Algorithms and Data Structures, Fall Introduction. Rasmus Pagh. Based on slides by Kevin Wayne, Princeton. Monday, August 29, 11

Transcription

1 Algorithms and Data Structures, Fall 2011 Introduction Rasmus Pagh Based on slides by Kevin Wayne, Princeton

2 Why study algorithms? Their impact is broad and far-reaching. Internet. Web search, packet routing, distributed file sharing,... Biology. Human genome project, protein folding,... Computers. Circuit layout, file system, compilers,... Computer graphics. Movies, video games, virtual reality,... Security. Cell phones, e-commerce, voting machines,... Multimedia. MP3, JPG, DivX, HDTV, face recognition,... Social networks. Recommendations, news feeds, advertisements,... Physics. N-body simulation, particle collision simulation,... 2

3 Why study algorithms? To solve problems that could not otherwise be addressed. Ex. Network connectivity. [stay tuned] 3

4 Why study algorithms? To become a proficient programmer. I will, in fact, claim that the difference between a bad programmer and a good one is whether he considers his code or his data structures more important. Bad programmers worry about the code. Good programmers worry about data structures and their relationships. Linus Torvalds (creator of Linux) Algorithms + Data Structures = Programs. Niklaus Wirth 4

5 Why study algorithms? They may unlock the secrets of life and of the universe. Computational models are replacing mathematical models in scientific inquiry. E = mc 2 F = ma 2 F = Gm 1m 2 r 2 2m 2 + V(r) Ψ(r) = E Ψ(r) for (double t = 0.0; true; t = t + dt) for (int i = 0; i < N; i++) { bodies[i].resetforce(); for (int j = 0; j < N; j++) if (i!= j) bodies[i].addforce(bodies[j]); } 20 th century science (formula based) 21 st century science (algorithm based) Algorithms: a common language for nature, human, and computer. A. Wigderson 5

6 Who am I? Rasmus Pagh, associate professor. Have been interested in algorithms half my life, since high school. PhD from Aarhus University, Topic: Hashing. Worked at ITU since then. Office 4D26. Research in algorithms and data structures applications in databases and data mining. Previous teaching in databases and algorithms. 6

7 Who are you? As an experiment, I invite you to present yourselves at the Student presentations forum in LearnIT. E.g., you may write about: What study line you are on. Your bachelor degree. Your expectations for the course. This will only be accessible to students in the course. Background surveys: Please fill in two questions on your self-assessment of background in programming and mathematics. Available in LearnIT. 7

8 Resources Textbook: Algorithms 4 th edition by R. Sedgewick and K. Wayne, Addison- Wesley Professional, 2011, ISBN X. Great website: algs4.cs.princeton.edu Algorithms F O U R T H E D I T I O N Available on paper or on Kindle. R O B E R T S E D G E W I C K K E V I N W AYNE Online: Amazon,... Brick-and-mortar: Academic Books KUA (and, I think, ITU this week) Web resources: Schedule, slides, exercises (itu.dk/people/pagh/ads11/) LearnIT (learnit.itu.dk) 8

9 Survival info Teaching assistant Konstantin Kutzkov, 4D21. Exercises each week Will revisit 1-2 selected problem from the previous week. Work on exercises related to the lecture of the day. Work at home on exercises is highly recommended (best way to find out if you get it ). Mandatory hand-ins. About 5 will be posed during the course - at least 4 must be approved. Hand-in must be submitted via LearnIT, learnit.itu.dk. It is allowed to work and submit in pairs. Exam Written, probably 4 hours, on January 11. 9

10 This course Provides a basic toolbox of algorithms every serious programmer needs. Need to understand to use well! Assess scalability. Learn a structured approach to design and analysis of complex software. New way of thinking about programs. Difficult! Supports exciting specializations: Scalable computing (Efficient AI Programming, Algorithm Design) Modern AI (Efficient AI Programming, Advanced AI in Games) Databases (Database Tuning, Building Database Systems) 10

11 Course goals Clearly explain the mechanics of computations and data structures involving manipulation of references, nested loops and recursion, specified in natural language, in abstract pseudocode or in concrete programming language (Java). Implement abstractly specified computations and data structures in an imperative programming language (Java). Analyze time and space usage of algorithms/programs. Argue for correctness of programs using invariants. Assess scalability of a given single-threaded software application, using asymptotic analysis. Choose among and make use of the most important algorithms and data structures in libraries, based on knowledge of their complexity. Design algorithms for ad hoc problems by using and combining known algorithms and data structures. Account for and describe the most important hardware and programming language factors influencing the speed at which a program runs. 11

