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Chapter 20 Computational complexity

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This chapter discusses n Algorithmic efficiency n A commonly used measure: computational complexity n The effects of algorithm choice on execution time.

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Measuring program efficiency n There are many factors that affect execution time: u Hardware u Operating system u System environment u Programming language and compiler u Run-time system or interpreter u Algorithm(s) u Data on which the program is run

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Time complexity n We limit our attention to the algorithm(s) and data. n Our measure of program efficiency is called computational complexity or time complexity. n Each instance of a problem has some inherent size. n Execution time often depends on size. n For the most part, we want to know the worst case behavior of an algorithm. n We measure time cost by the number of primitive steps the algorithm performs in solving the problem.

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Time complexity (cont.) n Time complexity of a method M is a function of t M from the natural numbers N to the positive reals R +. t M : N-> R + such that t M (n) is the maximum number of steps for method M to solve a problem of size n.

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Comparing different algorithms n Let f and g be functions from the natural numbers to the positive reals, n f,g: N-> R + We say f is O-dominated by g, and write f g, provided: u there is a positive integer n o, and u there is a positive real c, such that u for all n n o, f(n) c g(n).

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Time complexity

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Relation observations n f, g, h, f, g, h : N-> R + n The relation is reflexive: for any function f, f f. n The relation is transitive:f g and g h imply f h. n The relation is not antisymmetric: there are functions such that f g and g f, but f g. n The relation is not total: there are functions such that neither f g nor g f.

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Relation observations (cont.) n If f g and g f, f and g are said to have the same magnitude; f g. n If f g but not g f, then f g. n [f + g](n) = f(n) + g(n). n [f g](n) = f(n) g(n). n [max(f,g)](n) = max(f(n), g(n)).

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Computational rules n If c 1 and c 2 are positive real constants, then f c 1 f+c 2. n If f is a polynomial function, f(n) = c k n k +c k-1 n k-1 + … +c0, where c k > 0, then f n k. n If f g and f g,then f f g g. n If f g and f g,then f +f max(g,g). Also f +g max(f,g). n If a and b are > 1, log a n log b n. n 1 log a n n nlog a n n 2 n 3 n 4... 2 n.

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Complexity classes n Set of functions that are O- dominated by f. O(f) = { g | g f } (big oh of f) n Set of functions that O-dominate f. (f) = { g | f g } (omega of f) n Set of functions that have the same magnitude as f. (f) = { g | g f } (theta of f)

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Complexity classes (cont.) n If t M = c, then M is constant ( (1)). n If t M = n, then M is linear ( (n)). n If t M = n 2, then M is quadratic ( (n 2 )). n If not t M = n k, then M is exponential ((n k )).

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Problem complexity n Some problems have a known complexity. i.e. sorting can be done in n log n time. n Some problems are known to be exponential. They are called intractable. n Some problems are unsolvable. n Observing the complexity of a problem is generally difficult because it involves proving something about all possible methods for solving the problem.

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Cost of a method n To determine the time-cost of a method, we must count the steps performed in the worst case. n A method with constant time complexity needs to use only a fixed amount of its data. n Any method that, in the worst case, must examine all its data is at least linear. n Methods that are better than linear typically require the data to have some organization or structure.

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Cost of a statement n A simple statement requires a constant number of steps. n The number of steps of a method invocation is the number of steps in the method the statement invokes. n The worst case number of steps required by a conditional is the maximum of the number of steps required by each alternative.

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Cost of a statement (cont.) n A sequence of statements takes the maximum of the each statements steps. n Time-cost of a loop is the number of steps required by the loop body multiplied by the number of times the loop body is executed. n Recursion time-cost is determined by the depth of recursion.

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Example public double average(StudentList students) { int i, sum, count; count = students.size(); sum = 0; i = 0; while ( i < count) { sum = sum + students.get(i).finalExam(); i = i+1; } return (double)sum / (double)count; } n This method is (n).

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Example (cont.) Suppose get(i) is also linear (requires : ic 1 +c 0 steps). value of i: 012…n-1 steps: c 0 c 1 +c 0 2c 1 +c 0 …(n-1)c 1 +c 0 n The method is quadratic (n 2 ).

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Example boolean hasDuplicates (List list) { int i; int j; int n; boolean found; n = list.size(); found = false; for (i = 0; i < n-1 && !found; i=i+1) for (j = i+1; j < n && !found; j=j+1) found = list.get(i).equals( list.get(j)); return found; }

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Example (cont.) n The outer loop is performed n-1 times. The number of iterations depends on the second loop. value of i: 012…n-3n-2 steps: n-1n-2n-3…21 n Summing these altogether, we get ½(n 2 -n). n The method is quadratic (n 2 ).

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Analyzing different algorithms n Given a list of integers, find the maximum sum of the elements of a sublist of zero of more contiguous elements. listsublistmaxsum (-2,4,-3,5,3,-5,1)(4,-3,5,3)9 (2,4,5)(2,4,5)11 (-1,-2,3)(3)3 (-1,-2,-3)()0

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maxSublistSum int maxSublistSum (IntegerList list) The maximum sum of a sublist of the given list. if list.getInt(i)>0 for some i, 0<=i<list.size(), then max{ list.getInt(i)+…+list.getInt(j) | 0<=i<=j<list.size() } else 0

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maxSublistSum (cont.)

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n This makes it (n 3 ). It can be improved to (n 2 ) with a little adjustment.

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Recursive method n Base cases are lists of length 0 and 1. n If we have longer list, we divide it in half and consider possible cases. (divide and conquer) n There are 3 possible cases: u The sublist with the max sum is in the left half of the list (x 0 … x mid ) u The sublist with the max sum in the right half of the list (x mid+1 … x n-1 ) u The sublist with the max sum overlaps the middle of the list (includes x mid and x mid+1 ).

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maxSublistSum private int maxSublistSum (IntegerList list) The maximum sum of a sublist of the given list. That is, the maximum sum of a sublist of(list.get(first), …, list.get(last)). require: 0 <= first <= last < list.size() n The principle method goes something like this: int maxSublistSum (IntegerList list) { if (list.size() == 0) return 0; else return maxSublistSum(List,0,list.size()-1); }

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n The method is (n log n).

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Iterative method n Having found the maximum sum in the first i of the list, we look at i+1.

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Iterative method n There are 2 possibilities for the maximum sum sublist of the first i+1 elements: u it doesnt include x i, in which case it is the same as the maximum sum sublist of the first i elements. u it does include x i. n There are 2 possibilities if the maximum sum sublist ends with x i : u It consists only of x i. u It includes x i-1, in which case it is x i appended to the maximum sum sublist ending with x i-1.

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n The method is (n).

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Weve covered n Time complexity. n Comparing time-cost functions. n Computing time-cost for simple examples.

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Glossary

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O(1) – constant time The time is independent of n O(log n) – logarithmic time Usually the log is to the base 2 O(n) – linear time O(n*logn)

O(1) – constant time The time is independent of n O(log n) – logarithmic time Usually the log is to the base 2 O(n) – linear time O(n*logn)

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