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Dr. Scott Schaefer

Data Structures and Algorithms CSCE 221H

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Staff Instructor

Dr. Scott Schaefer HRBB 527B Office Hours: T 8:30am – 9:30am, R 9:30am-10:30am

(or by appointment) TA

Keishla Ortiz Lopez HRBB 410A Office Hours: MW 1pm-3pm

Peer Teacher Kuo-Wei Chiao

What will you learn? Analysis of Algorithms Stacks, Queues, Deques Vectors, Lists, and Sequences Trees Priority Queues & Heaps Maps, Dictionaries, Hashing Skip Lists Binary Search Trees Sorting and Selection Graphs

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Prerequisites CSCE 121 “Introduction to Program Design and

Concepts” or (ENGR 112 “Foundations of Engineering” and CSCE 113 “Intermediate Programming & Design”)

CSCE 222 “Discrete Structures” or MATH 302 “Discrete Mathematics” (either may be taken concurrently with CSCE 221)

Textbook

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Grading

3% Labs 12% Homework 5% Culture Assignments 30% Programming Assignments 10% Quizzes 20% Midterm 20% Final

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Assignments

Turn in code/homeworks via CSNET (get an account if you don’t already have one) Due by 11:59pm on day specified All programming in C++ Code, proj file, sln file, and Win32 executable Make your code readable (comment) You may discuss concepts, but coding is

individual (no “team coding” or web)

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Late Policy

Penalty = m: number of minutes late

days late

perc

enta

ge p

enal

ty%6.57

m

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Late Policy

Penalty = m: number of minutes late

days late

perc

enta

ge p

enal

ty%6.57

m

%25

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Late Policy

Penalty = m: number of minutes late

days late

perc

enta

ge p

enal

ty%6.57

m

%50

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Late Policy

Penalty = m: number of minutes late

days late

perc

enta

ge p

enal

ty%6.57

m

%100

Labs

Several structured labs at the beginning of the semester with (simple) exercisesGraded on completionTime to work on homework/projects

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Homework

Approximately 5 Written/Typed responses Simple coding if any

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Programming Assignments

About 5 throughout the semester Implementation of data structures or

algorithms we discusses in class Written portion of the assignment

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Quizzes

Approximately 10 throughout the semester Short answer, small number of questions Will only be given in class or lab

Must be present to take the quiz

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Culture Assignments

Two research seminar reportsOne before Spring break, one after

Biography of a famous Computer Scientist5 minute presentationSignup this week by sending me an email

http://faculty.cs.tamu.edu/schaefer/teaching/221_Spring2015/Assignments/culture.html

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Academic Honesty

Assignments are to be done on your ownMay discuss concepts, get help with a

persistent bugShould not copy work, download code, or

work together with others unless specifically stated otherwise

We use a software similarity checkerhttp://moss.stanford.edu

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Class Discussion Board

piazza.com/tamu/spring2015/csce221200

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Asymptotic Analysis

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Running Time The running time of an algorithm typically grows with the

input size. Average case time is often difficult to determine. We focus on the worst case running time.

Crucial to applications such as games, finance, and robotics Easier to analyze

Run

ning

Tim

e

1ms

2ms

3ms

4ms

5ms

Input InstanceA B C D E F G

worst case

best case

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Experimental Studies

Write a program implementing the algorithm

Run the program with inputs of varying size and composition

Use a method like clock() to get an accurate measure of the actual running time

Plot the results0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 50 100Input Size

Tim

e (m

s)

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Stop Watch Example

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Limitations of Experiments

It is necessary to implement the algorithm, which may be difficult

Results may not be indicative of the running time on other inputs not included in the experiment.

In order to compare two algorithms, the same hardware and software environments must be used

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Theoretical Analysis

Uses a high-level description of the algorithm instead of an implementation

Characterizes running time as a function of the input size, n.