12 Running time As soon as an Analytic Engine exists, it will necessarily guide the future course of the science. Whenever any result is sought by its aid, the question will arise By what course of calculation can these results be arrived at by the machine in the shortest time? Charles Babbage (1864) how many times do you have to turn the crank? Analytic Engine 12

13 Reasons to analyze algorithms Predict performance. Compare algorithms. this course Provide guarantees. Develop new algorithms. Algorithm Design (fall semesters) Primary practical reason: avoid performance bugs. client gets poor performance because programmer did not understand performance characteristics 13

14 The challenge Q. Will my program be able to solve a large practical input? Why is my program so slow? Why does it run out of memory? Key insight. [Knuth 1970s] Use scientific method to understand performance. 14

15 Example: 3-sum 3-sum. Given N distinct integers, how many triples sum to exactly zero? % more 8ints.txt % java ThreeSum < 8ints.txt 4 a[i] a[j] a[k] sum Context. Deeply related to problems in computational geometry. 15

16 3-sum: brute-force algorithm public class ThreeSum { public static int count(int[] a) { int N = a.length; int count = 0; for (int i = 0; i < N; i++) for (int j = i+1; j < N; j++) for (int k = j+1; k < N; k++) if (a[i] + a[j] + a[k] == 0) count++; return count; } check each triple we ignore any integer overflow } public static void main(string[] args) { int[] a = In.readInts(args[0]); StdOut.println(count(a)); } 16

17 Measuring the running time Q. How to time a program? A. Manual. % java ThreeSum 1Kints.txt tick tick tick 70 % java ThreeSum 2Kints.txt 528 % java ThreeSum 4Kints.txt 4039 tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick tick 17

18 Measuring the running time Q. How to time a program? A. Automatic. public class Stopwatch Stopwatch() create a new stopwatch double elapsedtime() time since creation (in seconds) public static void main(string[] args) { int[] a = In.readInts(args[0]); Stopwatch stopwatch = new Stopwatch(); StdOut.println(ThreeSum.count(a)); double time = stopwatch.elapsedtime(); } client code 18

19 Measuring the running time Q. How to time a program? A. Automatic. public class Stopwatch Stopwatch() create a new stopwatch double elapsedtime() time since creation (in seconds) public class Stopwatch { private final long start = System.currentTimeMillis(); } public double elapsedtime() { long now = System.currentTimeMillis(); return (now - start) / ; } implementation (part of stdlib.jar) 19

20 Empirical analysis Run the program for various input sizes and measure running time. N time (seconds) , , , , ,000? 20

21 Data analysis Standard plot. Plot running time T (N) vs. input size N. standard plot 50 running time T(N) K 2K 4K 8K problem size N 21

22 Data analysis Log-log plot. Plot running time vs. input size N using log-log scale. log-log plot straight line of slope 3 lg(t(n)) lg(t (N)) = b lg N + c b = c = T (N) = a N b, where a = 2 c 1K 2K 4K 8K lgn Regression. Fit straight line through data points: a N b. power law slope Hypothesis. The running time is about N seconds. 22

23 Prediction and validation Hypothesis. The running time is about N seconds. Predictions. "order of growth" of running time is about N 3 [stay tuned] 51.0 seconds for N = 8, seconds for N = 16,000. Observations. N time (seconds) 8, , , , validates hypothesis! 23

24 Doubling hypothesis Doubling hypothesis. Quick way to estimate b in a power-law relationship. Run program, doubling the size of the input. N time (seconds) ratio lg ratio , , , , seems to converge to a constant b 3 Hypothesis. Running time is about a N b with b = lg ratio. Caveat. Cannot identify logarithmic factors with doubling hypothesis. 24

25 Doubling hypothesis Doubling hypothesis. Quick way to estimate b in a power-law hypothesis. Q. How to estimate a? A. Run the program! N time (seconds) 8, , , = a a = Hypothesis. Running time is about N 3 seconds. almost identical hypothesis to one obtained via linear regression 25

26 Experimental algorithmics System independent effects. Algorithm. Input data. determines exponent b in power law System dependent effects. Hardware: CPU, memory, cache, Software: compiler, interpreter, garbage collector, System: operating system, network, other applications, helps determines constant a in power law Bad news. Difficult to get precise measurements. Good news. Much easier and cheaper than other sciences. e.g., can run huge number of experiments 26

