sizes.
- Lower Bound : The minimum amount of space/time taken by the algorithm is the lower bound. It is shown as a function of best cases of time/storage usage over all the possible input sizes.
Asymptotic Notations
Big-Oh Notation [O]
- The Big Oh notation is used to define the upper bound of an algorithm.
- Given a non negative funtion f(n) and other non negative funtion g(n), we say that $f(n) = O(g(n)$ if there exists a positive number $n_0$ and a positive constant $c$, such that \[ f(n) \le c.g(n) \ \ \forall n \ge n_0 \]
- So if growth rate of g(n) is greater than or equal to growth rate of f(n), then $f(n) = O(g(n))$.