Takes into account all possible inputs Allows us to evaluate the speed of an

algorithm independent of the hardware/software environment

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Important Functions Seven functions that often

appear in algorithm analysis:

Constant 1 Logarithmic log n Linear n N-Log-N n log n Quadratic n2

Cubic n3

Exponential 2n

1)(nLog

n

)(nLogn

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Important Functions Seven functions that often

appear in algorithm analysis:

Constant 1 Logarithmic log n Linear n N-Log-N n log n Quadratic n2

Cubic n3

Exponential 2n

)(nLogn

2n

3nn2

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Why Growth Rate Mattersif runtime

is... time for n + 1 time for 2 n time for 4 n

c lg n c lg (n + 1) c (lg n + 1) c(lg n + 2)

c n c (n + 1) 2c n 4c n

c n lg n~ c n lg n

+ c n2c n lg n +

2cn4c n lg n +

4cn

c n2 ~ c n2 + 2c n 4c n2 16c n2

c n3 ~ c n3 + 3c n2 8c n3 64c n3

c 2n c 2 n+1 c 2 2n c 2 4n

runtimequadrupleswhen problemsize doubles

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Comparison of Two Algorithms

insertion sort isn2 / 4

merge sort is2 n lg n

sort a million items?insertion sort takes

roughly 70 hourswhile

merge sort takesroughly 40 seconds

This is a slow machine, but if100 x as fast then it’s 40 minutesversus less than 0.5 seconds

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Constant Factors

The growth rate is not affected by constant factors or lower-order terms

Examples 102n 105 is a linear function 105n2 108n is a quadratic function

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Big-Oh Notation Given functions f(n) and

g(n), we say that f(n) is O(g(n)) if there are positive constantsc and n0 such that

f(n) cg(n) for n n0

Example: 2n 10 is O(n) 2n 10 cn (c 2) n 10 n 10(c 2) Pick c 3 and n0 10

1

10

100

1,000

10,000

1 10 100 1,000n

3n

2n+10

n

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Big-Oh Example

Example: the function n2 is not O(n)

n2 cn n c The above inequality

cannot be satisfied since c must be a constant

1

10

100

1,000

10,000

100,000

1,000,000

1 10 100 1,000n

n 2̂100n10nn

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More Big-Oh Examples7n-2

7n-2 is O(n)need c > 0 and n0 1 such that 7n-2 c•n for n n0

this is true for c = 7 and n0 = 1

3n3 + 20n2 + 53n3 + 20n2 + 5 is O(n3)need c > 0 and n0 1 such that 3n3 + 20n2 + 5 c•n3 for n n0

this is true for c = 4 and n0 = 21 3 log n + 5

3 log n + 5 is O(log n)need c > 0 and n0 1 such that 3 log n + 5 c•log n for n n0

this is true for c = 8 and n0 = 2

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Big-Oh and Growth Rate The big-Oh notation gives an upper bound on the growth

rate of a function The statement “f(n) is O(g(n))” means that the growth rate

of f(n) is no more than the growth rate of g(n) We can use the big-Oh notation to rank functions according

to their growth rate

f(n) is O(g(n)) g(n) is O(f(n))g(n) grows more

Yes No

f(n) grows more No YesSame growth Yes Yes

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Big-Oh Rules If is f(n) a polynomial of degree d, then f(n) is

O(nd), i.e.,1. Drop lower-order terms2. Drop constant factors

Use the smallest possible class of functions Say “2n is O(n)” instead of “2n is O(n2)”

Use the simplest expression of the class Say “3n 5 is O(n)” instead of “3n 5 is O(3n)”

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Computing Prefix Averages We further illustrate asymptotic analysis with two

algorithms for prefix averages The i-th prefix average of an array X is average of the

first (i 1) elements of X:A[i] X[0] X[1] … X[i])/(i+1)

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Prefix Averages (Quadratic)The following algorithm computes prefix averages in quadratic time by applying the definition

Algorithm prefixAverages1(X, n)Input array X of n integersOutput array A of prefix averages of X #operations A new array of n integers nfor i 0 to n 1 do n

s X[0] nfor j 1 to i do 1 2 … (n

1)s s X[j] 1 2 … (n

1)A[i] s (i 1) n

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Arithmetic Progression

The running time of prefixAverages1 isO(1 2 …n)

The sum of the first n integers is n(n 1) 2 Thus, algorithm prefixAverages1 runs in O(n2) time

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Prefix Averages (Linear)The following algorithm computes prefix averages in linear time by keeping a running sum

Algorithm prefixAverages2(X, n)Input array X of n integersOutput array A of prefix averages of X #operationsA new array of n integers ns 0 1for i 0 to n 1 do n

s s X[i] nA[i] s (i 1) n

return A 1Algorithm prefixAverages2 runs in O(n) time

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