Outline and Reading. Analysis of Algorithms 1
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1 Outline and Reading Algorithms Running time ( 3.1) Pseudo-code ( 3.2) Counting primitive operations ( 3.4) Asymptotic notation ( 3.4.1) Asymptotic analysis ( 3.4.2) Case study ( 3.4.3) Analysis of Algorithms 1
2 Algorithms A procedure for solving a mathematical problem (as of finding the greatest common divisor) in a finite number of steps that frequently involves repetition of an operation A step-by-step procedure for solving a problem or accomplishing some end especially by a computer Analysis of Algorithms 2
3 Algorithms Algorithm Can be performed by both humans and machines Can be expressed in any language Can be as abstract or as concrete as needed Program Performed by computers Must me implemented in a programming language Must be detailed and specific Analysis of Algorithms 3
4 Algorithms - example Find maximum element in an array array a[0..n] max=a[0] for each i from 1 to n if a[i] > max max=a[i] Note the informal language Analysis of Algorithms 4
5 Algorithms - important issues Correctness and efficiency Efficiency criteria Time complexity how much time does the algorithm require how does running time depend on input Space complexity how much memory does the algorithm require how do memory requirements depend on input Analysis of Algorithms 5
6 Running Time The running time of an algorithm varies with the input and typically grows with the input size Average case difficult to determine We focus on the worst case running time Easier to analyze Crucial to applications such as games, finance and robotics Running Time best case average case worst case Input Size Analysis of Algorithms 6
7 Experimental Studies 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 Time (ms) Input Size Analysis of Algorithms 7
8 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
9 Theoretical Analysis Uses a high-level description of the algorithm instead of an implementation Takes into account all possible inputs Allows us to evaluate the speed of an algorithm independent of the hardware/software environment Analysis of Algorithms 9
10 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 Example: find max element of an array 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 10
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 Method call 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 11
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 12
13 Primitive Operations Examples: Evaluating an expression Assigning a value to a variable Indexing into an array Calling a method Returning from a method a+b ; a>b X=5 A[2].. max(a) return x Analysis of Algorithms 13
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 arraymax(a, n) currentmax A[0] for i 1 to n 1 do if A[i] > currentmax then currentmax A[i] return currentmax expand Analysis of Algorithms 14
15 Counting Primitive Operations Worst case scenario - array sorted in ascending order Algorithm arraymax(a, n) currentmax A[0] 1 index + 1 assignment 2 i 1 1 assignment 1 while i <= (n 1) do 1 subtract + 1 compare (n-1+1) *2 if A[i] > currentmax then 1 index + 1 compare (n-1)*2 currentmax A[i] 1 index + 1 assignment (n-1)*2 i i +1 1 addition + 1 assignment (n-1)*2 return currentmax 1 return 1 Total 8n-2 Analysis of Algorithms 15
16 Counting Primitive Operations Best case scenario - array sorted in descending order Algorithm arraymax(a, n) currentmax A[0] 1 index + 1 assignment 2 i 1 1 assignment 1 while i <= (n 1) do 1 subtract + 1 compare (n-1+1) *2 if A[i] > currentmax then 1 index + 1 compare (n-1)*2 currentmax A[i] never executed i i +1 1 addition + 1 assignment (n-1)*2 return currentmax 1 return 1 Total 6n Analysis of Algorithms 16
17 Estimating Running Time Algorithm arraymax executes 8n 2 primitive operations in the worst case, 6n in best case Let T(n) be the actual running time of arraymax. We have 6n T(n) 8n 2 The running time T(n) is bounded by two linear functions Average running time hard to determine, depends on statistics of input set Analysis of Algorithms 17
18 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 18
19 Growth Rate Functions - Constant f(n)=c Performs a single task a fixed number of times Running time independent of n E.g. get array element at given index Usually we make the constant 1 f(n)=1 Analysis of Algorithms 19
20 Growth Rate Functions - Linear f(n)=n Perform a single task for each of n elements E.g. find a given element in an array Analysis of Algorithms 20
21 Growth Rate Functions - Quadratic f(n)=n 2 Perform a single task for all pairs of inputs E.g. nested loops Analysis of Algorithms 21
22 Growth Rate Functions - Polynomial f(n)=n k Less frequent for k>2 E.g. matrix multiplication k=3 Analysis of Algorithms 22
