COMP3251 Lecture 1: Introduction Welcome to COMP3251 Instructor: - HUANG, Zhiyi (zhiyi@cs.hku.hk) - 423 Chow Yei Ching Building - Consultation Hours: Tuesday, 4:00pm to 5:00pm - Other time slots may be available by appointment Tutors: - SIU, Chak Fung Je (je siu1@hku.hk) - HU, Qi (hooch@connect.hku.hk) - LOU, Yuke (louyuke@connect.hku.hk) - Consultation Hours: TBD 2 ff ff COMP3251 Zhiyi Huang What are algorithms? An algorithm is a set of precise step-by-step rules for solving a speci c computational problem. Examples: Multiplying two numbers, say, 1234 and 5678. In primary school, we learnt this as “the correct way” of calculating the multiplication of two numbers. In this course: - How good is this algorithm? - Can we do better? ⨉ 5 6 7 8 1 2 3 4 2 1 7 1 1 3 5 6 7 2 7 1 2 0 3 4 5 6 8 7 0 0 6 6 5 2 4 fi COMP3251 Zhiyi Huang An alternative multiplication algorithm Examples: Multiplying two numbers, say, 1234 and 5678. 1) Partition (1234) into two numbers (12) and (34). 2) Partition (5678) into two numbers (56) and (78). 3) Multiply (12) and (56) and get (672). 4) Multiply (34) and (78) and get (2652). 5) Multiply (12)+(34)=(46) and (56)+(78)=(134) and get (6164). 6) Subtract the rst two numbers from the third one and get (6164)-(672)-(2652)=(2840). 7) Sum up (6720000) (1st number padded with four 0’s), (2652) (2nd number), and (284000) (4th number padded with two 0’s), and get (6720000) + (2652) + (284000) = (7006652). 5 fi COMP3251 Zhiyi Huang An alternative multiplication algorithm Examples: Multiplying two numbers, say, 1234 and 5678. 1) Partition (1234) into two numbers (12) and (34). 2) Partition (5678) into two numbers (56) and (78). 3) (12)that andyou (56)cannot and get (672). It isMultiply expected make any sense of the algorithm thisand point, than(2652). the nal answer is correct. 4) Multiplyat(34) (78)other and get •5) After a few classes, youand will(56)+(78)=(134) be able to make sense of the Multiply (12)+(34)=(46) and get (6164). alternative multiplication algorithm. 6) Subtract the • By the end of the semester, you will be able to design algorithms like this one (and other fancy techniques too). 7) Sum up (6720000) (1st number padded with four 0’s), (2652) (2nd number), and (284000) (4th number padded with two 0’s), and get (6720000) 6 fi fi COMP3251 Zhiyi Huang Why study algorithms? Useful for all areas in CS networking database machine learning computer vision COMP3251 8 Zhiyi Huang Driving force behind tech innovations Page Rank COMP3251 9 Zhiyi Huang Driving force behind tech innovations Google Map Driving Direction COMP3251 10 Zhiyi Huang Driving force behind tech innovations Online Advertisement COMP3251 11 Zhiyi Huang Online Advertisement is Big! COMP3251 12 Zhiyi Huang Novel “lens” for areas outside CS Auctions COMP3251 Markets 13 Zhiyi Huang What you will get from this course? • Become a better programmer • Sharpen your analytical and mathematical skills • Ace the algorithm questions in your job interviews • Think “algorithmically” COMP3251 14 Zhiyi Huang Textbook Lecture notes: - Self-contained - Available on Moodle before each lecture Textbook (required supplementary readings): - Algorithms S. Dasgupta, C. H. Papadimitriou, and U. Vazirani Other resources (optional): - Algorithm Design E. Tardos and J. Kleinberg example - Introduction to Algorithms T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein - Online course by Tim Roughgarden COMP3251 15 Zhiyi Huang Course Topics Algorithm design paradigms: - Divide and conquer (DPV Ch. 2) - Greedy (DPV Ch. 5) - Dynamic programming (DPV Ch. 6) Graph algorithms: - Graph decomposition (DPV Ch. 3) - Shortest path (DPV Ch. 4) - Max ow (DPV Ch. 7.2, supplementary materials will be available on Moodle) Complexity theory: - NP-completeness and reduction (DPV Ch. 8) fl COMP3251 16 Zhiyi Huang Assessment • Homework assignments (40%) - 5 assignments - Count best 4 out of 5 - No late submission: If you happen to be extremely busy before one of the deadlines, that’s your worst 1 out of 5 - About 4 problems in each assignment • Midterm (10%) - 90 min, semi-open book (2 sheets of A4), in class - Most likely in the week after the reading week • Final (50%) - 3 hours, 4-5 questions, semi-open book (2 sheets of A4) COMP3251 17 Zhiyi Huang Collaboration Policy for Homework Assignments YES: Discuss with friends, acknowledge them, write your own solutions. NO: Copy solutions word-for-word or with minor modi cations (from your friends, from the Internet, from past-year, etc.) Note: I strongly recommend that you try to complete the homework assignments on your own. 18 Zhiyi Huang fi COMP3251 3251 (Regular class) vs. 3252 (Advanced Class) • COMP3251 focuses on - How basic algorithms work - Applications of algorithms • COMP3252 emphasizes more on - Problem solving skills - Mathematical proofs COMP3251 19 Zhiyi Huang Rest of this lecture: Review of basic running time analysis Basics of running time analysis Goal: Given an algorithm A, upper bound its running time. Example: We may want to prove that for any input of size n, algorithm A runs for at most 5 n2 + 7 n + 8 steps. But: • It is often very di cult to get a precise bound. • The precise bounds are often meaningless. - We generally count the number of steps in a pseudo-code speci cation of an algorithm. - The number of steps will be di erent if we implement the algorithm in di erent high-level programming languages. 21 ff ffi ff fi COMP3251 Zhiyi Huang Big-O notation Goal: Given an algorithm A, upper bound its running time. The main mathematical tool: Asymptotic (rough) analysis for comparing non-negative functions, say, f(n) and g(n). Intuitively Notation Read as De nition f(n) “is smaller than” g(n) f(n) = O(g(n)) f(n) is order g(n) limn→∞ f(n)/g(n) ≤ c for some constant c f(n) ≤ c g(n) when n is large enough It is a rough analysis because: • we do not mind what c is (as long as it is a constant); • we do not insist the inequality holds for all n, but it should hold for all large enough n. fi COMP3251 22 Zhiyi Huang Asymptotic bounds for some common functions Polynomial: For any polynomial p(n) = ad nd + ad-1 nd-1 + … + a1 n + a0 where ad > 0, and ai ≥ 0 for d > i ≥ 0, we have p(n) = O(nd). Why? - ni ≤ nd for any integer 0 ≤ i < d - So p(n) ≤ (ad + ad-1 + … + a1 + a0) nd - In fact, we can show that p(n) ≤ (ad + 1) nd for su ciently large n, but the above inequality is good enough. 23 Zhiyi Huang ffi COMP3251 Asymptotic bounds for some common functions Logarithmic vs. polynomial: For any b > 1 and any d > 0, we have logb n = O(nd). n0.2 ln n COMP3251 24 Zhiyi Huang Asymptotic bounds for some common functions Logarithmic vs. polynomial: For any b > 1 and any d > 0, we have logb n = O(nd). Polynomial vs. exponential: For any r > 1 and any d > 0, we have nd = O(r n). COMP3251 25 Zhiyi Huang
