Data Structures and Algorithms CSCE 221 H Dr

Data Structures and Algorithms CSCE 221 H Dr. Scott Schaefer 1

Staff n n n Instructor u Dr. Scott Schaefer u HRBB 304 A u Office Hours: TR 8 am-9 am (or by appointment) TA u Harish Kumar u EABB u Office Hours: TR noon-1 pm Peer Teachers u Nathan Brockway, Dominick Fabian u Peer Teacher Central (HRBB 129) 2/38

What will you learn? n n n n n 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 3/38

Prerequisites n CSCE 121 “Introduction to Program Design and Concepts” n CSCE 222 “Discrete Structures” or MATH 302 “Discrete Mathematics” (either may be taken concurrently with CSCE 221) 4/38

Textbook 5/38

Grading 3% Labs n 12% Homework n 5% Culture Assignments n 30% Programming Assignments n 10% Quizzes n 20% Midterm n 20% Final n 6/38

Assignments Turn in code/homeworks via Google Classroom (invitation code gpj 6 ff) n Due by 11: 59 pm on day specified n All programming in C++ n Code, proj file, sln file, and Win 32 executable n Make your code readable (comment) n You may discuss concepts, but coding is individual (no “team coding” or web) n 7/38

Late Policy Penalty = m: number of minutes late percentage penalty n days late 8/46

Late Policy Penalty = m: number of minutes late percentage penalty n days late 9/46

Late Policy Penalty = m: number of minutes late percentage penalty n days late 10/46

Late Policy Penalty = m: number of minutes late percentage penalty n days late 11/46

Labs n Several structured labs at the beginning of the semester with (simple) exercises u Graded on completion u Time to work on homework/projects 12/38

Homework Approximately 5 n Written/Typed responses n Simple coding if any n 13/38

Programming Assignments About 5 throughout the semester n Implementation of data structures or algorithms we discusses in class n Written portion of the assignment n 14/38

Quizzes Approximately 10 throughout the semester n Short answer, small number of questions n Will only be given in class or lab u Must be present to take the quiz n 15/38

Culture Assignments Two research seminar reports u One before Spring break, one after n Biography of a famous Computer Scientist u 5 minute presentation u Signup this week by sending me an email n n http: //faculty. cs. tamu. edu/schaefer/teaching/221_Fall 2018/Assignments/culture. html 16/38

Academic Honesty Assignments are to be done on your own u May discuss concepts, get help with a persistent bug u Should not copy work, download code, or work together with others unless specifically stated otherwise n We use a software similarity checker u http: //moss. stanford. edu n 17/38

Class Discussion Board n piazza. com/tamu/fall 2018/csce 221200 18/38

Asymptotic Analysis 19/38

Running Time n n 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. u u Crucial to applications such as games, finance, and robotics Easier to analyze worst case 5 ms Running Time n 4 ms 3 ms best case 2 ms 1 ms A B C D E Input Instance F G 20/38

Experimental Studies n n 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 results 21/38

Stop Watch Example 22/38

Limitations of Experiments It is necessary to implement the algorithm, which may be difficult n Results may not be indicative of the running time on other inputs not included in the experiment. n In order to compare two algorithms, the same hardware and software environments must be used n 23/38

Theoretical Analysis Uses a high-level description of the algorithm instead of an implementation n Characterizes running time as a function of the input size, n. n Takes into account all possible inputs n Allows us to evaluate the speed of an algorithm independent of the hardware/software environment n 24/38

Important Functions n Seven functions that often appear in algorithm analysis: u u u u Constant 1 Logarithmic log n Linear n N-Log-N n log n Quadratic n 2 Cubic n 3 Exponential 2 n 25/38

Important Functions n Seven functions that often appear in algorithm analysis: u u u u Constant 1 Logarithmic log n Linear n N-Log-N n log n Quadratic n 2 Cubic n 3 Exponential 2 n 26/38

