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Computer Science 212

Computer Science 212

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Computer Science 212

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  1. Computer Science 212 • Title: Data Structures and Algorithms • Instructor: Harry Plantinga • Text: Algorithm Design: Foundations, Analysis, and Internet Examples by Goodrich and Tamassia

  2. Computer Science 212Data Structures and Algorithms • The Heart of Computer Science • Data structures • Study of important algorithms • Algorithm analysis • Algorithm design techniquesPlus • Intelligent systems • Course Information: http://cs.calvin.edu/curriculum/cs/212/Grades on moodle

  3. Why study DS & Algorithms? • Some problems are difficult to solve and good solutions are known • Some “solutions” don’t always work • Some simple algorithms don’t scale well • Data structures and algorithms make good tools for addressing new problems • Fun! Joy! Esthetic beauty!

  4. Place in the curriculum • The study of interesting areas of computer science: upper-level electives • 108, 112: basic tools (like algebra to mathematics) • 212: More advanced tools. Introduction to the science of computing. (A little more mathematical than other core CS courses, but not as rigorous as a real math course.)

  5. Example: Search and Replace • Goal: replace one string with another • Which version is better? #!/usr/bin/perl my $a = shift; my $b = shift; my $input = ""; while (<STDIN>) { $input .= $_; } while ($input =~ m/$a/) { $input =~ s/$a/$b/; } #!/usr/bin/perl my $a = shift; my $b = shift; my $input = ""; while (<STDIN>) { $input .= $_; } $input =~ s/$a/$b/g;

  6. Empirical Analysis • Implement • Gather runtime data for various inputs • Analyze runtime vs. size • Example: suppose we test swap1 • Input sizes: 1, 2, 4, 8, 16, 32, 64, 128, 256 • Runtimes: 70, 110, 250, 560, 810, 1750, 6510, 25680, 102050

  7. swap1 runtime analysis

  8. swap2 runtime analysis

  9. Note that slope gives growth rate of runtimeNote that first few points are inflated by startup costs

  10. Empirical runtime analysis • What can we say about the runtimes of these two algorithms? • Could give runtime as a function of input size, for a particular implementation • But would that apply on different hardware? • What about a different implementation in C++ instead of Perl? • Is there anything we can say that is true in general? • Is version 2 better than version 1 in general? Why?

  11. Empirical analysis: conclusions • Empirical analyses are implementation-dependent • You have to use representative sample data • You can learn something about the growth rate from the slope or curve of the runtime graph • Bad algorithms can’t be fixed with faster computers • Big companies (e.g. Microsoft) are not immune to bad algorithms

  12. What’s the runtime? fori 0 ton  1 do forj 0 ton  1 do ifA[i]  A[j] then A[j]  A[i]

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

  14. Example: find max element of an array AlgorithmarrayMax(A, n) Inputarray A of n integers Outputmaximum element of A currentMaxA[0] fori1ton  1do ifA[i]  currentMaxthen currentMaxA[i] returncurrentMax Pseudocode (§1.1) • High-level description of an algorithm • More structured than English prose • Less detailed than a program • Preferred notation for describing algorithms • Hides program design issues

  15. Control flow if…then…[else …] while … do … repeat…until… for…do… Indentation replaces braces Method declaration Algorithm method (arg [, arg…]) Input… Output… Method call var.method (arg [, arg…]) Return value returnexpression Expressions Assignment(like  in Java) Equality testing(like  in Java) n2 Superscripts and other mathematical formatting allowed Pseudocode Details

  16. Questions… • Can a program be asymptotically faster on one type of CPU vs another? • Do all CPU instructions take equally long?

  17. 2 1 0 The Random Access Machine (RAM) Model • A CPU • An potentially unbounded bank of memory cells, each of which can hold an arbitrary number or character • Memory cells are numbered and accessing any cell in memory takes unit time.

  18. Basic computations performed by an algorithm Identifiable in pseudocode Largely independent from any programming language Exact definition not important (constant number of machine cycles per statement) Assumed to take a constant amount of time in the RAM model Examples: Evaluating an expression Assigning a value to a variable Indexing into an array Calling a method Returning from a method Primitive Operations

  19. By inspecting the pseudocode, we can determine the maximum number of primitive operations executed by an algorithm, as a function of the input size AlgorithmarrayMax(A, n) # operations currentMaxA[0] 2 fori1ton 1do 2+n ifA[i]  currentMaxthen 2(n 1) currentMaxA[i] 2(n 1) { increment counter i } 2(n 1) returncurrentMax 1 Total 7n 1 Counting Primitive Operations (§1.1)

  20. Algorithm arrayMax executes 7n 1 primitive operations in the worst case. Define: a = Time taken by the fastest primitive operation b = Time taken by the slowest primitive operation Let T(n) be worst-case time of arrayMax.Thena (7n 1) T(n)b (7n 1) Hence, the running time T(n) is bounded by two linear functions Estimating Running Time

