Analysis of Algorithms Input Output Algorithm Outline Running

Analysis of Algorithms Input Output Algorithm

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

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

What is “goodness”? 1. Correctness 2. Minimum use of “time” + “space” How can we quantify it? 1. Measure 2. Count 3. Estimate Analysis of Algorithms 4

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

Analysis of Algorithms 6

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 7

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

Pseudocode Example: find max High-level description element of an array of an algorithm More structured than Algorithm array. Max(A, n) English prose Input array A of n integers Less detailed than a Output maximum element of A program Preferred notation for current. Max A[0] for i 1 to n 1 do describing algorithms if A[i] current. Max then Hides program design current. Max A[i] issues return current. Max Analysis of Algorithms 9

Pseudocode Details Method call Control flow n n n if … then … [else …] while … do … repeat … until … for … do … Indentation replaces braces Method declaration Algorithm method (arg [, arg…]) Input … Output … var. method (arg [, arg…]) Return value return expression Expressions Assignment (like in Java) Equality testing (like in Java) n 2 Superscripts and other mathematical formatting allowed Analysis of Algorithms 10

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 0 2 1 Memory cells are numbered and accessing any cell in memory takes unit time. Analysis of Algorithms 11

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

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

Analyzing pseudocode (by counting) 1. 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. 2. Multiply this count with the number of times this line is executed. 3. Sum up over all lines. Analysis of Algorithms 14

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 array. Max(A, n) current. Max A[0] for i 1 to n 1 do if A[i] current. Max then current. Max A[i] { increment counter i } return current. Max Cost 2 2 2 1 Total Analysis of Algorithms Times 1 n (n 1) 1 8 n 3 15

Estimating Running Time Algorithm array. Max executes 8 n 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 array. Max. We have a (8 n 2) T(n) b(8 n 2) Hence, the running time T(n) is bounded by two linear functions Analysis of Algorithms 16

Insertion Sort Hint: 1. observe each line i can be implemented using a constant number of RAM instructions, Ci 2. 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 17

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

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

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 20

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

What about the Average Case? Idea: n n 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 22

See Time. Complexity Spreadsheet Analysis of Algorithms 23

What is “goodness”? 1. Correctness 2. Minimum use of “time” + “space” How can we quantify it? 1. Measure wall clock time 2. Count operations 3. Estimate Computational Complexity Analysis of Algorithms 24

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 25

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

Growth Rate of Running Time Changing the hardware/ software environment n n 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 array. Max Analysis of Algorithms 27

Which growth rate is best? T(n) = 1000 n + n 2 or T(n) = 2 n + n 3 Analysis of Algorithms 28

Growth Rates Growth rates of functions: n n n Linear n Quadratic n 2 Cubic n 3 In a log-log chart, the slope of the line corresponds to the growth rate of the function Analysis of Algorithms 29

Functions Graphed Using “Normal” Scale g(n) = 1 g(n) = n lg n g(n) = 2 n g(n) = n 2 g(n) = lg n g(n) = n 3 30 Analysis of Algorithms

Constant Factors The growth rate is not affected by n n constant factors or lower-order terms Examples n n 102 n + 105 is a linear function 105 n 2 + 108 n is a quadratic function Analysis of Algorithms 31

Big-Oh Notation Given functions f(n) and g(n), we say that f(n) is O(g(n)) if there are positive constants c and n 0 such that f(n) cg(n) for n n 0 Example: 2 n + 10 is O(n) n n 2 n + 10 cn (c 2) n 10/(c 2) Pick c 3 and n 0 10 Analysis of Algorithms 32

f(n) is O(g(n)) iff f(n) cg(n) for n n 0 Analysis of Algorithms 33

Big-Oh Notation (cont. ) Example: the function n 2 is not O(n) n n 2 cn n c The above inequality cannot be satisfied since c must be a constant Analysis of Algorithms 34

More Big-Oh Examples 7 n-2 is O(n) need c > 0 and n 0 1 such that 7 n-2 c • n for n n 0 this is true for c = 7 and n 0 = 1 3 n 3 + 20 n 2 + 5 is O(n 3) need c > 0 and n 0 1 such that 3 n 3 + 20 n 2 + 5 c • n 3 for n n 0 this is true for c = 4 and n 0 = 21 3 log n + 5 is O(log n) need c > 0 and n 0 1 such that 3 log n + 5 c • log n for n n 0 this is true for c = 8 and n 0 = 2 Analysis of Algorithms 35

