# Analysis of Algorithms - Western University

CS2210 Data Structures and Algorithms Lecture 2: Analysis of Algorithms Asymptotic notation Instructor: Olga Veksler 2004 Goodrich, Tamassia Outline Comparing algorithms Pseudocode Theoretical Analysis of Running time Primitive Operations Counting primitive operations Asymptotic analysis of running time Analysis of Algorithms 2 Comparing Algorithms

Given 2 or more algorithms to solve the same problem, how do we select the best one? Some criteria for selecting an algorithm 1) 2) 3) Is it easy to implement, understand, modify? How long does it take to run it to completion? How much of computer memory does it use? Software engineering is primarily concerned with the first criteria In this course we are interested in the second and third criteria Analysis of Algorithms 3 Comparing Algorithms Time complexity Space complexity

The amount of time that an algorithm needs to run to completion The amount of memory an algorithm needs to run We will occasionally look at space complexity, but we are mostly interested in time complexity in this course Thus in this course the better algorithm is the one which runs faster (has smaller time complexity) Analysis of Algorithms 4 How to Calculate Running time Most algorithms transform input objects into output objects 5 3 1 2 input object sorting algorithm

1 2 3 5 output object The running time of an algorithm typically grows with the input size idea: analyze running time as a function of input size Analysis of Algorithms 5 How to Calculate Running Time Even on inputs of the same size, running time can be very different Example: algorithm that finds the first prime number in an array by scanning it left to right Idea: analyze running time in the best case worst case average case

Analysis of Algorithms 6 How to Calculate Running Time Best case running time is usually useless Average case time is very useful but often difficult to determine We focus on the worst case running time Easier to analyze Crucial to applications such as games, finance and robotics best case average case worst case 120 100 Running Time

80 60 40 20 0 1000 Analysis of Algorithms 2000 3000 4000 I nput Size 7 Experimental Evaluation of Running Time 9000 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 8000 7000 Time (ms) 6000 5000 4000 3000 2000 1000 0 Analysis of Algorithms 0 50 100 I nput Size 8

Limitations of Experiments Experimental evaluation of running time is very useful but 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 9 Theoretical Analysis of Running Time Uses a pseudo-code 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 Allows us to evaluate the speed of an algorithm independent of the hardware/software environment Analysis of Algorithms 10 Pseudocode In this course, we will Example: find max mostly use pseudocode to element of an array describe an algorithm Pseudocode is a high-level description of an algorithm Algorithm arrayMax(A, n) Input: array A of n integers More structured than English prose Output: maximum element of A

Less detailed than a currentMax A[0] program for i 1 to n 1 do Preferred notation for describing algorithms if A[i] currentMax then Hides program design currentMax A[i] issues return currentMax Analysis of Algorithms 11 Pseudocode Details Control flow if then [else ] while do repeat until for do Indentation replaces braces

Method declaration Algorithm method (arg, arg) Input Output Algorithm arrayMax(A, n) Input: array A of n integers Output: maximum element of A currentMax A[0] for i 1 to n 1 do if A[i] currentMax then currentMax A[i] return currentMax Analysis of Algorithms 12 Pseudocode Details Method call var.method (arg [, arg]) Return value return expression Expressions Assignment (like in Java) Equality testing

(like in Java) n2 superscripts and other mathematical formatting allowed Algorithm arrayMax(A, n) Input: array A of n integers Output: maximum element of A currentMax A[0] for i 1 to n 1 do if A[i] currentMax then currentMax A[i] return currentMax Analysis of Algorithms 13 Primitive Operations For theoretical analysis, we will count primitive or basic operations, which are simple computations performed by an algorithm Basic operations are:

