Algorithm Paradigms High Level Approach To solving a Class of Problems

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.

Incremental Paradigm Applicability : Can be applied to develop an algorithm to solve a problem for which –the input is a sequence and –the output can be computed incrementally from the last output and the next term of the input. Pseudocode –S n = Incremental(x 1, x 2,.. x n ) Calculate S 0 for initial sequence x 1 or empty sequence for i = initial index to n –S i+1 = S i  x i+1 (  represents arbitrary operation/function) Examples –Sum, Maximum, Minimum of numerical sequence, insertion sort

Divide and Conquer Paradigm Applicability –Can be applied to develop an algorithm to solve a problem for which –the problem can be divided into smaller problems of the same type and –the output can be computed from solutions to the smaller problems in top down direction until division results in “atomic” problems which can be solved directly. Pseudocode –Answer = D&C( Input ) If Atomic( Input ) then Return Answer(Input) Else Divide Input into Input 1 and Input 2 –Answer 1 = D&C( Input 1 ) – Answer 2 = D&C( Input 2 ) Return Answer(Input) = Combination( Answer 1, Answer 2 ) End If Examples : Merge Sort, Quick Sort, C(n,k)

Dynamic Programming Paradigm Applicability –Can be applied to develop an algorithm to solve a problem for which –the problem can be divided into smaller problems of the same type and –the output can be computed from solutions to the smaller problems in bottom up direction by solving “atomic” problems first, and by –storing solutions to all problems in a table to be used in solving larger problems. Pseudocode –Given P.Input, construct table T for DP( P ) where X is index of problem P to be solved –For every atomic problem A with index Y, calculate DP( A ) and store in table T(Y). –While T( X ) is null, Identify an intermediate problem I with index V for which T( V ) is null and T(U) is non null for every immediate subproblem I’ with index U of I. Store DP( I ) in T(V) as a combination of T( U ) entries Examples : Knapsack, C(n,k), Edit Distance, Optimum Search Tree

Greedy Paradigm Applicability –Can be applied to solve a problem for which the solution can be obtained by –making a sequence of choices or selections, subject to some constraint to optimize some objective function. Pseudocode –While a solution has not been found do Selection : select or make decision to maximize improvement in objective function. Feasibility : accept selection or decision if it does not violate the problem constraint. Solution : check to see if a solution has been reached. Examples : Shortest Path, Minimum Spanning Tree, File Compression by Huffman Encoding

Backtrack Programming Paradigm Applicability –Applicable to problems where the output is a sequence over an alphabet A (possibly ordered) with a possible constraint on the sequence and a possible objective function to be optimized Pseudocode –Finds an optimum sequence over A subject to constraint C corresponding to a path from root to leaf in a search tree T –Expand(s : String) curvalue = value of string s /* number of colors, length of path, sum of sizes */ If s is a solution (path to leaf of T) then –if curvalue better_than best then »best = curvalue »s* = s Else –for each a in such that Constraint(s+a) »Expand (s + a) End if –Backtrack( P : problem input ) best = value of expand ( ) Output s*, best Examples : Vertex Coloration, Knapsack, Longest Path