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 36

Questions Is T(n) = 9 n 4 + 876 n = O(n 4)? Is T(n) = 9 n 4 + 876 n = O(n 3)? Is T(n) = 9 n 4 + 876 n = O(n 27)? T(n) = n 2 + 100 n = O(? ) T(n) = 3 n + 32 n 3 + 767249999 n 2 = O(? ) Analysis of Algorithms 37

Big-Oh Rules (shortcuts) If is f(n) a polynomial of degree d, then f(n) is O(nd), i. e. , 1. 2. Drop lower-order terms Drop constant factors Use the smallest possible class of functions n Say “ 2 n is O(n)” instead of “ 2 n is O(n 2)” Use the simplest expression of the class n Say “ 3 n + 5 is O(n)” instead of “ 3 n + 5 is O(3 n)” Analysis of Algorithms 38

Rank From Fast to Slow… T(n) = O(n 4) T(n) = O(n log n) T(n) = O(n 2 log n) T(n) = O(2 n) T(n) = O(log n) T(n) = O(n + 2 n) Note: Assume base of log is 2 unless otherwise instructed i. e. log n = log 2 n Analysis of Algorithms 39

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 40

Prefix Averages (Quadratic) The following algorithm computes prefix averages in quadratic time by applying the definition Algorithm prefix. Averages 1(X, n) Input array X of n integers Output array A of prefix averages of X #operations A new array of n integers n for i 0 to n 1 do n s X[0] n for j 1 to i do 1 + 2 + …+ (n 1) s s + X[j] 1 + 2 + …+ (n 1) A[i] s / (i + 1) n return A 1 Analysis of Algorithms 41

Arithmetic Progression The running time of prefix. Averages 1 is O(1 + 2 + …+ n) The sum of the first n integers is n(n + 1) / 2 n There is a simple visual proof of this fact Thus, algorithm prefix. Averages 1 runs in O(n 2) time Analysis of Algorithms 42

Prefix Averages (Linear) The following algorithm computes prefix averages in linear time by keeping a running sum Algorithm prefix. Averages 2(X, n) Input array X of n integers Output array A of prefix averages of X A new array of n integers s 0 for i 0 to n 1 do s s + X[i] A[i] s / (i + 1) return A #operations n 1 n n n 1 Algorithm prefix. Averages 2 runs in O(n) time Analysis of Algorithms 43

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 44

Big-Omega Definition: f(n) is (g(n)) if there is a constant c > 0 and an integer constant n 0 1 such that f(n) c • g(n) for n n 0 An asymptotic lower Bound Analysis of Algorithms 45

Big-Theta Definition: f(n) is (g(n)) if there are constants c’ > 0 and c’’ > 0 and an integer constant n 0 1 such that c’ • g(n) f(n) c’’ • g(n) for n n 0 Here we say that g(n) is an asymptotically tight bound for f(n). Analysis of Algorithms 46

Big-Theta (cont) Based on the definitions, we have the following theorem: n f(n) = Θ(g(n)) if and only if f(n) = O(g(n)) and f(n) = Ω(g(n)). For example, the statement n 2/2 + lg n = Θ(n 2) is equivalent to n 2/2 + lg n = O(n 2) and n 2/2 + lg n = Ω(n 2). 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 47

Prove n 2/2 + lg n = Θ(n 2). Proof. To prove this claim, we must determine positive constants c 1, c 2 and n 0, s. t. c 1 n 2<= n 2/2 + lg n <= c 2 n n n n c 1 <= ½ + (lg n) / n 2 <= c 2 (divide thru by n 2) Pick c 1 = ¼ , c 2 = ¾ and n 0 = 2 For n 0 = 2, 1/4 <= ½ + (lg 2) / 4 <= ¾, TRUE When n 0 > 2, the (½ + (lg 2) / 4) term grows smaller but never less than ½, therefore n 2/2 + lg n = Θ(n 2). Analysis of Algorithms 48

Intuition for Asymptotic Notation Big-Oh n f(n) is O(g(n)) if f(n) is asymptotically less than or equal to g(n) big-Omega n f(n) is (g(n)) if f(n) is asymptotically greater than or equal to g(n) big-Theta n f(n) is (g(n)) if f(n) is asymptotically equal to g(n) Analysis of Algorithms 49

A word of caution… Asymptotic analysis is fragile n Constants do matter! Don’t play games! Modern processors are not RAM Analysis of Algorithms 50

A word of caution… In practice, performance comes from Algorithmic Analysis Careful Engineering Systematic Evaluation Analysis of Algorithms 51