Algorithm paradigms
- 1. Algorithm Paradigms By G Suresh Babu
- 2. Algorithm An algorithm is a step-by-step procedure for solving a problem. Paradigm “Pattern of thought” which governs scientific apprehension during a certain period of time
- 3. Paradigms for Algorithm Design To aid in designing algorithms for new problems, we create a taxonomy of high level patterns or paradigms, for the purpose of structuring a new algorithm along the lines of one of these paradigms. A paradigm can be viewed as a very high level algorithm for solving a class of problems
- 4. Various algorithm Paradigms Brute force paradigm Divide and conquer paradigm Backtracking paradigm Greedy paradigm Dynamic programming paradigm
- 5. Brute force paradigm Brute force is one of the easiest and straight forward approach to solve a problem Based on trying all possible solutions It is useful for solving small–size instances of a problem Generally most expensive approach Examples: Sequential search, factors of number
- 6. Divide and conquer paradigm Based on dividing problem into sub problems Approach 1. Divide problem into smaller sub problems Sub problems must be of same type Sub problems do not need to overlap 2. Solve each sub problem recursively 3. Combine solutions to solve original problem Examples : Merge Sort, Quick Sort.
- 7. Backtracking paradigm Backtracking is a technique used to solve problems with a large search space, by systematically trying and eliminating possibilities Based on depth-first recursive search Examples : Find path through maze, eight queens problem
- 8. Finding Path through maze Path A Path B
- 9. Eight Queens Problem Find an arrangement of 8 queens on a single chess board such that no two queens are attacking one another. 1) Place a queen on the first available square in row 1. 2) Move onto the next row, placing a queen on the first available square there (that doesn't conflict with the previously placed queens).
- 10. Greedy paradigm Optimal solution :A solution which minimizes or maximizes objective function is called optimal solution (i.e. BEST solution) This technique suggest devising an algorithm in no of steps, each stage depends on a particular input which gives “optimal solution”. Examples : 0/1 knapsack, kruskals & prims, counting money, Dijkstra’s shortest path algorithm
- 11. Counting money Suppose you want to count certain amount of money, using the fewest possible bills and coins A greedy algorithm to do this would be: 1. At each step, take the largest possible bill or coin that does not overshoot Example: To make 17Rs from 1,5,10 rupee denominations a)add 10 rupee note b)add 5 rupee note c)add two 1 rupee notes
- 12. Dynamic programming paradigm It is also an optimization technique. Based on remembering past results Approach a) Divide problem into smaller sub problems Sub problems must be of same type Sub problems must overlap b)Solve each sub problem recursively May simply look up solution c)Combine solutions into to solve original problem Store solution to problem Examples: All pairs shortest path, Fibonacci series
- 13. Fibonacci series Fibonacci numbers – Fibonacci(0) = 1 – Fibonacci(1) = 1 – Fibonacci(n) = Fibonacci(n-1) + Fibonacci(n-2) Recursive algorithm to calculate Fibonacci(n) – If n is 0 or 1, return 1 – Else compute Fibonacci(n-1) and Fibonacci(n-2) – Return their sum
- 14. Dynamic programming comes to rescue! Each computation stores the result in memory for future references. Instead of calling Fib(n-1) and Fib(n-2) we are first checking if they are already present in memory or not.
- 15. Summary Wide range of strategies Choice depends on Properties of problem Expected problem size Available resources Any queries ???
- 16. Thank you