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Analysis of Algorithms

Analysis of Algorithms. Input. Algorithm. Output. An algorithm is a step-by-step procedure for solving a problem in a finite amount of time. Running Time (§3.1) . Most algorithms transform input objects into output objects.

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Analysis of Algorithms

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  1. Analysis of Algorithms Input Algorithm Output An algorithm is a step-by-step procedure for solving a problem in a finite amount of time.

  2. Running Time (§3.1) • Most algorithms transform input objects into output objects. • The running time of an algorithm typically grows with the input size. • Average case time is often difficult to determine. • We focus on the worst case running time. • Easier to analyze • Crucial to applications such as games, finance and robotics Analysis of Algorithms

  3. 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 Analysis of Algorithms

  4. 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

  5. Theoretical Analysis • Uses a high-level 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

  6. 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 (§3.2) • 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

  7. 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

  8. 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

  9. Seven Important Functions (§3.3) • Seven functions that often appear in algorithm analysis: • Constant  1 • Logarithmic  log n • Linear  n • N-Log-N  n log n • Quadratic  n2 • Cubic  n3 • Exponential  2n • In a log-log chart, the slope of the line corresponds to the growth rate of the function Analysis of Algorithms

  10. Basic computations performed by an algorithm 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 in the RAM model 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

  11. 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) # operations currentMaxA[0] 2 fori1ton 1do 2n ifA[i]  currentMaxthen 2(n 1) currentMaxA[i] 2(n 1) { increment counter i } 2(n 1) returncurrentMax 1 Total 8n 2 Counting Primitive Operations (§3.4) Analysis of Algorithms

  12. 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 worst-case time of arrayMax.Thena (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

  13. 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

  14. 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

  15. Big-Oh Notation (§3.4) • 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

  16. Big-Oh Example • 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

  17. 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

  18. 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

  19. Big-Oh Rules • 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

  20. Asymptotic Algorithm Analysis • The asymptotic analysis of an algorithm determines the running time in big-Oh notation • To perform the asymptotic analysis • 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 • Example: • We determine 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

  21. 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 Analysis of Algorithms

  22. 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

  23. 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

  24. 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

  25. Math you 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

  26. 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)  c•g(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 c’•g(n)  f(n)  c’’•g(n) for n  n0 Analysis of Algorithms

  27. 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

  28. 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)  c•g(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)  c•g(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) <c•g(n) for n  n0 Let c = 5 and n0 = 1 Analysis of Algorithms

  29. Notación Asintótica • Q, O, W, o, w • Se usa para describir los tiempos de ejecución de algoritmos • En lugar de usar tiempos de ejecución exactos, se usa Q(n2) • Definido para funciones cuyo dominio es el conjunto de números naturales • Determina conjuntos de funciones, en la práctica se usa para comparar dos funciones Analysis of Algorithms

  30. notación -  Para una función dada g(n), definimos (g(n)) el conjunto de funciones (g(n)) = { f(n): existen constantes positivas c1, c2 and n0 tal que 0 c1g(n) f(n)c2g(n), para todo n n0} Decimos que g(n) es una cota ajustada asintótica para f(n) Analysis of Algorithms

  31. Ejemplo • 10n2-3n = Q(n2) • Para qué constantes n0, c1, y c2 funciona? • Hacemos c1 un poco menor que el coeficiente predominante, y c2 un poco mayor • Para comparar órdenes de crecimiento, miramos los términos predominantes Analysis of Algorithms

  32. Notación - O Para una función dada g(n), definimos O(g(n))el conjunto de funcionesO(g(n)) = {f(n): existen constantes positivasc y n0 tal que 0 f(n)cg(n) para todo n n0} Decimos queg(n) es una cota superior asintótica para f(n) Analysis of Algorithms

  33. notación -  Para una función dada g(n), definimos (g(n)) el conjunto de funciones (g(n)) = {f(n): existen constantes positivasc y n0 tal que 0 cg(n)f(n) para todo n n0} Decimos queg(n) es una cotas inferior asintótica para f(n) Analysis of Algorithms

