Comp Sci 2500-1A: Algorithms

Mon/Wed/Fri 9:00am - 9:50am 121 Butler-Carlton Hall

http://web.mst.edu/~guozh/cs2500

Zhishan Guo310 Compupter Science Building

guozh@mst.eduOffice Hours: Mon 10:00-11:00, Thu 17:00-18:00

Or By Appointment

Comp Sci 2500-1B: Algorithms

Tue/Thu 3:30pm - 4:45pm 295 Toomey

http://web.mst.edu/~guozh/cs2500

Zhishan Guo310 Compupter Science Building

guozh@mst.eduOffice Hours: Mon 10:00-11:00, Thu 17:00-18:00

Or By Appointment

Teaching Assistant

• Ayushi Mathur

• Email: amrd6@mst.edu

• Office Hours:

– Mo 15:00 - 16:00,

– Fr 14:00 - 16:00

– 302 Engineering Research Lab (ERL),

– or by appointment

Textbook & References• Introduction to Algorithms, 3rd Ed. by Cormen, Leiserson, Rivest, &

Stein (CLRS), McGraw Hill, 2009. • Lecture slides will be put online

– Thanks to Profs. Mark Foskey, Ming Lin, Dinesh Manocha, Ketan Mayer-Patel, David Plaisted, Jack Snoeyink, Dr. Umamaheswari Devi (all UNC-Chapel Hill), Prof. Jeff Erickson (UIUC), and Prof. Simone Silvestri (MST).

OTHER REFERENCES:

Algorithmics: The Spirit of Computing, Harel

How to Solve It, Polya.

The Design and Analysis of Computer Algorithms, Aho, Hopcroft and Ullman.

Algorithms, Sedgewick.

Algorithmics: Theory & Practice, Brassard & Bratley.

Writing Efficient Programs & Programming Pearls, Bentley.

The Science of Programming, by Gries.

The Craft of Programming, by Reynolds.

Prerequisites

• Data Structure + Programming + Calculus

• Assume that you know, or can recall with a quick review, the materials in the following chapters. – Chapter 0, 1, and 2

– Section 3.2: growth of functions

– Chapter 10: elementary data structures

Course Roadmap

(Weeks)

• Algorithmic Basics (1~2)

• Divide and Conquer (2~3)

• Search Trees (1~2)

• Graph Algorithms (2~3)

• Dynamic Programming (1~2)

• Greedy Algorithms (2~3)

• Exams & Other (~2)

Course Work & Grades

• Homework: 20%

(total of 6-8, mostly design & analysis)

• Programming Assignment: 5%

• Class Participation: 5% + up to half a grade bonus

(Quiz: very basic materials)

• Mid-Term Exams: 30%

(total of 2, in class)

• Final Exam: 30%

• Programming Projects: 10% + up to 5% bonus

• 90% -> A, 80% -> B, 70% -> C, 60% -> D

“Half” closed book, NO collaboration

Homework Assignments

• Due at the beginning of each class on the due date given

• No late homework will be accepted• Lowest score will be dropped• Can discuss in group, but must write/formulate

solutions alone (failure to explain your solution orally to the instructor = cheat)

• Be neat, clear, precise, formal– You’ll be graded on correctness, simplicity, elegance

& clarity

Course Project

• Due on the last day of class• No late project report or code will be accepted• You are responsible for defining/proposing the course

project. You are encouraged to discuss with me or the TA and/or submit a proposal earlier than Nov 11.

• It can be either some implementations of core algorithms we cover (you are responsible for showing the correctness of your code), or some algorithmic study to any open problem.

• Group work is allowed (max=3) though contribution of each member must be clarified in the final report.