Identifiable in pseudocode Largely independent from the programming language Exact definition not important (we will see why later) Assumed to take a constant amount of time Analysis of Algorithms 14 Primitive Operations Examples of primitive operations: Evaluating an expression Assigning a value to a variable cnt+1 Indexing into an array Calling a method mySort(A,n) Returning from a method return(cnt) Analysis of Algorithms x2+ey

cnt A[5] 15 Counting Primitive Operations By inspecting the pseudocode, we can determine the maximum number of primitive operations executed by an algorithm, as a function of the input size Algorithm arrayMax(A, n) currentMax A[0] for i 1 to n 1 do if A[i] currentMax then currentMax A[i] { increment counter i } return currentMax 2 2+n 2(n 1) 2(n 1) 2(n 1) 1 Total Analysis of Algorithms 7n 1 16

Estimating Running Time 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. Then a (7n 1) T(n) b(7n 1) Hence, the running time T(n) is bounded by two linear functions Analysis of Algorithms 17 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) Thus we focus on the big-picture which is the growth rate of an algorithm The linear growth rate of the running time T(n) is an intrinsic property of algorithm arrayMax algorithm arrayMax grows proportionally with n, with its true running time being n times a constant factor that depends on the specific computer Analysis of Algorithms 18 Constant Factors The growth rate is not affected by Examples constant factors or lower-order terms 102n 105 is a linear function 105n2 108n is a quadratic function

How do we get rid of the constant factors to focus on the essential part of the running time? Analysis of Algorithms 19 Big-Oh Notation Motivation Or asymptotic analysis The big-Oh notation is used widely to characterize running times and space bounds The big-Oh notation allows us to ignore constant factors and lower order terms and focus on the main components of a function which affect its growth Analysis of Algorithms 20 Big-Oh Notation Definition Given functions f(n) and g(n), we say that

f(n) is O(g(n)) if there are positive constants c and n0 such that 80 70 3n 60 2n+10 50 n 40 f(n) cg(n) for n n0 30 Example: 2n 10 is O(n) 20 2n 10 cn (c 2) n 10 n 10(c 2) Pick c 3 and n0 10 10

0 0 5 Analysis of Algorithms 10 15 n 20 25 30 21 Big-Oh Example Example: the function n2 is not O(n)

100,000 n^2 90,000 100n 80,000 n2 cn nc The above inequality cannot be satisfied since c must be a constant 10n 70,000 n 60,000 50,000 40,000 30,000 20,000 10,000 0 0 100 Analysis of Algorithms

200 n 300 400 500 22 More Big-Oh Examples 7n-2 7n-2 is O(n) need c > 0 and n0 1 such that 7n-2 cn 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 cn3 for n n0 this is true c = 4 and n0 = 21 3 log n for +5 3 log n + 5 is O(log n)

need c > 0 and n0 1 such that 3 log n + 5 clog n for n n0 this is true for c = 8 and n0 = 2 Analysis of Algorithms 23 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 f(n) is O(g(n)) g(n) is O(f(n)) g(n) grows more Yes No f(n) grows more

No Yes Same growth Yes Yes Analysis of Algorithms 24 Big-Oh Rules f n a0 a1 n a2 n 2 ... ad n d If is f(n) a polynomial of degree d, then f(n) is O(nd), i.e., 1. 2. Use the smallest possible class of functions Drop lower-order terms Drop constant factors 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 25 Asymptotic Algorithm Analysis The asymptotic analysis of an algorithm determines the running time in big-Oh notation To perform the asymptotic analysis Example: We find the worst-case number of primitive operations executed as a function of the input size We express this function with big-Oh notation 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 anyhow, we can disregard them when counting primitive operations Analysis of Algorithms 26 Important Functions Often appear in algorithm analysis: Constant 1 Logarithmic log n Linear n N-Log-N n log n Quadratic n2 Cubic n3 Exponential 2n Analysis of Algorithms 27 Important Functions Growth

Rates n log(n ) n nlog(n ) n2 n3 2n 8 3 8 24 64 512 256 16 4

16 64 256 4096 65536 32 5 32 160 1024 32768 4.3x109 64 6 64 384 4096 262144 1.8x101

9 128 7 128 896 16384 209715 3.4x103 8 2 256 8 256 2048 65536 167772 1.2x107 7 18 Analysis of Algorithms 28 Growth Rates Illustration