  34. Relación entre Q, W, O • Para dos funciones cualesquiera g(n) and f(n), f(n)=(g(n)) si y solo si f(n)=O(g(n)) y f(n)=(g(n)). • Por tanto, (g(n)) =O(g(n)) ÇW(g(n)) Analysis of Algorithms

  35. Tiempos de ejecución • “El tiempo de ejecución es O(f(n))” Þ El peor caso es O(f(n)) • “El tiempo de ejecución es W(f(n))” Þ El mejor caso es W(f(n)) • Se puede decir “El pero caso de tiempo de ejecución es W(f(n))” • Significa que el peor caso de ejecución es dado por alguna función desconocida g(n)ÎW(f(n)) Analysis of Algorithms

  36. Ejemplo • Ordenación por inserción toma un tiempo Q(n2) en el peor caso, por eso la ordenación (como problema) es O(n2) • Cualquier algoritmo de ordenación debe comprobar cada elemento, por ello la ordenación es W(n) • En efecto, usando la ordenación por mezcla, la ordenación es Q(n lg n) en el peor caso Analysis of Algorithms

  37. Notación asintótica en Ecuaciones • Se usa para reemplazar expresiones que contienen términos de bajo orden • Ejemplo, 4n3 + 3n2 + 2n + 1 = 4n3 + 3n2 + (n) = 4n3 + (n2) = (n3) • En ecuaciones, (f(n)) siempre significa una funciónanónima g(n) Î(f(n)) • En el ejemplo anterior, (n2) representa 3n2 + 2n + 1 Analysis of Algorithms

  38. Notación-o Para una función dada g(n), definimos o(g(n)) al conjunto de funciones o(g(n)) = {f(n): para una constante positiva c > 0, existe una constante n0 > 0 tal que 0 f(n)<cg(n)para todo n n0} f(n) se hace insignificante relativo a g(n)cuando n tiende a infinito: lim [f(n) / g(n)] = 0 n Decimos que g(n) es una cota superior para f(n)que no es asintótica. Analysis of Algorithms

  39. notación-w Para una función dada g(n), definimos w(g(n)) al conjunto de funciones w(g(n)) = {f(n): para una constante positiva c > 0, existe una constante n0 > 0 tal que 0 cg(n)<f(n)para todo n n0} f(n) se hace grande relativo a g(n)cuando n tiende a infinito : lim [f(n) / g(n)] =  n Decimos que g(n) es una cota inferior para f(n)que no es asintótica. Analysis of Algorithms

  40. Medición de Velocidad

  41. Medición de la Velocidad Medir el tiempo de un algoritmo en un ordenador. Qué necesitamos?

  42. Necesidades lenguaje de programación -> programa ordenador compilador javac datos a usar Peor, mejor y promedio de los casos reloj

  43. Tiempo en Java long startTime = System.currentTimeMillis(); // devuelve tiempo en milisegundos desde 1/1/1970 GMT // codigo long elapsedTime = System.currentTimeMillis() - startTime; Analysis of Algorithms

  44. Problema Precisión del reloj: asumimos 100 milisegundos Repetimos el trabajo varias veces para conseguir que el tiempo total sea >= 1 seg con lo que se consigue una precisión de 10%. Analysis of Algorithms

  45. Tiempo preciso long startTime = System.currentTimeMillis(); long counter; do { counter++; doSomething(); } while (System.currentTimeMillis() - startTime < 1000) long elapsedTime = System.currentTimeMillis() - startTime; float timeForMethod = ((float) elapsedTime)/counter; Analysis of Algorithms

  46. Esquema del programa long startTime = System.currentTimeMillis(); long counter; do { counter++; // inicializar a[] InsertionSort.insertionSort(a); } while (System.currentTimeMillis() - startTime < 1000) long elapsedTime = System.currentTimeMillis() - startTime; float timeForMethod = ((float) elapsedTime)/counter; Analysis of Algorithms

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