Communication

• Visit instructor / TA during office hours, by appointment, or email correspondence

• All lecture notes and most of handouts are posted at the course website:

http://web.mst.edu/~guozh/cs2500

• Major messages are put on course website & canvas

• Discussions -- face-to-face in groups, or on canvas

• Student grades can be checked with the instructor / TA

Basic Courtesy

• Write/print assignments neatly & formally

• Please do not read newspaper & other materials, or browse web in class

• When coming to the class late or leaving early, please take an aisle seat quietly

• Remain quiet, except asking questions or answering questions posed by instructors– no whispers or private conversation

THANK YOU!!!

How to Succeed in this Course

• Start early on all assignments. DON'T procrastinate.

• Complete all reading before class.

• Participate in class.

• Think in class.

• Review after each class.

• Be formal and precise on all problems sets and exams

Weekly Reading Assignment

Chapters 0, 1, 2, 3 and Appendix A

(Textbook: CLRS)

Solving a Computational Problem

• Problem definition & specification

– specify input, output and constraints

• Algorithm design & analysis

– devise a correct & efficient algorithm

• Implementation planning

• Coding, testing and verification

Our Focus

Primary Focus

Develop thinking ability

– formal thinking

(proof techniques & analysis)

– problem solving skills

(algorithm design and application)

About Coding/Programming…

Goals

• Be very familiar with a collection of core algorithms.

• Be fluent in algorithm design paradigms: divide & conquer, greedy algorithms, dynamic programming, (randomization), (approximation methods).

• Be able to analyze the correctness and runtime performanceof a given algorithm.

• Be familiar with the inherent complexity (lower bounds & intractability) of some problems.

• Be intimately familiar with basic data structures.

• Be able to apply techniques in practical problems.

What Will We Be Doing

• Examine interesting problems

• Devise algorithms for solving them

• Prove their correctness

• Analyze their runtime performance

• Study data structures & core algorithms

• Learn problem-solving techniques

• Applications in real-world problems

The Flipped Classroom Style will be very limited.

Congressional Apportionment

Article I, Section 2 of the United States Constitution requires that Representatives and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers. . . The Number of Representatives shall not exceed one for every 30,000, but each State shall have at Least one Representative. . .

The Huntington-Hill Method

Currently, n = 50 and R = 435.

Pseudo-code

• Well-written pseudocode reveals the internal structure of the algorithm but hides irrelevant implementation details, making the algorithm much easier to understand, analyze, debug, and implement.

• The precise syntax of pseudocode is a personal choice,

but the overriding goal should be clarity and precision.

Ideally, pseudocode should allow any competent

programmer to implement the underlying algorithm,

quickly and correctly, in their favorite programming

language, without understanding why the algorithm

works.

Algorithms

• A tool for solving a well-specified computational problem

• Example: sortinginput: A sequence of number

output: An ordered permutation of input

issues: correctness, efficiency, storage, etc.

AlgorithmInput Output

for j=2 to length(A)

do key=A[j]

i=j-1

while i>0 and A[i]>key

do A[i+1]=A[i]

i--

A[i+1]=key

Example: Insertion Sort

for j 2 to length(A)

do key=A[j]

i j-1

while i>0 and A[i]>key

do A[i+1] A[i]

i--

A[i+1] key

Correctness Proofs

• Proving (beyond “any” doubt) that an algorithm is correct.– Prove that the algorithm produces correct output

when it terminates. Partial Correctness.

– Prove that the algorithm will necessarily terminate. Total Correctness.

• Techniques– Proof by Construction.

– Proof by Induction.

– Proof by Contradiction.

Loop Invariant

• Logical expression with the following properties.

– Holds true before the first iteration of the loop –Initialization.

– If it is true before an iteration of the loop, it remains true before the next iteration – Maintenance.

– When the loop terminates, the invariant ― along with the fact that the loop terminated ― gives a useful property that helps show that the loop is correct – Termination.

• Similar to mathematical induction.