27 Mathematical models for running time Total running time: sum of cost frequency for all operations. Need to analyze program to determine set of operations. Cost depends on machine, compiler. Frequency depends on algorithm, input data. Donald Knuth 1974 Turing Award In principle, accurate mathematical models are available. 27

28 Cost of basic operations operation example nanoseconds integer add a + b 2.1 integer multiply a * b 2.4 integer divide a / b 5.4 floating-point add a + b 4.6 floating-point multiply a * b 4.2 floating-point divide a / b 13.5 sine Math.sin(theta) 91.3 arctangent Math.atan2(y, x) Running OS X on Macbook Pro 2.2GHz with 2GB RAM 28

29 Cost of basic operations operation example nanoseconds variable declaration int a c1 assignment statement a = b c2 integer compare a < b c3 array element access a[i] c4 array length a.length c5 1D array allocation new int[n] c6 N 2D array allocation new int[n][n] c7 N 2 string length s.length() c8 substring extraction s.substring(n/2, N) c9 string concatenation s + t c10 N Novice mistake. Abusive string concatenation. 29

30 Example: 1-sum Q. How many instructions as a function of input size N? int count = 0; for (int i = 0; i < N; i++) if (a[i] == 0) count++; operation frequency variable declaration 2 assignment statement 2 less than compare N + 1 equal to compare array access increment N N N to 2 N 30

31 Example: 2-sum Q. How many instructions as a function of input size N? int count = 0; for (int i = 0; i < N; i++) for (int j = i+1; j < N; j++) if (a[i] + a[j] == 0) count++; operation frequency (N 1) = 1 N (N 1) 2 N = 2 variable declaration N + 2 assignment statement N + 2 less than compare ½ (N + 1) (N + 2) equal to compare ½ N (N 1) array access N (N 1) tedious to count exactly increment ½ N (N 1) to N (N 1) 31

32 Simplification 1: cost model Cost model. Use some basic operation as a proxy for running time. int count = 0; for (int i = 0; i < N; i++) for (int j = i+1; j < N; j++) if (a[i] + a[j] == 0) count++; operation frequency (N 1) = 1 N (N 1) 2 N = 2 variable declaration N + 2 assignment statement N + 2 less than compare ½ (N + 1) (N + 2) equal to compare ½ N (N 1) array access N (N 1) cost model = array accesses increment ½ N (N 1) to N (N 1) 32

33 Simplification 2: tilde notation Estimate running time (or memory) as a function of input size N. Ignore lower order terms. - when N is large, terms are negligible - when N is small, we don't care N 3 /6 Ex 1. ⅙ N N + 16 ~ ⅙ N 3 Ex 2. ⅙ N N 4/ ~ ⅙ N 3 Ex 3. ⅙ N 3 - ½ N 2 + ⅓ N ~ ⅙ N 3 166,666,667 N 3 /6 N 2 /2 + N /3 166,167,000 discard lower-order terms (e.g., N = 1000: 500 thousand vs. 166 million) N 1,000 Leading-term approximation Technical definition. f(n) ~ g(n) means lim N f (N) g(n) = 1 33

34 Simplification 2: tilde notation Estimate running time (or memory) as a function of input size N. Ignore lower order terms. - when N is large, terms are negligible - when N is small, we don't care operation frequency tilde notation variable declaration N + 2 ~ N assignment statement N + 2 ~ N less than compare ½ (N + 1) (N + 2) ~ ½ N 2 equal to compare ½ N (N 1) ~ ½ N 2 array access N (N 1) ~ N 2 increment ½ N (N 1) to N (N 1) ~ ½ N 2 to ~ N 2 34

35 Example: 2-sum Q. Approximately how many array accesses as a function of input size N? int count = 0; for (int i = 0; i < N; i++) for (int j = i+1; j < N; j++) if (a[i] + a[j] == 0) count++; "inner loop" A. ~ N 2 array accesses (N 1) = 1 N (N 1) 2 N = 2 Bottom line. Use cost model and tilde notation to simplify frequency counts. 35

36 Example: 3-sum Q. Approximately how many array accesses as a function of input size N? int count = 0; for (int i = 0; i < N; i++) for (int j = i+1; j < N; j++) for (int k = j+1; k < N; k++) if (a[i] + a[j] + a[k] == 0) count++; "inner loop" A. ~ ½ N 3 array accesses. N 3 = N(N 1)(N 2) 3! 1 6 N 3 Bottom line. Use cost model and tilde notation to simplify frequency counts. 36

