Algorithm Paradigms

By G Suresh Babu

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

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

Various algorithm Paradigms

Brute force paradigm

Divide and conquer paradigm

Backtracking paradigm

Greedy paradigm

Dynamic programming paradigm

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

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 recursively3. Combine solutions to solve original problem

Examples : Merge Sort, Quick Sort.

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

Junction

Finding Path through maze

Path B

Path

A

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).

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

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

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 overlapb)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

Fibonacci seriesFibonacci 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

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.

Summary

Wide range of strategies

Choice depends on Properties of problem Expected problem size Available resources

Any queries ???

Thank you