• Invariant: at the start of each for loop, A[1…j-1] consists of elements originally in A[1…j-1] but in sorted order; all other elements are unchanged.

for j=2 to length(A)

do key=A[j]

i=j-1

while i>0 and A[i]>key

do A[i+1]=A[i]

i--

A[i+1]=key

Example: Insertion Sort

for j 2 to length(A)

do key=A[j]

i j-1

while i>0 and A[i]>key

do A[i+1] A[i]

i--

A[i+1] key

• Invariant: at the start of each for loop, A[1…j-1] consists of elements originally in A[1…j-1] but in sorted order; all other elements are unchanged

for j=2 to length(A)

do key=A[j]

i=j-1

while i>0 and A[i]>key

do A[i+1]=A[i]

i--

A[i+1]=key

Initialization: j = 2, the invariant trivially holds because A[1] is a sorted array. √

Example: Insertion Sort

for j 2 to length(A)

do key=A[j]

i j-1

while i>0 and A[i]>key

do A[i+1] A[i]

i--

A[i+1] key

• Invariant: at the start of each for loop, A[1…j-1] consists of elements originally in A[1…j-1] but in sorted order; all other elements are unchanged

for j=2 to length(A)

do key=A[j]

i=j-1

while i>0 and A[i]>key

do A[i+1]=A[i]

i--

A[i+1]=key

Maintenance: the inner while loop finds the position iwith A[i] <= key, and shifts A[j-1], A[j-2], …, A[i+1] right by one position. Then key, formerly known as A[j], is placed in position i+1 so that A[i] A[i+1] < A[i+2].

A[1…j-1] sorted + A[j] A[1…j] sorted

Example: Insertion Sort

for j 2 to length(A)

do key=A[j]

i j-1

while i>0 and A[i]>key

do A[i+1] A[i]

i--

A[i+1] key

• Invariant: at the start of each for loop, A[1…j-1] consists of elements originally in A[1…j-1] but in sorted order; all other elements are unchanged

for j=2 to length(A)

do key=A[j]

i=j-1

while i>0 and A[i]>key

do A[i+1]=A[i]

i--

A[i+1]=key

Termination: the loop terminates, when j=n+1. Then the invariant states: “A[1…n] consists of elements originally in A[1…n] but in sorted order.” √

Example: Insertion Sort

for j 2 to length(A)

do key=A[j]

i j-1

while i>0 and A[i]>key

do A[i+1] A[i]

i--

A[i+1] key

Running time

• Depends on input (e.g., sorted/reversely)

• Depends on input size (5 elements vs 500K)

– Parameterize in input size (n)

• Want upper bounds (generally)

– Guarantee to the user

for j 2 to length(A)

do key=A[j]

i j-1

while i>0 and A[i]>key

do A[i+1] A[i]

i--

A[i+1] key

Analysis

• Worst-Case (usually)

T(n) = max time on any input of size n

• Average-Case (sometimes)

T(n) = expected time over all inputs of size n

(Need assumption of…)

• Best-Case

A Simple Example – Linear Search INPUT: a sequence of n numbers, key to search for.

OUTPUT: true if key occurs in the sequence, false otherwise.

LinearSearch(A, key)1 i 1

2 while i ≤ n and A[i] != key

3 do i++

4 if i n

5 then return true

6 else return false

A Simple Example – Linear Search INPUT: a sequence of n numbers, key to search for.

OUTPUT: true if key occurs in the sequence, false otherwise.

n

i 21

LinearSearch(A, key) cost times1 i 1 c1 1

2 while i ≤ n and A[i] != key c2 x

3 do i++ c3 x-1

4 if i n c4 1

5 then return true c5 1

6 else return false c6 1

A Simple Example – Linear Search INPUT: a sequence of n numbers, key to search for.

OUTPUT: true if key occurs in the sequence, false otherwise.

n

i 21

LinearSearch(A, key) cost times1 i 1 c1 1

2 while i ≤ n and A[i] != key c2 x

3 do i++ c3 x-1

4 if i n c4 1

5 then return true c5 1

6 else return false c6 1

x ranges between 1 and n+1.