37 Estimating a discrete sum Q. How to estimate a discrete sum? A1. Take the Discrete Mathematics course. A2. Replace the sum with an integral, and use calculus! Ex N. N i=1 i N x=1 xdx 1 2 N 2 Ex /2 + 1/ /N. N i=1 1 i N x=1 1 dx = ln N x Ex 3. 3-sum triple loop. N i=1 N j=i N k=j 1 N x=1 N y=x N z=y dz dy dx 1 6 N 3 37

38 Mathematical models for running time In principle, accurate mathematical models are available. In practice, Formulas can be complicated. Advanced mathematics might be required. Exact models best left for experts. costs (depend on machine, compiler) TN = c1 A + c2 B + c3 C + c4 D + c5 E A = array access B = integer add C = integer compare D = increment E = variable assignment frequencies (depend on algorithm, input) Bottom line. We use approximate models in this course: T(N) ~ c N 3. 38

39 Common order-of-growth classifications Good news. the small set of functions 1, log N, N, N log N, N 2, N 3, and 2 N suffices to describe order-of-growth of typical algorithms. log-log plot 512T exponential cubic quadratic linearithmic linear 64T time 8T 4T 2T T logarithmic constant 1K 2K 4K 8K 512K size Typical orders of growth 39

40 Common order-of-growth classifications growth rate name typical code framework description example T(2N) / T(N) 1 constant a = b + c; statement add two numbers 1 log N logarithmic while (N > 1) { N = N / 2;... } divide in half binary search ~ 1 N linear for (int i = 0; i < N; i++) {... } loop find the maximum 2 N log N linearithmic [see sorting lecture] divide and conquer mergesort ~ 2 N 2 quadratic for (int i = 0; i < N; i++) for (int j = 0; j < N; j++) {... } double loop check all pairs 4 N 3 cubic for (int i = 0; i < N; i++) for (int j = 0; j < N; j++) for (int k = 0; k < N; k++) {... } triple loop check all triples 8 2 N exponential [not in this course] exhaustive search check all subsets T(N) 40

41 Practical implications of order-of-growth effect on a program that growth rate name description runs for a few seconds time for 100x size for 100x more data faster computer 1 constant independent of input size log N logarithmic nearly independent of input size N linear optimal for N inputs a few minutes 100x N log N linearithmic nearly optimal for N inputs a few minutes 100x N 2 quadratic not practical for large problems several hours 10x N 3 cubic not practical for medium problems several weeks 4 5x 2 N exponential useful only for tiny problems forever 1x 41

42 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Binary search. Compare key against middle entry. Too small, go left. Too big, go right. Equal, found lo hi 42

43 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Binary search. Compare key against middle entry. Too small, go left. Too big, go right. Equal, found lo mid hi 42

44 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Successful search. Binary search for lo hi 43

45 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Successful search. Binary search for lo mid hi 43

46 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Successful search. Binary search for lo hi 44

47 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Successful search. Binary search for lo mid hi 44

48 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Successful search. Binary search for lo hi 45

49 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Successful search. Binary search for lo mid hi 45

50 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Successful search. Binary search for 33. lo = hi

51 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Successful search. Binary search for 33. lo = hi mid return 4 46

52 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Unsuccessful search. Binary search for lo hi 47

53 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Unsuccessful search. Binary search for lo mid hi 47

54 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Unsuccessful search. Binary search for lo hi 48

55 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Unsuccessful search. Binary search for lo mid hi 48

56 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Unsuccessful search. Binary search for lo hi 49

57 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Unsuccessful search. Binary search for lo mid hi 49

58 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Unsuccessful search. Binary search for 34. lo = hi

59 Binary search Goal. Given a sorted array and a key, find index of the key in the array? Unsuccessful search. Binary search for 34. lo = hi mid return -1 50

60 Binary search: Java implementation Trivial to implement? First binary search published in 1946; first bug-free one published in Java bug in Arrays.binarySearch() not fixed until public static int binarysearch(int[] a, int key) { int lo = 0, hi = a.length-1; while (lo <= hi) { int mid = lo + (hi - lo) / 2; if (key < a[mid]) hi = mid - 1; else if (key > a[mid]) lo = mid + 1; else return mid; } return -1; } one 3-way compare Invariant. If key appears in the array a[], then a[lo] key a[hi]. 51

61 Binary search: mathematical analysis Proposition. Binary search uses at most 1 + lg N compares to search in a sorted array of size N. Def. T (N) # compares to binary search in a sorted subarray of size at most N. Binary search recurrence. T (N) T (N / 2) + 1 for N > 1, with T (1) = 1. Pf sketch. left or right half possible to implement with one 2-way compare (instead of 3-way) 52