  21. Theoretical analysis: conclusion • We can count the number of RAM-equivalent statements executed as a function of input size • All remaining implementation dependencies amount to multiplicative constants, which we will ignore • (But watch out for statements that take more than constant time, such as $input =~ s/$a/$b/g) • Linear, quadratic, etc. runtime is an intrinsic property of an algorithm

  22. What’s the runtime? int n; cin >> n; for (int i=0; i<n; i++) for (int j=0; j<n; j++) for (int k=0; k<n; k++) { cout << "Hello, "; cout << "greetings, "; cout << "bonjour, "; cout << "guten tag, "; cout << "Здравствуйте, "; cout << "你好 "; cout << " world!"; }

  23. What’s the runtime? int n; cin >> n; if (n<1000) for (int i=0; i<n; i++) for (int j=0; j<n; j++) for (int k=0; k<n; k++) cout << "Hello\n"; else for (int j=0; j<n; j++) for (int k=0; k<n; k++) cout << "world!\n";

  24. Function Growth Rates • Growth rates of functions: • Linear  n • Quadratic  n2 • Cubic  n3 • In a log-log chart, the slope of the line corresponds to the growth rate of the function

  25. Constant factors • The growth rate is not affected by • constant factors or • lower-order terms • Examples • 102n+105is a linear function • 105n2+ 108nis a quadratic function

  26. Asymptotic (big-O) Notation (§1.2) • Given functions f(n) and g(n), we say that f(n) is O(g(n))if there are positive constantsc and n0 such that f(n)cg(n) for n n0 • Example: 2n+10 is O(n) • 2n+10cn • (c 2) n  10 • n  10/(c 2) • Pick c = 3 and n0 = 10

  27. Example • Example: the function n2is not O(n) • n2cn • n c • The above inequality cannot be satisfied since c must be a constant

  28. More Big-O Examples • 7n-2 7n-2 is O(n) need c > 0 and n0 1 such that 7n-2  c•n for n  n0 this is true for c = 7 and n0 = 1 • 3n3 + 20n2 + 5 3n3 + 20n2 + 5 is O(n3) need c > 0 and n0 1 such that 3n3 + 20n2 + 5  c•n3 for n  n0 this is true for c = 4 and n0 = 21 • 3 log n + log log n 3 log n + log log n is O(log n) need c > 0 and n0 1 such that 3 log n + log log n  c•log n for n  n0 this is true for c = 4 and n0 = 2

  29. Asymptotic analysis of functions • Asymptotic analysis is equivalent to • ignoring multiplicative constants • ignoring lower-order terms • “for large enough inputs” • Big-O and growth rate • Big-O gives an upper bound on the growth rate of a function • Think of it as <= [asymptotically speaking]

  30. Big-O Rules • If is f(n) a polynomial of degree d, then f(n) is O(nd), i.e., • Drop lower-order terms • Drop constant factors • Use the smallest possible class of functions, if possible • Say “2n is O(n)”instead of “2n is O(n2)” • (The former is a stronger statement) • Use the simplest expression of the class • Say “3n+5 is O(n)”instead of “3n+5 is O(3n)”

  31. Asymptotic Algorithm Analysis • Asymptotic analysis: determine the runtime in big-O notation • To perform the asymptotic analysis • Find the worst-case number of primitive operations executed as a function of the input size • Express this function with big-O notation • Example: • We determine that algorithm arrayMax executes at most 7n 1 primitive operations • We say that algorithm arrayMax“runs in O(n) time” • Since constant factors and lower-order terms are eventually dropped anyway, we can disregard them when counting primitive operations

  32. Computing Prefix Averages • Runtime analysis example: 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) • Computing the array A of prefix averages of another array X has applications to financial analysis

  33. Prefix Averages (Quadratic) • The following algorithm computes prefix averages in quadratic time by applying the definition AlgorithmprefixAverages1(X, n) Inputarray X of n integers Outputarray A of prefix averages of X #operations A new array of n integers n fori0ton 1do n sX[0] n forj1toido 1 + 2 + …+ (n 1) ss+X[j] 1 + 2 + …+ (n 1) A[i]s/(i+ 1)n returnA 1

  34. Prefix Averages (Linear) • The following algorithm computes prefix averages in linear time by keeping a running sum AlgorithmprefixAverages2(X, n) Inputarray X of n integers Outputarray A of prefix averages of X #operations A new array of n integers n s 0 1 fori0ton 1do n ss+X[i] n A[i]s/(i+ 1)n returnA 1 • Algorithm prefixAverages2 runs in O(n) time

  35. The running time of prefixAverages1 isO(1 + 2 + …+ n) The sum of the first n integers is n(n+ 1) / 2 There is a simple visual proof of this fact Thus, algorithm prefixAverages1 runs in O(n2) time Arithmetic Progression

  36. What’s the runtime? int n; cin >> n; for (int i=0; i<n; i++) for (int j=0; j<n; j++) for (int k=0; k<n; k++) cout << “Hello world!\n”; 2n3+n2+n+2? O(n3) runtime What if the last line is replaced by: string *s=new string(“Hello world!\n”); O(n3) time and space

