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Algorithm

Algorithm. An algorithm is a finite set of steps required to solve a problem. An algorithm must have following properties: Input: An algorithm must have zero or more quantities as input whcih are externally supplied. Output:

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Algorithm

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  1. Algorithm An algorithm is a finite set of steps required to solve a problem. An algorithm must have following properties: • Input: An algorithm must have zero or more quantities as input whcih are externally supplied. • Output: An algorithm must produce one or more output after processing set of statements. • Definiteness: Each instruction must be clear and ditinct. • Finiteness: The algorithm must terminate after a finite number of steps. • Effectiveness: Each operations must be definite also it should be feasible Visit for more Learning Resources

  2. Example of an algorithm 1. Start 2. Accept size for an array : (Read size) 3. Accept array elements values from user i.e. Array elements. 4. Accept element to be searched from user i.e. Read Value 5. Set i=0,flag=0 6. Compare A[i] with value If(A[i] is a value) Set flag=1 go to step 8 Else Move to next data element i= i+1; 7. If (i<=n) go to step 6 8. If(flag=1) then Print found and Return i as position Else Print not found 9. Stop.

  3. Algorithm Design Approches • Top-Down Approach: • Bottom-Up Approach:

  4. Top-Down Approach: • A top-down approach starts with identifying major components of system or program decomposing them into their lower level components & iterating until desired level of module complexity is achieved . • In this we start with topmost module & incrementally add modules that is calls. • It takes the form of step wise procedure. • In this solution is divided into sub task and each sub task further divided into smallest subtask. • The sub task are then combined into single solution.

  5. Bottom-Up Approach: • It is inverse of top down method. • A bottom-up approach starts with designing most basic or primitive component & proceeds to higher level components. • Starting from very bottom , operations that provide layer of abstraction are implemented. • The programmer may write code to perform basic operations then combined those to make a modules ,whcih are finally combined to form overall system structure.

  6. Time complexity • Time complexity of program / algorithm is the amount of computer time that it needs to run to completion. • While calculating time complexity, we develop frequency count for all key statements which are important.

  7. Consider three algorithms given below:- • Algorithm A:- a=a+1 • Algorithm B:- for x=1 to n step • a=a+1 • Loop • Algorithm C:- for x=1 to n step 1 • for y=1 to n step 2 • a=a+1 • loop

  8. Frequency count for algorithm A is 1 as a=a+1 statement will execute only once. • Frequency count for algorithm B is n as a=a+1 is a key statement executes “n‟ times as loop runs “n‟ times. • Frequency count for algorithm C is n2 as a=a+1 is a key statement executes n2 times as the inner loop runs n times, each time the outer loop runs and the outer loop also runs for n times.

  9. Space complexity • Space complexity of a program / algorithm is the amount of memory that it needs to run to completion. • The space needed by the program is the sum of the following components. • Fixed space requirements:- • Fixed space is not dependent on the characteristics of the input and outputs. • Fixed space consists of space for simple variable, fixed size variables, etc. • Variable space requirements:- • Variable space includes space needed by variables whose size depends upon the particular problem being solved, referenced variables and the stack space required for recursion on particular instance of variables. • e.g. Additional space required where function uses recursion.

  10. Algorithm analysis: • There are different ways of solving problem & there are different algorithms which can be designed to solve a problem. • There is difference between problem & algorithm. • A problem has single problem statement that describes it in general terms. • However there are different ways to solve a problem & some solutions may be more efficient than others.

  11. There are different types of time complexities which can be analyzed for an algorithm: • Best Case Time Complexity: • Worst Case Time Complexity: • Average Case Time Complexity:

  12. Best Case Time Complexity: • It is measure of minimum time that algorithm will require for input of size “n‟. • Running time of many algorithms varies not only for inputs of different sizes but also input of same size. • For example in running time of some sorting algorithms, sorting will depend on ordering of input data. Therefore if input data of “n‟ items is presented in sorted order, operations performed by algorithm will take least time.

  13. Worst Case Time Complexity: • It is measure of maximum time that algorithm will require for input of size “n‟. • Therefore if various algorithms for sorting are taken into account & say “n‟ input data items are supplied in reverse order for any sorting algorithm, then algorithm will require n2 operations to perform sort which will correspond to worst case time complexity of algorithm.

  14. Average Case Time Complexity: • The time that an algorithm will require to execute typical input data of size “n‟ is known as average case time complexity. • We can say that value that is obtained by averaging running time of an algorithm for all possible inputs of size “n‟ can determine average case time complexity.

  15. Big O notation • Big O notation is used in Computer Science to describe the performance or complexity of an algorithm. • Big O specifically describes the worst-case scenario, and can be used to describe the execution time required or the space used (e.g. in memory or on disk) by an algorithm.

  16. O(1) • O(1) describes an algorithm that will always execute in the same time (or space) regardless of the size of the input data set. • E.g Push and POP operation for a stack • O(N) • O(N) describes an algorithm whose performance will grow linearly and in direct proportion to the size of the input data set. • Big O notation will always assume the upper limit. • E.g. Linear search with unsorted data. • O(N2) • O(N2) represents an algorithm whose performance is directly proportional to the square of the size of the input data set. • This is common with algorithms that involve nested iterations over the data set. • E.g. Comparing two dimensional arrays of size n.

  17. O(log N). Logarithmic Time The iterative halving of data sets described in the binary search example produces a growth curve that peaks at the beginning and slowly flattens out as the size of the data sets increase E.g. Binary search: an input data set containing 10 items takes one second to complete, a data set containing 100 items takes two seconds, and a data set containing 1000 items will take three seconds. O(N log N). Logarithmic Time • E.G. More advanced sorting algorithms: quick sort,merge sort.

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