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Analysis of Algorithms

Analysis of Algorithms. Input. Output. Algorithm. Outline. Running time Pseudo-code Counting primitive operations Asymptotic notation Asymptotic analysis. How good is Insertion-Sort?. How can you answer such questions?. What is “goodness”?. Correctness Minimum use of “time” + “space”.

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Analysis of Algorithms

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  1. Analysis of Algorithms Input Output Algorithm

  2. Outline • Running time • Pseudo-code • Counting primitive operations • Asymptotic notation • Asymptotic analysis Analysis of Algorithms

  3. How good is Insertion-Sort? • How can you answer such questions? Analysis of Algorithms

  4. What is “goodness”? • Correctness • Minimum use of “time” + “space” • Measure • Count • Estimate How can we quantify it? Analysis of Algorithms

  5. 1) Measure it • Write a program implementing the algorithm • Run the program with inputs of varying size and composition • Use a method like System.currentTimeMillis() to get an accurate measure of the actual running time • Plot the results Analysis of Algorithms

  6. Analysis of Algorithms

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

  8. 2) Count Primitive Operations • The Idea • Write down the pseudocode • Count the number of “primitive operations” Analysis of Algorithms

  9. 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 • 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 Analysis of Algorithms

  10. 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 Analysis of Algorithms

  11. 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. Analysis of Algorithms

  12. Random Access Model (RAM) • Time complexity (running time) = number of instructions executed • Space complexity = the number of memory cells accessed Analysis of Algorithms

  13. Basic computations performed by an algorithm Identifiable in pseudocode Largely independent from the programming language Exact definition not important (we will see why later) Examples: Evaluating an expression Assigning a value to a variable Indexing into an array Calling a method Returning from a method Primitive Operations Analysis of Algorithms

  14. Analyzing pseudocode (by counting) • For each line of pseudocode, count the number of primitive operations in it. Pay attention to the word "primitive" here; sorting an array is not a primitive operation. • Multiply this count with the number of times this line is executed. • Sum up over all lines. Analysis of Algorithms

  15. 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) Cost Times currentMaxA[0] 2 1 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 1 Total 8n 3 Counting Primitive Operations Analysis of Algorithms

  16. Algorithm arrayMax executes 8n 2 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 the actual worst-case running time of arrayMax. We havea (8n 2) T(n)b(8n 2) Hence, the running time T(n) is bounded by two linear functions Estimating Running Time Analysis of Algorithms

  17. Insertion Sort • Hint: • observe each line i can be implemented using a constant number of RAM instructions, Ci • for each value of j in the outer loop, we let tj be the number of times that the while loop test in line 4 is executed.  Analysis of Algorithms

  18. Insertion Sort Question: So what is the running time? Analysis of Algorithms

  19. Time Complexity may depend on the input! • Best Case:? • Worst Case:? • Average Case:? Analysis of Algorithms

  20. Complexity may depend on the input! • Best Case: Already sorted. tj=1. Running time = C * N (a linear function of N) • Worst Case: Inverse order. tj=j. Analysis of Algorithms

  21. Neglecting the constants, the worst-base running time of insertion sort is proportional to1 + 2 + …+ n The sum of the first n integers is n(n+ 1) / 2 Thus, algorithm insertion sort required almost n2 operations. Arithmetic Progression Analysis of Algorithms

  22. What about the Average Case? • Idea: • Assume that each of the n! permutations of A is equally likely. • Compute the average over all possible different inputs of length N. • Difficult to compute! • In this course we focus on Worst Case Analysis! Analysis of Algorithms

  23. See TimeComplexity Spreadsheet Analysis of Algorithms

  24. What is “goodness”? • Correctness • Minimum use of “time” + “space” • Measure wall clock time • Count operations • Estimate Computational Complexity How can we quantify it? Analysis of Algorithms

  25. Asymptotic Analysis • Uses a high-level description of the algorithm instead of an implementation • Allows us to evaluate the speed of an algorithm independent of the hardware/software environment • Estimate the growth rate of T(n) • A back of the envelope calculation!!! Analysis of Algorithms

  26. The idea • Write down an algorithm • Using Pseudocode • In terms of a set of primitive operations • Count the # of steps • In terms of primitive operations • Considering worst case input • Bound or “estimate” the running time • Ignore constant factors • Bound fundamental running time Analysis of Algorithms

