COMP 550 Algorithms Analysis MonWedFri 9 05 am

COMP 550: Algorithms & Analysis Mon/Wed/Fri 9: 05 am - 9: 55 am (FB 009) http: //www. cs. unc. edu/~zsguo/comp 550. html Zhishan Guo SN 126 zsguo@cs. unc. edu Office Hours: After Class on Mon. & Wed. Or By Appointment UNC Chapel Hill Z. Guo

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 UNC Chapel Hill Our Focus

Primary Focus Develop thinking ability – formal thinking (proof techniques & analysis) – problem solving skills (algorithm design and application) About Coding/Programming… UNC Chapel Hill

Goals • Be very familiar with a collection of core algorithms. • Be fluent in algorithm design paradigms: divide & conquer, greedy algorithms, randomization, dynamic programming, approximation methods. • Be able to analyze the correctness and runtime performance of 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. UNC Chapel Hill

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 inverted Classroom Style will be very limited. UNC Chapel Hill

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. . . UNC Chapel Hill

The Huntington-Hill Method Currently, n = 50 and R = 435. UNC Chapel Hill

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. UNC Chapel Hill

Textbook & References • Introduction to Algorithms, 3 rd Ed. by Cormen, Leiserson, Rivest, & Stein (CLRS), Mc. Graw 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, and Prof. Jeff Erickson (UIUC). OTHER REFERENCES: · · · Algorithmics: The Spirit of Computing, Harel · · Algorithms, Sedgewick. · The Craft of Programming, by Reynolds. How to Solve It, Polya. The Design and Analysis of Computer Algorithms, Aho, Hopcroft and Ullman. Algorithmics: Theory & Practice, Brassard & Bratley. Writing Efficient Programs & Programming Pearls, Bentley. The Science of Programming, by Gries. UNC Chapel Hill

Prerequisites • Data Structure + Discrete Mathematics • 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 UNC Chapel Hill

Course Roadmap • • Algorithmic Basics (4) Divide and Conquer (5 -6) Randomized Algorithms (9 -11) Search Trees (5) Graph Algorithms (5) Dynamic Programming (3) Greedy Algorithms (6) Special Topics* (1 -2) • M 13+W 15+F 14 = 42 UNC Chapel Hill

Course Work & Grades • Homework: 25% (total of 6 -9, mostly design & analysis) • Quizzes: 5% (very basic materials) • Mid-Term Exams: 40% (total of 3, 50 min each, in class) • Final Exam: 30% • Active Class Participation: up to half a grade bonus • Programming Projects: up to 10% bonus UNC Chapel Hill

Examinations • Quizzes: throughout the semester • Midterms: 3 in class • Final: 8 am - 11 am on May 4 th, 2015 “Half” closed book, NO collaboration UNC Chapel Hill

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 UNC Chapel Hill

Course Project • Due on 23: 59 Apr 27 (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 and/or submit a proposal earlier than Mar 30. • 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 though contribution of each member must be clarified in the final report. UNC Chapel Hill

Communication • Visit instructor during office hours, by appointment, or email correspondence • All lecture notes and most of handouts are posted at the course website: http: //www. cs. unc. edu/~zsguo/comp 550. html • Major messages are notified by email alias + on course website • Discussions -- face-to-face in groups, or on Piazza • Student grades can be checked with the instructor UNC Chapel Hill

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!!! UNC Chapel Hill

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 UNC Chapel Hill

Weekly Reading Assignment Chapters 0, 1, 2, 3 and Appendix A (Textbook: CLRS) UNC Chapel Hill

Algorithms • A tool for solving a well-specified computational problem Input Algorithm Output • Example: sorting input: A sequence of number output: An ordered permutation of input issues: correctness, efficiency, storage, etc. UNC Chapel Hill

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 UNC Chapel Hill

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. UNC Chapel Hill

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. UNC Chapel Hill

Example: Insertion Sort • 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. UNC Chapel Hill 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 • 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 n 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. √ UNC Chapel Hill

Example: Insertion Sort for j=2 to length(A) • Invariant: at the start of do key=A[j] each for loop, A[1…j-1] i=j-1 consists of elements while i>0 and A[i]>key do A[i+1]=A[i] originally in A[1…j-1] but in i-sorted order; all other A[i+1]: =key elements are unchanged n Maintenance: the inner while loop finds the position i with 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 UNC Chapel Hill

Example: Insertion Sort • 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 n 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. ” √ UNC Chapel Hill

Running time • Depends on input (e. g. , sorted/reversely) • Depends on input size (5 elements vs 500 K) 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 – Parameterize in input size (n) • Want upper bounds (generally) – Guarantee to the user UNC Chapel Hill

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 UNC Chapel Hill

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. Linear. Search(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. Linear. Search(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 cost times 1 1 1 1 x x-1 1 Assign a cost of 1 to all statement executions. Now, the running time ranges between 1+ 1+ 1 = 4 – best case and 1+ (n+1)+ n + 1 = 2 n+4 – 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. Linear. Search(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 cost times 1 1 1 1 x x-1 1 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 + 1 = n+3

Correctness Proof of Linear Search • Use Loop Invariant for the while loop: Linear. Search(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 UNC Chapel Hill w. If the algm. terminates, then it produces correct result. w. Initialization. w. Maintenance. w. Termination. w. 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]. Linear. Search(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 UNC Chapel Hill w. If the algm. terminates, then it produces correct result. w. Initialization. w. Maintenance. w. Termination. w. Argue that it terminates.

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. UNC Chapel Hill

Comparisons of Algorithms • Sorting – insertion sort: (n 2) – merge sort: (n 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… UNC Chapel Hill

Effect of faster machines Ops/sec: 1 M 2 M Gain n*n alg 1000 1414 1. 4 n log n alg 62700 118600 1. 9 The number of items that can be sorted in one second using an algorithm taking exactly n 2 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!) UNC Chapel Hill