62 Binary search: mathematical analysis Proposition. Binary search uses at most 1 + lg N compares to search in a sorted array of size N. Def. T (N) # compares to binary search in a sorted subarray of size at most N. Binary search recurrence. T (N) T (N / 2) + 1 for N > 1, with T (1) = 1. Pf sketch. left or right half possible to implement with one 2-way compare (instead of 3-way) T (N) T (N / 2) + 1 T (N / 4) T (N / 8) T (N / N) = 1 + lg N given apply recurrence to first term apply recurrence to first term stop applying, T(1) = 1 52

63 An N 2 log N algorithm for 3-sum Algorithm. Sort the N numbers. For each pair of numbers a[i] and a[j], binary search for -(a[i] + a[j]). input sort Analysis. Order of growth is N 2 log N. Step 1: N 2 with insertion sort. Step 2: N 2 log N with binary search. binary search (-40, -20) 60 (-40, -10) 30 (-40, 0) 40 (-40, 5) 35 (-40, 10) 30 (-40, 40) 0 (-10, 0) 10 (-20, 10) 10 ( 10, 30) -40 ( 10, 40) -50 ( 30, 40) -70 only count if a[i] < a[j] < a[k] to avoid double counting 53

64 Comparing programs Hypothesis. The N 2 log N three-sum algorithm is significantly faster in practice than the brute-force N 3 one. N time (seconds) N time (seconds) 1, , , , ThreeSum.java 1, , , , , , , ThreeSumFast.java Bottom line. Typically, better order of growth faster in practice. 54

65 Types of analyses Best case. Lower bound on cost. Determined by easiest input. Provides a goal for all inputs. Worst case. Upper bound on cost. Determined by most difficult input. Provides a guarantee for all inputs. Average case. Expected cost for random input. Need a model for random input. Provides a way to predict performance. Ex 1. Array accesses for brute-force 3 sum. Best: ~ ½ N 3 Average: ~ ½ N 3 Worst: ~ ½ N 3 Ex 2. Compares for binary search. Best: ~ 1 Average: ~ lg N Worst: ~ lg N 55

66 Types of analyses Best case. Lower bound on cost. Worst case. Upper bound on cost. Average case. Expected cost. Actual data might not match input model? Need to understand input to effectively process it. Approach 1: design for the worst case. Approach 2: randomize, depend on probabilistic guarantee. 56

67 Commonly-used notations notation provides example shorthand for used to 10 N N log N Tilde leading term ~ 10 N 2 10 N2 10 N N + 37 provide approximate model Big Theta asymptotic growth rate Θ(N 2 ) ½ N 2 10 N 2 5 N N log N + 3N classify algorithms Big Oh Θ(N 2 ) and smaller O(N 2 ) Big Omega Θ(N 2 ) and larger Ω(N 2 ) 10 N N 22 N log N + 3 N ½ N 2 N 5 N N log N + 3 N develop upper bounds develop lower bounds Common mistake. Interpreting big-oh as an approximate model. 57

68 Tilde notation vs. big-oh notation We use tilde notation whenever possible. Big-Oh notation suppresses leading constant. Big-Oh notation only provides upper bound (not lower bound). time/memory time/memory values represented by O(f(N)) f(n) values represented by ~ c f(n) c f(n) input size input size 58

69 Examples Ex 1. Our brute-force 3-sum algorithm takes Θ(N 3 ) time. Ex 2. Conjecture: worst-case running time for any 3-sum algorithm is Ω(N 1.99 ). Ex 3. Insertion sort uses O(N 2 ) compares to sort any array of N elements; it uses ~ N compares in the best case (already sorted) and ~ ½N 2 compares in the worst case (reverse sorted). 59

70 Memory usage Memory usage can be analyzed using tilde notation. We will start doing this next week, as we start to study data structures. Read book for a primer on memory usage in Java. 60

71 Turning the crank: summary Empirical analysis. Execute program to perform experiments. Assume power law and formulate a hypothesis for running time. Model enables us to make predictions. Mathematical analysis. Analyze algorithm to count frequency of operations. Use tilde notation to simplify analysis. Model enables us to explain behavior. Scientific method. Mathematical model is independent of a particular system; applies to machines not yet built. Empirical analysis is necessary to validate mathematical models and to make predictions. 61

72 Next week We start considering simple abstract data types, their possible implementations, and their performance. Stacks, queues, unbounded arrays Linked lists, extendible arrays (amortized analysis) 62