So, the running time ranges between

c1+ c2+ c4 + c5 – best case

and

c1+ c2(n+1)+ c3n + c4 + c6 – worst case

A Simple Example – Linear Search INPUT: a sequence of n numbers, key to search for.

OUTPUT: true if key occurs in the sequence, false otherwise.

n

i 21

Assign a cost of 1 to all statement executions.

Now, the running time ranges between

1+ 1+ 1 + 1 = 4 – best case

and

1+ (n+1)+ n + 1 + 1 = 2n+4 – worst case

LinearSearch(A, key) cost times1 i 1 1 1

2 while i ≤ n and A[i] != key 1 x

3 do i++ 1 x-1

4 if i n 1 1

5 then return true 1 1

6 else return false 1 1

A Simple Example – Linear Search INPUT: a sequence of n numbers, key to search for.

OUTPUT: true if key occurs in the sequence, false otherwise.

n

i 21

If we assume that we search for a random item in the list,

on an average, Statements 2 and 3 will be executed n/2 times.

Running times of other statements are independent of input.

Hence, average-case complexity is

1+ n/2+ n/2 + 1 + 1 = n+3

LinearSearch(A, key) cost times1 i 1 1 1

2 while i ≤ n and A[i] != key 1 x

3 do i++ 1 x-1

4 if i n 1 1

5 then return true 1 1

6 else return false 1 1

Worst-Case time & Order of Growth

• Depends on computer (software vs. hardware)

• BIG IDEA - Asymptotic analysis

– Ignore machine dependent constants

– Look at the growth of T(n) as n -> +inf

– Notation: we can ignore the lower-order terms, since they are relatively insignificant for very large n. We can also ignore leading term’s constant coefficients, since they are not as important for the rate of growth in computational efficiency for very large n.

Correctness Proof of Linear Search• Use Loop Invariant for the while loop:

LinearSearch(A, key)1 i 1

2 while i ≤ n and A[i] != key

3 do i++

4 if i n

5 then return true

6 else return false

If the algm. terminates, then it produces correct result.

Initialization.

Maintenance.

Termination.

Argue that it terminates.

Correctness Proof of Linear Search• Use Loop Invariant for the while loop:

– At the start of each iteration of the while loop, the search key is not in the subarray A[1…i-1].

LinearSearch(A, key)1 i 1

2 while i ≤ n and A[i] != key

3 do i++

4 if i n

5 then return true

6 else return false

If the algm. terminates, then it produces correct result.

Initialization.

Maintenance.

Termination.

Argue that it terminates.

Comparisons of Algorithms

• Sorting– Insertion sort: (n2) = c1n

2

– Merge sort: (n log n) = c2n log n

For 106 numbers, insertion sort takes 5.56 hrs on a supercomputer using machine language and 16.67 min on a PC using C/C++ with merge sort.

Why Order of Growth Matters?Computer speeds double every two years…

Order of growth matters

• Computer A: IntelCore i7 (2015), 1011 instructions per second, executes Insertion sort

• Computer B: Intel 386 (1985) 107 instructions per second, executes Merge sort

• We also assume that the young and cool owner of Computer A is a better programmer than the nostalgic owner of Computer B, thus c1 < c2, and in particular c1 = 2 and c2 = 50.

• We consider a sequence of n = 108 elements

Order of growth matters

• No matter the technological advancements, designing efficient algorithms is the key to achieving good and satisfactory performance.

Effect of faster machines

The number of items that can be sorted in one second

using an algorithm taking exactly n2 time as compared to

one taking n lg n time, assuming 1 million and 2 million

operations per second. Notice that, for the n lg n algorithm,

doubling the speed almost doubles the number of items

that can be sorted. (Order of growth matters!)

Ops/sec: 1M 2M Gain

n*n alg 1000 1414 1.4

n log n alg 62700 118600 1.9