Design and Analysis of Algorithms Unit 1 - Introduction

1. The Design and
Analysis of
Algorithms
by Anany Levitin

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CHAPTER 1: INTRODUCTIONCHAPTER 1: INTRODUCTION
 What is an Algorithm
 Steps in Designing and Implementing
an Algorithm
 Important Problem Types
 Fundamental Data Structures

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Important problem types
 Sorting
 Searching
 String processing
 Graph problems
 Combinatorial problems
 Geometric problems
 Numerical problems

4. Sorting
 The sorting problem is to rearrange the
items of a given list in nondecreasing
order.
 Specially chosen piece of information that
guides sorting is called a key.
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5. Why Sort?
 ranking Internet search results
 ranking students by their GPAscores.
 sorting makes many questions about the
list easier to answer.
The most important of them is searching
sorting is used as an auxiliary step in several
important algorithms ,
 e.g., geometric algorithms and Data Compression,
Greedy Approach
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6. Different Algorithms Different
Time Complexities
 Good sorting algorithms that sort an arbitrary
array of size n using about n log2 n
comparisons
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7. Properties of Sorting Algorithms
 Stable Algorithm
 preserves the relative order of any two equal
elements in its input.
 In- place Algorithm
algorithm is does not require extra memory,
except, possibly, for a few memory units.
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9. Searching
 The searching problem deals with
finding a given value, called a search
key, in a given set or a multiset
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10.  For searching, too, there is no single
algorithm that fits all situations best.
 Some algorithms work faster with more
memory requirement
 Some are very fast but applicable only to
sorted arrays
 Applications where the underlying data
may change frequently relative to the
number of searches to be considered
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11. String Processing
 A string is a sequence of characters
from an alphabet
 Applications: searching for a given word in
a text called string matching
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12. String Processing
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13. Graph Problems
 A graph can be thought of as a
collection of points called vertices,
some of which are connected by line
segments called edges. (informal
definition)
 Applications:
 transportation
 Communication
 social and economic networks
 project scheduling
 games.
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14.  Basic graph algorithms:
Graph-traversal algorithms
 Shortest-path algorithms
 Topological sorting for graphs with directed
edges
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15. Example Problems:
 Graph-coloring problem
Event scheduling
 Traveling Salesman Problem (TSP)
route planning
Circuit board and VLSI chip fabrication
X-ray crystallography
genetic engineering
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17. Colour the graph with smallest number of colours
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18. Combinatorial Problems
 Problems that ask, explicitly or implicitly,
to find a combinatorial object that satisfies
certain constraint:
 a permutation
a combination
a subset
Additional properties:
 Maximum Value
 Minimum Value
 Eg. TSP
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19.  Difficulties faced in solving combinatorial
problems:
 The number of combinatorial objects typically
grows extremely fast with a problem’s size
 There are no known algorithms for solving most
such problems exactly in an acceptable amount
of time. (This statement is neither proved nor
disproved)
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20. Geometric Problems
 Geometric algorithms deal with
geometric objects such as points,
lines, and polygons
 Applications:
Computer graphics
Robotics
 Tomography
 Classic problems:
Closest-pair problem
 Convex-hull problem 20

21. Numerical Problems
 Numerical problems involve mathematical
objects of continuous nature: solving
equations and systems of equations,
computing definite integrals, evaluating
functions, and so on.
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22.  Majority of such mathematical problems
can be solved only approximately
 Problems require manipulating real
numbers, which can be represented in a
computer only approximately
 A large number of arithmetic operations -
lead to an accumulation of the round-off
error - drastically distort an output
produced by a seemingly sound algorithm
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23.  Requirement of Applications:
information storage, retrieval
transportation through networks
presentation to users
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24. Check if you are Clear with:
 Sorting
 Searching
 String processing
 Graph problems
 Combinatorial problems
 Geometric problems
 Numerical problems
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25. Have a SPLENDID Day..!!!
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