Download presentation

Presentation is loading. Please wait.

Published byAvery Hayes Modified over 4 years ago

1
Chapter 20 Computational complexity

2
This chapter discusses n Algorithmic efficiency n A commonly used measure: computational complexity n The effects of algorithm choice on execution time.

3
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

4
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.

5
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.

6
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).

7
Time complexity

8
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.

9
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)).

10
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.

11
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)

12
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 )).

14
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.

15
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.

16
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.

17
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.

18
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).

19
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 ).

20
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; }

21
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 ).

22
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

23
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

24
maxSublistSum (cont.)

26
n This makes it (n 3 ). It can be improved to (n 2 ) with a little adjustment.

27
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 ).

28
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); }

32
n The method is (n log n).

33
Iterative method n Having found the maximum sum in the first i of the list, we look at i+1.

34
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.

36
n The method is (n).

37
Weve covered n Time complexity. n Comparing time-cost functions. n Computing time-cost for simple examples.

38
Glossary

Similar presentations

Presentation is loading. Please wait....

OK

Algorithm Analysis (Big O)

Algorithm Analysis (Big O)

© 2018 SlidePlayer.com Inc.

All rights reserved.

By using this website, you agree with our use of **cookies** to functioning of the site. More info in our Privacy Policy and Google Privacy & Terms.

Ads by Google

Ppt on standing order act materials Marketing mix ppt on sony picture Ppt on demographic structure of neighbourhood Maths ppt on limits and derivatives Ppt on wireless technology Ppt on water chemistry Ppt on construction company profile Ppt on natural numbers symbol Ppt on revolution of the earth and seasons video Ppt on bluetooth download