PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 1 CS 311 Design and Algorithms Analysis Dr. Mohamed Tounsi

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 2 Problems and Algorithms Abstractly, a problem is just a function p : Problem instances -> Solutions Example: Sorting a list of integers Problem instances = lists of integers Solutions = sorted lists of integers p : ` L-> Sorted version of L An algorithm for p is a program which computes p. There are four related questions which warrant consideration:

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 3 n Question 1: Given an algorithm A to solve a problem p, how “good” is A? n Question 2: Given a particular problem p for which several algorithms are known to exist, which is best? n Question 3: Given a problem p, how does one design good algorithms for p? Problems and Algorithms

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 4 Algorithm Analysis The main focus of algorithm analysis in this course will be upon the “quality” of algorithms already known to be correct. This analysis proceeds in two dimensions: n Time complexity: The amount of time which execution of the algorithm takes, usually specified as a function of the size of the input n Space complexity: The amount of space (memory) which execution of the algorithm takes, usually specified as a function of the size of the input. Such analyses may be performed both experimentally and analytically.

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 5 Designing Algorithms It is important to study the process of designing good algorithms in the first place. There are two principal approaches to algorithm design. n By problem: Study sorting algorithms, then scheduling algorithms, etc. n By strategy: Study algorithms by design strategy. Examples of design strategies include: Divide-and-conquer The greedy method Dynamic programming Backtracking Branch-and-bound

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 6 Algorithm Definition Algorithm Input Output An algorithm is a step-by-step procedure for solving a problem in a finite amount of time.

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 7 Running Time n Most algorithms transform input objects into output objects. n The running time of an algorithm typically grows with the input size. n Average case time is often difficult to determine. n We focus on the worst case running time. l Easier to analyze l Crucial to applications such as games, finance and robotics

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 8 Experimental Studies n Write a program implementing the algorithm n Run the program with inputs of varying size and composition Use a method like times() to get an accurate measure of the actual running time n Plot the results

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 9 Limitations of Experiments n It is necessary to implement the algorithm, which may be difficult n Results may not be indicative of the running time on other inputs not included in the experiment. n In order to compare two algorithms, the same hardware and software environments must be used

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 10 Theoretical Analysis n Uses a high-level description of the algorithm instead of an implementation n Characterizes running time as a function of the input size, n. n Takes into account all possible inputs n Allows us to evaluate the speed of an algorithm independent of the hardware/software environment

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 11 Pseudocode n High-level description of an algorithm n More structured than English prose n Less detailed than a program n Preferred notation for describing algorithms n Hides program design issues 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 Example: find max element of an array

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 12 Pseudocode Details n Control flow l if … then … [else …] l while … do … l repeat … until … l for … do … l Indentation replaces braces n Method declaration Algorithm method (arg [, arg…]) Input … Output … n Method call var.method (arg [, arg…]) n Return value return expression n Expressions Assignment (like  in Java) Equality testing (like  in Java) n 2 Superscripts and other mathematical formatting allowed

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 13 The Random Access Machine (RAM) Model n A CPU n An potentially unbounded bank of memory cells, each of which can hold an arbitrary number or character n Memory cells are numbered and accessing any cell in memory takes unit time.

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 14 Primitive Operations n Basic computations performed by an algorithm n Identifiable in pseudocode n Largely independent from the programming language n Exact definition not important (we will see why later) n Assumed to take a constant amount of time in the RAM model n Examples: l Evaluating an expression l Assigning a value to a variable l Indexing into an array l Calling a method l Returning from a method

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 15 Counting Primitive Operations n 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) # operations currentMax  A[0] 2 for i  1 to n  1 do 2  n if A[i]  currentMax then2(n  1) currentMax  A[i]2(n  1) { increment counter i }2(n  1) return currentMax 1 Total 7n  1

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 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

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 17 Growth Rate of Running Time n 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

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 18 Growth Rates n Growth rates of functions: Linear  n Quadratic  n 2 Cubic  n 3 n In a log-log chart, the slope of the line corresponds to the growth rate of the function

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 19 Constant Factors n The growth rate is not affected by l constant factors or l lower-order terms n Examples 10 2 n  10 5 is a linear function 10 5 n 2  10 8 n is a quadratic function

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 20 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: 2n  10 is O(n) 2n  10  cn (c  2) n  10 n  10  (c  2) Pick c  3 and n 0  10

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 21 Big-Oh Example Example: the function n 2 is not O(n) n 2  cn n  c The above inequality cannot be satisfied since c must be a constant

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 22 More Big-Oh Examples 7n-2 7n-2 is O(n) need c > 0 and n 0  1 such that 7n-2  cn for n  n 0 this is true for c = 7 and n 0 = 1 3n n n n is O(n 3 ) need c > 0 and n 0  1 such that 3n n  cn 3 for n  n 0 this is true for c = 4 and n 0 = 21 3 log n + log log n 3 log n + log log n is O(log n) need c > 0 and n 0  1 such that 3 log n + log log n  clog n for n  n 0 this is true for c = 4 and n 0 = 2

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 23 Big-Oh and Growth Rate n 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) 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 moreYesNo f(n) grows moreNoYes Same growthYes

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 24 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 n Use the smallest possible class of functions Say “ 2n is O(n) ” instead of “ 2n is O(n 2 ) ” n Use the simplest expression of the class Say “ 3n  5 is O(n) ” instead of “ 3n  5 is O(3n) ”