Why Growth Rate Matters if runtime is. . . time for n + 1 time for 2 n time for 4 n c lg (n + 1) c (lg n + 1) c(lg n + 2) cn c (n + 1) 2 c n 4 c n lg n ~ c n lg n + cn 2 c n lg n + 2 cn 4 c n lg n + 4 cn c n 2 ~ c n 2 + 2 c n 4 c n 2 16 c n 2 c n 3 ~ c n 3 + 3 c n 2 8 c n 3 64 c n 3 c 2 n c 2 n+1 c 2 2 n c 2 4 n runtime quadruples when problem size doubles 27/38

Comparison of Two Algorithms insertion sort is n 2 / 4 merge sort is 2 n lg n sort a million items? insertion sort takes roughly 70 hours while merge sort takes roughly 40 seconds This is a slow machine, but if 100 x as fast then it’s 40 minutes versus less than 0. 5 seconds 28/38

Constant Factors n The growth rate is not affected by u u n constant factors or lower-order terms Examples u u 102 n + 105 is a linear function 105 n 2 + 108 n is a quadratic function 29/38

Big-Oh Notation n n Given functions f(n) and g(n), we say that f(n) is O(g(n)) if there are positive constants c and n 0 such that f(n) cg(n) for n n 0 Example: 2 n + 10 is O(n) u u 2 n + 10 cn (c 2) n 10/(c 2) Pick c = 3 and n 0 = 10 30/38

Big-Oh Example n Example: the function n 2 is not O(n) u u u n 2 cn n c The above inequality cannot be satisfied since c must be a constant 31/38

More Big-Oh Examples 7 n-2 is O(n) need c > 0 and n 0 1 such that 7 n-2 c • n for n n 0 this is true for c = 7 and n 0 = 1 n 3 n 3 + 20 n 2 + 5 is O(n 3) need c > 0 and n 0 1 such that 3 n 3 + 20 n 2 + 5 c • n 3 for n n 0 this is true for c = 4 and n 0 = 21 n 3 log n + 5 is O(log n) need c > 0 and n 0 1 such that 3 log n + 5 c • log n for n n 0 this is true for c = 8 and n 0 = 2 32/38

Big-Oh and Growth Rate n n n 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 g(n) grows more f(n) grows more Same growth f(n) is O(g(n)) g(n) is O(f(n)) Yes No Yes 33/38

Big-Oh Rules n If is f(n) a polynomial of degree d, then f(n) is O(nd), i. e. , 1. 2. n Use the smallest possible class of functions u n Drop lower-order terms Drop constant factors Say “ 2 n is O(n)” instead of “ 2 n is O(n 2)” Use the simplest expression of the class u Say “ 3 n + 5 is O(n)” instead of “ 3 n + 5 is O(3 n)” 34/38

Computing Prefix Averages n n 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) 35/38

Prefix Averages (Quadratic) The following algorithm computes prefix averages in quadratic time by applying the definition Algorithm prefix. Averages 1(X, n) Input array X of n integers Output array A of prefix averages of X #operations A new array of n integers n for i 0 to n 1 do n s X[0] n for j 1 to i do 1 + 2 + …+ (n 1) s s + X[j] 1 + 2 + …+ (n 1) A[i] s / (i + 1) n return A 1 36/38

Arithmetic Progression n The running time of prefix. Averages 1 is O(1 + 2 + …+ n) The sum of the first n integers is n(n + 1) / 2 Thus, algorithm prefix. Averages 1 runs in O(n 2) time 37/38

Prefix Averages (Linear) The following algorithm computes prefix averages in linear time by keeping a running sum Algorithm prefix. Averages 2(X, n) Input array X of n integers Output array A of prefix averages of X A new array of n integers s 0 for i 0 to n 1 do s s + X[i] A[i] s / (i + 1) return A #operations n 1 n n n 1 Algorithm prefix. Averages 2 runs in O(n) time 38/38