UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis

Transcription

1 UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis Instructor: Prof. Andrew B. Kahng OHs: W 12:30-1:30pm, F 10:00-11:00am, and by appt Office: EBU3B 2134 Class URL:

2 Goals of This Course Reasoning about combinatorial and discrete structures Lists, sets, and multisets; Subsets and k-lists Rule of sum and rule of product Binomial coefficients Discrete probability; expectation; Bayes rule, conditioning, independence analysis of randomized algorithms Functions and their properties Decision trees complexity: lower bounds, average-case Graphs and digraphs Eulerian circuits; Hamiltonian cycles easy vs. hard Spanning trees, shortest paths greed, dynamic programming Rates of growth big-o asymptotics Problem-solving Zeitz book on class webpage; Polya How To Solve It Google interviews Prepare you for success in CSE 100, 101 and beyond

3 Discrete Math and Problem Solving Discrete Math = foundation for all CS Problem Solving = The Spirit of Computing Driven by real-world necessity Computational science (biology, chemistry, ) AI (intelligent vehicles, computational finance, ) Optimization (logistics, scheduling, ) Data science (web social big data and cloud/mobile) Problems are solved on computers with algorithms and data structures Tools: counting, recurrences, graph models All in CSE 21!

4 Class Structure Textbook: Bender and Williamson (online) Links to MIT, Harvard courses, boot camp Many other resources can be found online Platform iclickers for quiz, participation Please register on ted.ucsd.edu WeBWorK for HW HW #0 pipe-cleaner is due Thursday before class Piazza for discussion, announcements Class website: Grading 5% participation, 10% quiz, 15% HW, 30% MT, 40% Final

5 Class Structure (cont.) Dr. Rob Rubalcaba teaches Section A00 (MWF) Shared between the two sections TAs OHs Discussion sessions (M and W, 4pm and 5pm; Center 115) HWs NOT shared between the two sections Quizzes and Participation Midterm Final Questions?

6 Tips for Effective Group Discussion Take turns being the first one to talk Even if you all agree on the answer, don t stop there! Go over each wrong answer and explain why it is wrong Ask yourselves why somebody might be tempted to choose it. What s the trick? Even if your group-mate has said something very clearly and correctly, think about other ways to phrase it in your own words So, what I think you said was,... Your brain will remember it better if you say it too!

7 Today: CL Sections 1, 2 Question: Teams A and B play in a basketball tournament. The first team to win two games in a row or a total of three games wins the tournament. What is the number of ways in which the tournament can occur?

8 Today: CL Sections 1, 2 Question: Teams A and B play in a basketball tournament. The first team to win two games in a row or a total of three games wins the tournament. What is the number of ways in which the tournament can occur? Ideas.. The answer is even ( symmetry ) We can t have more than five games in total The possibilities can be organized as a tree

9 Sets and Lists Set : a collection of distinct objects where order does not matter Notation: {a, b, c, d} List (or string) : an ordered collection Notation: (a, b, c, d, a) or abcda (Whether repetition is allowed will be specified when referring to a list) Size or cardinality : A k-set is a set of size k S = {a, b, c, d} is a 4-set, and S = 4 A k-list is a list of size k L = (a, b, c, d, a) is a 5-list, and L = 5 Multiset : a collection of objects where order does not matter, but repetition does matter Example: {a, b, b, a, a}

10 Examples Consider the 4-set S = {w,x,y,z} I ve starting a list of all the 2-element subsets of S : {w,x}, {w,y}, {x,y}, {x,z}, Clicker question C1-1: How many of the following should be added to my list of 2-element subsets? {x,x} {x,w} {, x} {{x,y}} {z,w} (A): 0 (B): 1 (C): 2 (D): 3 (E) 4 or more

11 Examples How many 2-element subsets of the 4-set S = {w,x,y,z}? How many 2-lists of elements from S? How many 4-lists of elements from S? How many 2-lists of elements from S without repetition? How many 4-lists of elements from S without repetition? How many 4-lists of elements from S = {t,u,v,w,x,y,z}?