  37. What’s the runtime? int n; cin >> n; for (int i=0; i<n; i++) for (int j=0; j<n; j++) for (int k=0; k<n; k++) cout << “Hello world!\n”; for (int i=0; i<n; i++) for (int j=0; j<n; j++) for (int k=0; k<n; k++) cout << “Hello world!\n”; O(n3) + O(n3) = O(n3) Statements or blocks in sequence: add

  38. What’s the runtime? int n; cin >> n; for (int i=0; i<n; i++) for (int j=n; j>1; j/=2) cout << “Hello world!\n”; Loops: add up cost of each iteration (multiply loop cost by number of iterations if they all take the same time) log n iterations of n steps  O(n log n)

  39. What’s the runtime? int n; cin >> n; for (int i=1; i<=n; i++) for (int j=1; j<=i; j++) cout << “Hello world!\n”; Loops: add up cost of each iteration 1 + 2 + 3 + … + n = n(n+1)/2 = Q(n2)

  40. What’s the runtime? template <class Item> void insert(Item a[], int l, int r) { int i; for (i=r; i>l; i--) compexch(a[i-1],a[i]); for (i=l+2; i<=r; i++) { int j=i; Item v=a[i]; while (v<a[j-1]) { a[j] = a[j-1]; j--; } a[j] = v; } }

  41. Math you need to Review • properties of logarithms: logb(xy) = logbx + logby logb (x/y) = logbx - logby Logb xa = a logb x logba = logxa/logxb • properties of exponentials: a(b+c) = aba c abc = (ab)c ab /ac = a(b-c) b = a logab bc = a c*logab • Summations (Sec. 1.3.1) • Logarithms and Exponents (Sec. 1.3.2) • Proof techniques (Sec. 1.3.3) • Basic probability (Sec. 1.3.4)

  42. Relatives of Big-Oh • big-Omega • f(n) is (g(n)) if there is a constant c > 0 and an integer constant n0  1 such that f(n)  c•g(n) for n  n0 • big-Theta • f(n) is (g(n)) if there are constants c’ > 0 and c’’ > 0 and an integer constant n0  1 such that c’•g(n)  f(n)  c’’•g(n) for n  n0 • little-o • f(n) is o(g(n)) if, for any constant c > 0, there is an integer constant n0  0 such that f(n)  c•g(n) for n  n0 • little-omega • f(n) is (g(n)) if, for any constant c > 0, there is an integer constant n0  0 such that f(n)  c•g(n) for n  n0

  43. Intuition for Asymptotic Notation Big-Oh • f(n) is O(g(n)) if f(n) is asymptotically less than or equal to g(n) big-Omega • f(n) is (g(n)) if f(n) is asymptotically greater than or equal to g(n) big-Theta • f(n) is (g(n)) if f(n) is asymptotically equal to g(n) little-oh • f(n) is o(g(n)) if f(n) is asymptotically strictly less than g(n) little-omega • f(n) is (g(n)) if is asymptotically strictly greater than g(n)

  44. Example Uses of the Relatives of Big-Oh • 5n2 is (n2) f(n) is (g(n)) if there is a constant c > 0 and an integer constant n0  1 such that f(n)  c•g(n) for n  n0 let c = 5 and n0 = 1 • 5n2 is (n) f(n) is (g(n)) if there is a constant c > 0 and an integer constant n0  1 such that f(n)  c•g(n) for n  n0 let c = 1 and n0 = 1 • 5n2 is (n) f(n) is (g(n)) if, for any constant c > 0, there is an integer constant n0  0 such that f(n)  c•g(n) for n  n0 need 5n02  c•n0  given c, the n0 that satisfies this is n0  c/5  0

  45. Asymptotic Analysis: Review • What does it mean to say that an algorithm has runtime O(n log n)? • n: Problem size • Big-O: upper bound over all inputs of size n • “Ignore constant factor” (why?) • “as n grows large” O: like <= for functions (asymptotically speaking) W: like >= Q: like =

  46. Asymptotic notation: examples • Asymptotic runtime, in terms of O, W, Q? • Suppose the runtime for a function is • n2 + 2n log n + 40 • 0.0000001 n2+ 1000000n1.999 • n3 + n2 log n • n2.0001 + n2 log n • 2n+ 100 n2 • 1.00001n+ 100 n97

  47. Asymptotic comparisons • 0.0000001 n2 = O(1000000n1.999 )? • n1.000001 = O(n log n)? • 1.0001n = O(n943)? • lg n = Q(ln n)? (Evaluate the limit of the quotient of the functions) No – a polynomial with a higher power dominates one with a lower power No – all polynomials (n.000001) dominate any polylog (log n) No – all exponentials dominate any polynomial Yes – different bases are just a constant factor difference

  48. Estimate the runtime • Suppose an algorithm has runtime Q(n3) • suppose solving a problem of size 1000 takes 10 seconds. How long to solve a problem of size 10000? • Suppose an algorithm has runtime Q(n log n) • suppose solving a problem of size 1000 takes 10 seconds. How long to solve a problem of size 10000? runtime 10-8 n3; if n=10000, runtime 10000s = 2.7hr runtime 10-3 n lg n; if n=10000, runtime 133 secs