Running Time in ms (10-3 of sec) Maximum Problem Size (n) 1000 ms m (1 second) hour) 60000 ms (1 minute) 36*105 (1 n 1000 60,000 3,600,000 n2 32 245 1,897 2n

10 16 22 Analysis of Algorithms 29 Useful Big-Oh Rules If is f(n) a polynomial of degree d, then f(n) is O(nd) If d(n) is O(f(n)) and e(n) is O(g(n)) then d(n)+e(n) is O(f(n)+g(n)) d(n)e(n) is O(f(n) g(n)) If d(n) is O(f(n)) and f(n) is O(g(n)) then d(n) is O(g(n)) If p(n) is a polynomial in n then log p(n) is O(log(n)) Analysis of Algorithms 30 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])/(i+1) Computing the array A of prefix averages of another array X has applications to financial analysis 35 X A 30 25 20 15 10 5 0 Analysis of Algorithms 1 2 3

4 5 6 7 31 Prefix Averages (Quadratic) The following algorithm computes prefix averages in quadratic time by applying the definition Algorithm prefixAverages1(X, n) Input array X of n integers Output array A of prefix averages of X #operations A new array of n integers for i 0 to n 1 do s X[0] for j 1 to i do s s + X[j] A[i] s / (i + 1) return A Analysis of Algorithms n n

n 1 + 2 + + (n 1) 1 + 2 + + (n 1) n 1 32 Arithmetic Progression The running time of prefixAverages1 is O(1 2 n) The sum of the first n integers is n(n 1) 2 7 There is a simple visual proof of this fact 3 Thus, algorithm prefixAverages1 runs in O(n2) time 6 5

4 2 1 0 1 Analysis of Algorithms 2 3 4 5 6 33 Prefix Averages (Linear) The following algorithm computes prefix averages in linear time by keeping a running sum Algorithm prefixAverages2(X, n) Input array X of n integers Output array A of prefix averages of X #operations A new array of n integers s0 for i 0 to n 1 do s s + X[i] A[i] s / (i + 1)

return A n 1 n n n 1 Algorithm prefixAverages2 runs in O(n) time Analysis of Algorithms 34 More Examples Algorithm SumTripleArray(X, n) Input triple array X[ ][ ][ ] of n by n by n integers Output sum of elements of X #operations s0 1 for i 0 to n 1 do n for j 0 to n 1 do n+n++n=n2 for k 0 to n 1 do n2+n2++n2 = n3 s s + X[i][j][k] n2+n2++n2 = n3 return s 1

Algorithm SumTripleArray runs in O(n3) time Analysis of Algorithms 35 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) cg(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 cg(n) f(n) cg(n) for n n0 Analysis of Algorithms 36 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) Note that f(n) is (g(n)) if and only if g(n) is O(f(n)) big-Theta

f(n) is (g(n)) if f(n) is asymptotically equal to g(n) Note that f(n) is (g(n)) if and only if if g(n) is O(f(n)) and if f(n) is O(g(n)) Analysis of Algorithms 37 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) cg(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) cg(n) for n n0 let c = 1 and n0 = 1 5n2 is (n2) f(n) is (g(n)) if it is (n2) and O(n2). We have already seen the former, for the latter recall that f(n) is O(g(n)) if there is a constant c > 0 and an integer constant n0 1 such that f(n) < cg(n) for n n0 Let c = 5 and n0 = 1 Analysis of Algorithms 38

Math you need to Review Summations Logarithms and Exponents 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 39 Final Notes Even though in this course we focus on the asymptotic growth using big-Oh notation,

practitioners do care about constant factors occasionally Suppose we have 2 algorithms Algorithm A has running time 30000n Algorithm B has running time 3n2 Asymptotically, algorithm A is better than algorithm B However, if the problem size you deal with is always less than 10000, then the quadratic one is faster Analysis of Algorithms Running time A B 10000 problem size 40

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