23 Growth Rate Functions - Exponential f(n)=k n Resource allocation problems E.g. traveling salesman Not uncommon for brute force algorithms Usually k=2 Analysis of Algorithms 23
24 Growth Rate Functions - Logarithmic f(n)=log k n Cut the size of the problem by some fraction at each step E.g. binary search (20 questions) Usually k=2 Analysis of Algorithms 24
25 Growth Rate Functions - N-log-N f(n)=n. log n Break a problem into smaller subproblems, solve them independently, combine the solutions E.g. merge-sort Analysis of Algorithms 25
26 Growth Rates Analysis of Algorithms 26
27 Asymptotic Analysis How do f1(n), f2(n), and g(n) compare? f1(n) and f2(n) differ by a constant factor - we say they are asymptotically the same g(n) is asymptotically larger than f1 and f2 - it will become (eventually) larger than f1 and f2 multiplied by any constant Analysis of Algorithms 27
28 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 f(n) is no larger than g(n) Formal definition (set of functions) O(g(n))={f(n) c, n 0 >0 s.t. 0 f(n) cg(n), n n 0 } Analysis of Algorithms 28
29 Big-Ω Notation Similar to big-oh, but for lower bound f(n) is at least as large as g(n) Formal definition Ω(g(n))={f(n) c, n 0 >0 s.t. 0 cg(n) f(n), n n 0 } Analysis of Algorithms 29
30 Big-Θ Notation Tight bound f(n) is the same size as g(n) Formal definition Θ(g(n))={f(n) c1, c2, n 0 >0 s.t. 0 c1g(n) f(n) c1g(n), n n 0 } Analysis of Algorithms 30
31 Big-Oh Notation (cont.) Example: 2n + 10 is O(n) ν 2n + 10 cn Pick c = 3 and n 0 = 10 Other solutions: c = 4 and n 0 = 5 c = 12 and n 0 = 1 Example: the function n 2 is not O(n) ν n 2 cn ν n c The above inequality cannot be satisfied since c must be a constant Analysis of Algorithms 31
32 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 g(n) grows more f(n) grows more Same growth f(n) is O(g(n)) Yes No Yes g(n) is O(f(n)) No Yes Yes Analysis of Algorithms 32
33 Big-Oh Rules If is f(n) a polynomial of degree d, then f(n) is O(n d ), i.e., 1. Drop lower-order terms 2. Drop constant factors Use the smallest possible class of functions Say 2n is O(n) instead of 2n is O(n 2 ) Use the simplest expression of the class Say 3n + 5 is O(n) instead of 3n + 5 is O(3n) Analysis of Algorithms 33
34 Classes of Functions f O(1) O(log(n)) O(n) O(n log(n)) O(n 2 ) O(n k ), k>0 O(k n ), k>1 class constant logarithmic linear n-log-n quadratic polynomial exponential Increase in growth rate Analysis of Algorithms 34
35 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 determined that algorithm arraymax executes at most 8n 2 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 35
36 Excercises Prove that 2n 2 is NOT O(n) Proof by contradiction Assume 2n 2 is O(n) Find c, n 0 such that 2n 2 c n for n n 0 Or n c/2 for n n 0 For any c, n 0 there are (an infinity of) values of n that violate the inequality (e.g. n=c/2+1) So the assumption is wrong Analysis of Algorithms 36
37 Excercises Prove that 20n n log(n)+5 is O(n 3 ) Assume n 0 =1 10 n log(n) 10n 3 for n 1 5 5n 3 for n 1 So 20n n log(n)+5 20n 3 +10n 3 +5n 3 =35n 3 So the (c=35, n 0 =1) pair satisfies the big-o requirement Analysis of Algorithms 37
38 Excercises Prove that if f(n) is O(g(n)) and d(n) id O(h(n)), then f(n)+d(n) is O(g(n)+h(n)) This means we have (c f, n 0f ) and (c d, n 0d ) Let n 0 =max(n 0f, n 0d ) Let c= max(c f, c d ) So f(n)+d(n) c f g(n) + c d h(n) c(g(n) + h(n)) for n n 0 Analysis of Algorithms 38
39 Excercises Beware for(int i=0; i<n; i++) for(int j=0; j<n; j++) for(int k=0; k<5; k++) some task T T requires t time units What is the time complexity of this algorithm? Analysis of Algorithms 39
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 the average of its first i+1 elements Computing the array A of prefix averages of another array X has applications to financial analysis Analysis of Algorithms 40
41 Computing Prefix Averages X[0] X[1] X[2] X[3] X[4] A[0]= X[0] A[1]=(X[0]+X[1])/2 A[2]=(X[0] +X[1] +X[2])/3 A[3]=(X[0] +X[1] +X[2]+X[3])/ X A Analysis of Algorithms 41
42 Prefix Averages (Quadratic) 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 n for i 0 to n 1 do n s 0 n for j 0 to i do (n 1) s s + X[j] (n 1) A[i] s / (i + 1) n return A 1 Analysis of Algorithms 42
43 Arithmetic Progression The running time of prefixaverages1 is O( n) The sum of the first n integers is n(n + 1) / 2 ν There is a simple visual proof of this fact Thus, algorithm prefixaverages1 runs in O(n 2 ) time Analysis of Algorithms 43
44 Can we do better? Original Improved X[0] X[1] X[2] X[3] X[4] A[0]= X[0] A[1]=(X[0]+X[1])/2 A[2]=(X[0] +X[1] +X[2])/3 A[3]=(X[0] +X[1] +X[2]+X[3])/4 S =X[0] S+=X[1] S+=X[2] S+=X[3] A[0]=S/1 A[1]=S/2 A[2]=S/3 A[3]=S/4 Analysis of Algorithms 44
45 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 n s 0 1 for i 0 to n 1 do n s s + X[i] n A[i] s / (i + 1) n return A 1 Algorithm prefixaverages2 runs in O(n) time Analysis of Algorithms 45
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