12 Theorems Theorem 1: There are n k ways to form a k-list from the elements of an n-set. Theorem 4: There are n (n 1) (n 2) (n k + 1) ways to form a k-list from distinct elements of an n-set. Remember: distinct means without repetition That is, n (n 1) (n 2) (n k + 1) = n! / (n k)! Ways There are n! ways to order the n elements of an n-set Use k = n in Theorem 4 above

13 Cartesian Product and Lexicographic Order Given sets C 1, C 2,, C k Their Cartesian product is C 1 C 2 C k and has elements (x 1,x 2,,x k ) where x i C i, i = 1,, k Cardinalities: C 1 C 2 C k = C 1 C 2 C k Example: S = {a,b,c}; T = {1,2} S T = {(a,1), (a,2), (b,1), (b,2), (c,1), (c,2)} S T = 6 = 3 2

14 Cartesian Product and Lexicographic Order Given sets C 1, C 2,, C k Their Cartesian product is C 1 C 2 C k and has elements (x 1,x 2,,x k ) where x i C i, i = 1,, k Cardinalities: C 1 C 2 C k = C 1 C 2 C k Lexicographic order = dictionary order, given existence of an order on the elements of each set C i in the Cartesian product S = {a,b,c} with a < b < c. Then, S S in lexicographic order:

15 HW #1 (due Sunday April 6 11:59 PM) HW 1.1 (1 pts) Standard automobile license plates in Pennsylvania have 1 nonzero digit, followed by 3 letters, followed by 4 digits. How many different standard plates are possible in this system? HW 1.2 (3 pts) A vendor sells ice cream from a cart on a sidewalk in Mission Bay. He offers 4 different flavors and 5 different kinds of cones. (a) How many different single-scoop ice cream cones can you buy from this vendor? (b) How many different two-scoop ice cream cones, where, e.g., chocolate on top of strawberry is different than strawberry on top of chocolate? (c) What would be the answer to (b) if, e.g., chocolate-strawberry is considered the same as strawberry-chocolate? HW 1.3 (4 pts) There are 9 chairs in a row. 5 boys and 4 girls are to be seated in the chairs. In how many ways can this be done if (a) there are no restrictions; (b) if boys and girls must alternate; (c) if all the boys must sit next to each other (i.e., in consecutive chairs); and (d) if all the boys must sit next to each other, and all the girls must sit next to each other? HW 1.4 (4 pts) An n-digit number is a list of n 1 digits where the first digit is nonzero. (a) How many n-digit numbers are there (as a function of n)? (b) How many n-digit numbers contain no 1 s? (c) How many n-digit numbers contain at least one 2? (d) For n > 2, how many n-digit numbers contain at least one 2 and at least one 3, but no 7 s? HW 1.5 (2 pts) How many distinct anagrams can be created from the word MILLENNIUM if the new words do not need to be meaningful? (An anagram is a rearrangement of letters.) HW 1.6 (2 pts) (a) How many ways are there to form a list of three letters from the letters in the word MILLENNIUM, if no letter can be used more often than it appears in MILLENNIUM? (b) What about lists with four letters? HW 1.7 (3 pts) Consider words which only use the letters A, B, C, D, E. (a) How many four-letter words can be made using these letters? (b) How many four-letter words have exactly two vowels and two consonants? (c) Of the four-letter words with exactly two vowels and two consonants, how many have distinct vowels? HW 1.8 (3 pts) A monotone increasing number consists of digits taken from the set {1, 2,, 9}, with each digit greater than or equal to its neighbor digit to the left (if that digit exists). E.g., is a monotone increasing number with 13 digits. How many 7-digit monotone increasing numbers are there? HW1.9 (3 pts) An icosahedron has 20 faces. Each is an equilateral triangle. Five faces meet at each vertex. Determine the number of vertices V and the number of edges E in an icosahedron. Write the sum V + E for your answer.

16 Problems 1: Parity, Pigeonhole, Invariants P1.1 You have 7 glasses, all right-side up. In any single move, you can turn over exactly 4 glasses. Is there a sequence of moves that end up with all 7 glasses upsidedown? Explain. P1.2 Prove that if five points lie in a 2 x 2 square region, some two of them must be no more than sqrt(2) distance apart. P1.3 Given any small positive real > 0, prove that there is some integer n such that n is within of an integer value. P1.4 Given the set of integers S = {1, 2,, 200}. Prove that no matter how you choose 101 numbers from S, there will be two chosen numbers such that one evenly divides the other.

17 Problems 1: Parity, Pigeonhole, Invariants P1.5 Three frogs are initially located in the infinite plane at points (1,1), (1,0) and (0,0). When one frog jumps over another frog, its position is reflected about the other frog s position. E.g., if the frog at (0,0) jumps over the frog at (1,0), it lands at (2,0). Can one of the three frogs ever get to (1, 1)? Explain. P1.6 At a big international conference, many handshakes are exchanged. We call a person an odd person if she has exchanged an odd number of handshakes. Otherwise, she is called an even person. Explain why, at any moment, there is an even number of odd persons.