Strings
6. Strings
String. Sequence of characters. Ex. Natural languages, Java programs, genomic sequences, .
The digital information that underlies biochemistry, cell biology, and development can be represented by a simple string of G's, A's, T's and C's
Linear Programming
see ORF 307
Linear Programming
What is it? Quintessential tool for optimal allocation of scarce resources, among a number of competing activities. Powerful and general problem-solving method that encompasses: shortest path, network flow
B asic T e r m s
3 .1 E le m e n t a r y S o r t s
Ex: student record in a University.
Sort: rearrange sequence of objects into ascending order.
Reference: Chapter 6, Algorithms in Java, 3rd Edition, Robert Sedgewick.
Robert Sedgewick and Kevin Wayne Copy
S y m b o l T a b le
E le m e n t a r y S y m b o l T a b le s
Symbol table: key-value pair abstraction.
Insert a value with specified key.
Given a key, search for the corresponding value.
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DNS lookup.
Insert URL with specified IP address.
Given URL, f
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4 .5 S y m b o l T a b le A p p li c a t i o n s
Set. Unordered collection of distinct keys.
API for SET.
!
contains(key)
!
remove(key)
!
insert the key into the set
is the given key in the set?
remove the key from the set
return iterator over a
Minimum Spanning Tree
Minimum Spanning Tree
MST. Given connected graph G with positive edge weights, find a min weight set of edges that connects all of the vertices.
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Reference: Chapter 20, Algorithms in Java, 3rd Edi
S t r in g S e ar ch
S t r in g S ear chin g
String search. Given a pattern string, find first match in text.
Model. Can't afford to preprocess the text.
Parameters. N = length of text, M = length of pattern.
typically N > M
Pattern
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U n d ir ect ed G r aphs
Graph. Set of objects with pairwise connections.
Why study graph algorithms?
Interesting and broadly useful abstraction.
Challenging branch of computer science and discrete math.
Hundreds of graph algor
B-Trees
B-Trees
1
Motivation for B-Trees
Index structures for large datasets cannot be stored in main
memory
Storing it on disk requires different approach to efficiency
Assuming that a disk spins at 3600 RPM, one revolution
occurs in 1/60 of a second,
CmSc 250 Intro to Algorithms
An Example of Heapsort:
Given an array of 6 elements: 15, 19, 10, 7, 17, 16, sort it in ascending order using heap sort.
Steps:
1.
Consider the values of the elements as priorities and build the heap tree.
2.
Start deleteMin o
CSCE 3110
Data Structures & Algorithm
Analysis
Rada Mihalcea
http:/www.cs.unt.edu/~rada/CSCE3110
Graphs (I)
Reading: Chap.9, Weiss
What is a Graph?
A graph G = (V,E) is composed of:
V: set of vertices
E: set of edges connecting the vertices in V
An edge e
G e o m e t r ic se ar ch : O v e r v ie w
G e o m e t r i c A lg o r i t h m s
Types of data. Points, lines, planes, polygons, circles, .
This lecture. Sets of N objects.
Geometric problems extend to higher dimensions.
Good algorithms also extend to high
Optimize Judiciously
4.2 Hashing
More computing sins are committed in the name of efficiency (without necessarily achieving it) than for any other single reason including blind stupidity. - William A. Wulf
We should forget about small efficiencies, say ab
G e o m e t r i c A lg o r i t h m s
Reference: Chapters 26-27, Algorithms in C, 2nd Edition, Robert Sedgewick
Robert Sedgewick and Kevin Wayne Copyright 2006 http:/www.Princeton.EDU/~cos226
G e o m e t r ic se ar ch : O v e r v ie w
Types of data. Points
Desiderata
Reductions
Desiderata. Classify problems according to their computational requirements. Frustrating news. Huge number of fundamental problems have defied classification for decades.
Robert Sedgewick and Kevin Wayne Copyright 2006 http:/www.Prin
Pattern Matching
Regular Expressions
String search. Search for given string in a large text file. Regular expression. Natural and compact way to express multiple text patterns. Quintessential programmer's tool.
! !
Ex. Fragile X syndrome is a common cause
Mergesort and Quicksort
Mergesort and Quicksort
Two great sorting algorithms. Full scientific understanding of their properties has enabled us to hammer them into practical system sorts. Occupies a prominent place in world's computational infrastructure.
Sorting Applications
3.5 Applications
Applications.
!
Sort a list of names. Organize an MP3 library. Display Google PageRank results.
obvious applications
!
!
!
List RSS news items in reverse chronological order. Find the median. Find the closest pair. Bi
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Symbol Table Review
6.2 String Sets
Symbol table. Associate a value with a key. Search for value given key. Balanced trees use O(log N) key comparisons. Hashing uses O(1) probes, but probe proportional to key length.
! ! ! !
Q. Are key comparisons necessa
Running Time
Analysis of Algorithms
As soon as an Analytic Engine exists, it will necessarily guide the future course of the science. Whenever any result is sought by its aid, the question will arise - By what course of calculation can these results be ar
Symbol Table Review
4.4 Balanced Trees
Symbol table: key-value pair abstraction. Insert a value with specified key. Search for value given key. Delete value with given key.
! ! !
Randomized BST. O(log N) time per op. [unless you get ridiculously unlucky]
Data Compression
Data Compression
Compression reduces the size of a file: To save space when storing it. To save time when transmitting it. Most files have lots of redundancy.
! ! !
Who needs compression? Moore's law: # transistors on a chip doubles every
Directed Graphs
Directed Graphs
Digraph. Set of objects with oriented pairwise connections. Ex. One-way street, hyperlink.
Reference: Chapter 19, Algorithms in Java, 3rd Edition, Robert Sedgewick
Robert Sedgewick and Kevin Wayne Copyright 2006 http:/www.P
S y m b o l T a b le C h a lle n g e s
4 .3 B i n a r y S e a r c h T r e e s
Symbol table. Key-value pair abstraction.
Insert a value with specified key.
Search for value given key.
Delete value with given key.
!
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!
B in a r y s e a r c h t r e e s
Chal
O v e r v ie w
Com b in at or ial S ear ch
Exhaustive search. Iterate through all elements of a search space.
Backtracking. Systematic method for generating all solutions
to a problem, by successively augmenting partial solutions.
Applicability. Huge rang