  27. Growth Rate of Running Time • Changing the hardware/ software environment • Affects T(n) by a constant factor, but • Does not alter the growth rate of T(n) • The linear growth rate of the running time T(n) is an intrinsic property of algorithm arrayMax Analysis of Algorithms

  28. Which growth rate is best? • T(n) = 1000n + n2 or T(n) = 2n + n3 Analysis of Algorithms

  29. 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 Analysis of Algorithms

  30. Functions Graphed Using “Normal” Scale g(n) = n lg n g(n) = 1 g(n) = n2 g(n) = lg n g(n) = n g(n) = 2n g(n) = n3 Analysis of Algorithms

  31. 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 Analysis of Algorithms

  32. Big-Oh Notation • 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 Analysis of Algorithms

  33. f(n) is O(g(n)) iff f(n)cg(n) for n n0 Analysis of Algorithms

  34. Big-Oh Notation (cont.) • Example: the function n2is not O(n) • n2cn • n c • The above inequality cannot be satisfied since c must be a constant Analysis of Algorithms

  35. More Big-Oh 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 + 5 3 log n + 5 is O(log n) need c > 0 and n0 1 such that 3 log n + 5  c•log n for n  n0 this is true for c = 8 and n0 = 2 Analysis of Algorithms

  36. Big-Oh and Growth Rate • 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 Analysis of Algorithms

  37. Questions • Is T(n) = 9n4 + 876n = O(n4)? • Is T(n) = 9n4 + 876n = O(n3)? • Is T(n) = 9n4 + 876n = O(n27)? • T(n) = n2 + 100n = O(?) • T(n) = 3n + 32n3 + 767249999n2 = O(?) Analysis of Algorithms

  38. Big-Oh Rules (shortcuts) • 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 • Say “2n is O(n)”instead of “2n is O(n2)” • Use the simplest expression of the class • Say “3n+5 is O(n)”instead of “3n+5 is O(3n)” Analysis of Algorithms

  39. Rank From Fast to Slow… • T(n) = O(n4) • T(n) = O(n log n) • T(n) = O(n2) • T(n) = O(n2 log n) • T(n) = O(n) • T(n) = O(2n) • T(n) = O(log n) • T(n) = O(n + 2n) Note: Assume base of log is 2 unless otherwise instructed i.e. log n = log2 n Analysis of Algorithms

  40. Computing Prefix Averages • 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] • Computing the array A of prefix averages of another array X has applications to financial analysis Analysis of Algorithms

  41. 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 Analysis of Algorithms

  42. 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 Analysis of Algorithms

  43. 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 Analysis of Algorithms

  44. Math you may need to Review • Summations • Logarithms and Exponents • Proof techniques • Basic probability • properties of logarithms: logb(xy) = logbx + logby logb (x/y) = logbx - logby logbxa = alogbx 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 Analysis of Algorithms

  45. Big-Omega  • Definition: 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 • An asymptotic lower Bound Analysis of Algorithms

  46. Big-Theta  • Definition: 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 • Here we say that g(n) is an asymptotically tight bound for f(n). Analysis of Algorithms

  47. Big-Theta  (cont) • Based on the definitions, we have the following theorem: • f(n) = Θ(g(n)) if and only if f(n) = O(g(n)) and f(n) = Ω(g(n)). • For example, the statement n2/2 + lg n = Θ(n2) is equivalent to n2/2 + lg n = O(n2) and n2/2 + lg n = Ω(n2). • Note that asymptotic notation applies to asymptotically positive functions only, which are functions whose values are positive for all sufficiently large n. Analysis of Algorithms

  48. Prove n2/2 + lg n = Θ(n2). • Proof. To prove this claim, we must determine positive constants c1, c2 and n0, s.t. c1 n2<= n2/2 + lg n <= c2 n2 • c1 <= ½ + (lg n) / n2 <= c2 (divide thru by n2) • Pick c1 = ¼ , c2 = ¾ and n0 = 2 • For n0 = 2, 1/4  <= ½ + (lg 2) / 4 <= ¾, TRUE • When n0 > 2, the (½ + (lg 2) / 4) term grows smaller but never less than ½, therefore n2/2 + lg n = Θ(n2). Analysis of Algorithms

  49. 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) Analysis of Algorithms

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