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SEARCHING ALGORITHMS:

The Broad Picture

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Introduction

Classes

Some important algos explained

Bibliogaphy

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Searching Algorithm procedure written

for finding an item with specified

properties among a collection of items.

Items maybe stored in database as

records or as math formulae orprocedures or graphical entities.

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For virtual search spaces :

Used in constraint satisfaction problem

andMax-Min problem Linear Search,Brute Force Search, Binary Search, etc.

Used in local search methods whereelements of a graph include search space

Simulated Annealing, Tabu Search,

Genetic Programming, etc.

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Used in Tree Search algorithmswhere

vertices of tree are elements of searchspace and traversing is done in some

special order (breadth-first ,depth-first)

BacktrackingandBranch and Bound algo.

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For sub-structures of a given structure:

Used in Combinatorial Search algos that look

for specific sub-structure of a given discretestructure such as a graph, a string or finite

group Brents algo , Gale Shapley algo , etc.

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Used in Graph algorithms for finding specificsub-structure in a given graph- sub graphs,circuits, paths, etc. Djikshtras algo, Kruskalsalgo, Prims algo, etc.

Used in String Searchingalgorithms that searchfor patterns within strings

Boyer-Moore algo, Knuth-Morris-Pratt algo, BruteForce algo, Karp-Rabin algo, Deterministic FiniteAutomaton algo, etc.

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For quantum computers :

Used for designing quantum computers theoretically faster than linear or brute-force search Grovers algorithm.

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Simulated Annealing: It has analogy with

annealing in metallurgy, where materials are

heated and cooled to increase size of its crystals

and reduce its effects.

Following analogy, each step of SA algo

replaces current solution by a random nearby

solution.

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Nearby solution is chosen according to diff.

b/w corresponding function values and global

parameter T.

Dependency is such that current solution

changes almost randomly when T is large, but

goes downhillas T goes to zero.

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s s0; e E(s)

sbest s; ebest e

k 0

while k < kmax and e < emaxsnew neighbour(s)

enew E(snew)

ifP(e, enew, temp(k/kmax)) > random() then

s snew; e enewifenew > ebest then

sbest snew; ebest enew

k k + 1

return sbest

Pseudocode

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Branch and Bound :B&B is a

general algorithm for finding optimal solutions of

various optimization problems.

Goal is to find theminimumvalue of a

functionf(x), wherexranges over some

set Sofadmissible or candidate solutionsor

search space.

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It includes two steps:Firstly a splitting

procedure that in a given set Sreturns two or

more smaller sets. Its called branching.

Secondly, it computes upper and lower bounds

for theminimumvalue off(x)within a given

subset S. Its called bounding.

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Three main components of B&B algo:

1. Bounding function : provides lower bound for thebest solution value obtainable in the subspace,

2. Strategy: for selecting the live solution subspaceto be investigated in the current iteration, and

3. Branching rule : to be applied for subdividing

the subspace considered into two or moresubspaces to be investigated in subsequentiterations.

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Knuth-Morris-Pratt Algo :

searches for occurrences of a "word" W within a

main "text string" S by employing theobservation that when a mismatch occurs, the

word itself embodies sufficient information to

determine where the next match could begin,

thus bypassing re-examination of previouslymatched characters.

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Consider an attempt at a left position j, that is

when the window is positioned on the textfactory[j..j+m-1]. Assume that the first

mismatch occurs betweenx[i] andy[i+j] with 0

< i

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When shifting, it is reasonable to expect that a

prefix vof the pattern matches some suffix of the

portion u of the text.

If we want to avoid another immediate

mismatch, the character following the prefix v in

the pattern must be different from a. The longest

such prefix vis called the tagged borderofu.

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Now letkmpNext[i] be the length of the longestborder ofx[0 ..i-1] followed by a

character cdifferent fromx[i] and -1 if no such

tagged border exits, for 0

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The tablekmpNextcan be computed in O(m)

The searching phase can be performed

inO(m+n) time.

Algo performs at most 2n-1 text character

comparisons during the searching phase.

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BIBLIOGRAPHY:

KNUTHD.E.,MORRIS (Jr) J.H.,PRATTV.R.,

1977,Fast pattern matching in strings, SIAM

Journal on Computing

CORMEN, T.H., LEISERSON, C.E., RIVEST, R.L.,

1990.Introduction to Algorithms

http://www-igm.univ-mlv.fr/~lecroq/string/

www.wikipedia.org