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 25 Asymptotic Algorithm Analysis n The asymptotic analysis of an algorithm determines the running time in big-Oh notation n To perform the asymptotic analysis l We find the worst-case number of primitive operations executed as a function of the input size l We express this function with big-Oh notation n Example: We determine that algorithm arrayMax executes at most 7n  1 primitive operations We say that algorithm arrayMax “runs in O(n) time” n Since constant factors and lower-order terms are eventually dropped anyhow, we can disregard them when counting primitive operations

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 26 Computing Prefix Averages n 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

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 27 Prefix Averages (Quadratic) n 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 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

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 28 Arithmetic Progression The running time of prefixAverages1 is O(1  2  …  n) The sum of the first n integers is n(n  1)  2 l There is a simple visual proof of this fact Thus, algorithm prefixAverages1 runs in O(n 2 ) time

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 29 Prefix Averages (Linear) n 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 integersn s  0 1 for i  0 to n  1 don s  s  X[i]n A[i]  s  (i  1) n return A 1 n Algorithm prefixAverages2 runs in O(n) time

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 30 n properties of logarithms: log b (xy) = log b x + log b y log b (x/y) = log b x - log b y log b xa = alog b x log b a = log x a/log x b n properties of exponentials: a (b+c) = a b a c a bc = (a b ) c a b /a c = a (b-c) b = a log a b b c = a c*log a b n Summations n Logarithms and Exponents n Proof techniques n Basic probability Math you need to Review *

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 31 Relatives of Big-Oh n big-Omega l f(n) is (g(n)) if there is a constant c > 0 and an integer constant n 0  1 such that f(n)  cg(n) for n  n 0 n big-Theta l 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 n little-oh l f(n) is o(g(n)) if, for any constant c > 0, there is an integer constant n 0  0 such that f(n)  cg(n) for n  n 0 n little-omega l f(n) is (g(n)) if, for any constant c > 0, there is an integer constant n 0  0 such that f(n)  cg(n) for n  n 0

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 32 Intuition for Asymptotic Notation Big-Oh l f(n) is O(g(n)) if f(n) is asymptotically less than or equal to g(n) big-Omega l f(n) is (g(n)) if f(n) is asymptotically greater than or equal to g(n) big-Theta l f(n) is (g(n)) if f(n) is asymptotically equal to g(n) little-oh l f(n) is o(g(n)) if f(n) is asymptotically strictly less than g(n) little-omega l f(n) is (g(n)) if is asymptotically strictly greater than g(n)

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 33 Example Uses of the Relatives of Big-Oh f(n) is  (g(n)) if, for any constant c > 0, there is an integer constant n 0  0 such that f(n)  cg(n) for n  n 0 need 5n 0 2  cn 0  given c, the n 0 that satifies this is n 0  c/5  0 5n 2 is  (n) f(n) is  (g(n)) if there is a constant c > 0 and an integer constant n 0  1 such that f(n)  cg(n) for n  n 0 let c = 1 and n 0 = 1 5n 2 is  (n) f(n) is  (g(n)) if there is a constant c > 0 and an integer constant n 0  1 such that f(n)  cg(n) for n  n 0 let c = 5 and n 0 = 1 5n 2 is  (n 2 )

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 34 Best Worst and Average Case n The worst case complexity of the algorithm is the function defined by the maximum number of steps taken on any instance of size n n The best case complexity of the algorithm is the function defined by the minimum number of steps taken on any instance of size n n Each of these complexities defines a numerical function time vs size

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 35 Best Worst and Average Case (cont.)

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 36 Exact Analysis is Hard !! n We have agreed that the best, worst and average case complexity of an algorithm is a numerical function of the size of the instances n However it is difficult to work with exactly because it is typically very complicated n Thus it is usually cleaner and easier to talk about upper and lower bounds of the function n This is where the big O notation comes in

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 37 Formalization of the Concept of Order The next task is to formalize the notion of one function being more difficult to compute than another, subject to the following assumptions: n The argument n defining the instance size is sufficiently large n Positive constant multipliers are ignored.

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 38 Formalization of Concepts (cont) Definition Let f : n Complexity functions will never be negative for usable arguments. n In any case, behavior before no is not significant for the asymptotic mathematical analysis.

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 39 Big O Notation n Definition Let f : n It is said that “g is big-oh of f ”. n We write g = O( f ) or g in O( f ) n The intuition is that g is “smaller” than f ; i.e., that g represents a lesser complexity.

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 40 ,  and  n The value of n0 shown is the minimum possible value (any greater value would also work

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 41 Sequential Search void seqsearch(int n, const keytype S [ ], keytype x, index & location) { location = 1; while (location <= n && S[location] != x) location ++; if (location > n) location=0; } n T(n) = N

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 42 Binary Search void binsearch (int n, const keytype S[], keytype x, index& location) { index low, high, mid; low = 1; high = n; location = 0; while (low < = high && location = = 0){ mid = ∟(low + high)/2 ⌋ ; if (x = = S[mid]) location = mid; else if (x < S[ mid ]) high = mid - 1; else low = mid + 1; }

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 43 Matrix Multiplication void matrixmult (int n, const number A[][], const number B[][], number C[][]) { index i, j, k; for (i=1; j<= n; j++) for (j=1; j<= n; j++) { C[ i ] [ j ] = 0; for (k=1; k<= n; k++) C[ i ] [ j ] = C[ i ] [ j ] + A [ i ] [ k ] * B [ k ] [ j ]; }

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 44 Exchange Sort void exchangesort (int n, keytype S[]) { index i, j; for (i=1;i<= 1; i++) for (j=i+l; j<= n; j++) if (S[j] < S[i]) exchange S[i] and S[j]; }

PSU CS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi 45 Add Array Members number sum (int n, const number S[ ]) { index i; number result; result = 0; for (i = 1; i <= n; i++) result = result + S[i]; return result; }