This document discusses various algorithms and data structures including divide-and-conquer, greedy algorithms, dynamic programming, pattern matching, and tries. It provides examples of each technique and describes how to determine when a problem can be solved using each approach. The key characteristics of different trie variants like standard, compressed, and suffix tries are also outlined. Overall, the document presents fundamental algorithm design strategies and data structures.

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 After completing this chapter, the reader will be able to
understand the following:

 Basic tools needed to develop and analyze algorithms
 Methods to compute the efficiency of algorithms
 Ways to make a wise choice among many solutions for a
given problem

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Algorithm Analysis
Asymptoti c Notations (W, p, O)
Big Omega (W)

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Algorithm Analysis
Asymptoti c Notations (W, p, O)
Big Omega (W)

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 The following steps elaborate the general structure of the
divide-and-conquer strategy
 If the data size n of problem P is fundamental,
calculate the result of P(n) and go to step 4
 If the data size n of problem P is not fundamental,
divide the problem P(n) into equivalent subproblems
P(n1), P(n2), … P(ni) such that i ≥ 1
 Apply divide-and-conquer recursively to each
individual subproblem P(n1), P(n2), …,P(ni)
 Combine the results of all subproblems P(n1), P(n2),…,
P(ni) to get the final solution of P(n)

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 At level one, only one call to a partition is made
with n elements; at level two, Atmost two calls are
made with elements (n - 1), and so on
C(n) = O(n2)

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 General
 Knapsack Problem
 Elements of Greedy Strategy

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 To decide whether a problem can be solved using a
greedy strategy, the following elements should be
considered:
 Greedy-choice property
 Optimal substructure Greedy Method

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 Greedy-choice property A problem exhibits
greedy-choice property if a globally optimal
solution can be arrived at by making a locally
optimal greedy choice
 That is, we make the choice that seems best at
that time without considering the results from
the sub problems

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 The General Method
 Elements of Dynamic Programming
 Principle of Optimality
 Limitations of Dynamic Programming
 Knapsack Problem

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 The General Method
 Elements of Dynamic Programming
 Principle of Optimality
 Limitations of Dynamic Programming
 Knapsack Problem

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A dynamic programming solution has the following three
components:
 Formulate the answer as a recurrence relation or a
recursive algorithm
 Show that the number of different instances of your
recurrence is bounded by a polynomial
 Specify an order of evaluation for the recurrence

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To decide whether a problem can be solved using the dynamic
programming method, the following three elements of dynamic
programming should be considered:

 Optimal substructure
 Overlapping subproblems
 Memorization

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 A problem exhibits optimal substructure if an optimal
solution to the problem contains within it optimal
solutions to subproblems
 It also means that dynamic programming (and greedy
method) might apply

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 When a recursive algorithm revisits the same problem
repeatedly, it is said that the optimization problem has
overlapping sub problems
 This is beneficial for dynamic programming
 It solves each subproblem once and stores the answer in a
table
 This answer can be searched in constant time when required.
This is contradictory to the divide-and-conquer strategy
where a new problem is generated at each step of recursion

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 A problem exhibits optimal substructure if an optimal
solution to the problem contains within it optimal solutions
to subproblems
 It also means that dynamic programming (and greedy
method) might apply

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 However, it uses the control structure similar to the
recursive algorithm
 In a memorized recursive algorithm, an entry is
maintained in a table for the solution to each subproblem
 Initially, all entries contain a special value, which
indicates that the entry is not yet used
 For each subproblem, which is encountered for the first
time, its solution is computed and stored in the table
 Next time, for that subproblem, its entry is searched and
the value is used
 This can be implemented using hashing

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 The principle of optimality states that an optimal
sequence of decisions has the property that whatever
the initial state and decision are, the remaining
decisions must constitute an optimal decision
sequence with regard to the state resulting from the
first decision
Principle of Optimality

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 Pattern matching is the process of finding the
presence of a particular string (pattern) in the given
string (text)
Pattern Matching

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 Database search
 Search engine
 Text editors
 Intrusion detection
 Natural language processing
 Feature detection in digitized images
A few such applications are as follows:

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The most popular are the following:
Brute-force approach
Boyer–Moore algorithm
Knuth–Morris–Pratt algorithm
Robin–Karp algorithm
Text partitioning algorithm
Semi-numerical algorithm
Popular techniques for
string pattern search

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 String : A string is a finite sequence of symbols that are
chosen from a set or alphabet:
 Alphabet is a set of characters or symbols
 Substring: A substring or subsequence of a string is a
subset of the symbols in a string where the order of
elements is preserved
 Suffix: A suffix of S is a substring S[i, …, m − 1], where i
ranges between 0 and m − 1

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 This is a simple straight forward approach based on
the comparison of a pattern character by character
with a string

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 The steps involved in this approach are as
follows:
 Adjust the pattern P at beginning of the text
 Start moving from left to right and compare the
character of pattern to the corresponding character
in text
 Continue with step 2 until successful (all characters
of the pattern are matched) or unsuccessful (a
mismatch is detected)

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 Boyer and Moore have developed an efficient pattern
matching algorithm
 Instead of sliding by one character to the right at a time, in
Boyer–Moore approach, the sliding to the right is done in
longer steps
 The algorithm scans the character of pattern from right to
left beginning with the rightmost character
 If the text symbol compared with the rightmost
pattern symbol does not occur in the pattern at all, then the
pattern can be shifted by m positions (where m is length
of pattern)

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 The researchers Knuth, Morris, and Pattern
proposed a linear time algorithm for the string
matching problem
 In this approach, a matching time of O(n) is
achieved by avoiding comparisons with characters
of T that have previously been involved in
comparison with some element of the pattern P to
be matched so that backtracking is avoided

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 The KMP matcher finds the occurrence of the
pattern P in text T and returns the number of shifts
of P, after which the occurrence is found taking T, P,
and prefix function p as inputs

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 A compact data structure that represents a set of
strings (such as all the words in a text) known as tries
 A trie is a tree-based data structure for storing strings
to make pattern matching faster
 A trie helps in pattern matching in time that is
proportional to the length of the pattern
 Tries can be used to perform prefix query for
information retrieval
 Prefix query searches for the longest prefix of a given
string that matches a prefix of some string in the tries

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 There are variants of tries, which are listed as
follows:
 Standard tries
 Compressed trie
 Suffix tries

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 The standard trie for a set of strings S is an ordered tree
such that
 Each node but the root is labelled with a character;
 The children of a node are alphabetically ordered;
 The paths from the external nodes to the oot yield the
strings of S

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 Similar to the standard trie, a compressed is a tree-
based data structure
 For storing strings in order to make pattern matching
much faster
 This is an optimized approach for pattern matching
specially suitable for applications where time is a
more crucial factor

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Following are the unique characteristics of
compressed tire:
 A compressed trie (or Patricia trie) has internal
nodes of degree at least 2
 It is obtained from standard trie by compressing
chains of redundant nodes

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m e g h a v i n d
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 A suffix trie is a compressed Trie for all the suffixes
of a text
 This is a compressed Trie, and hence, possesses all
features a compressed trie and makes it more
powerful for making a search faster as it includes
all suffixes of a text

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