1 / 10

RUNNING TIME

RUNNING TIME. 10.4 – 10.5 (P. 551 – 555). RUNNING TIME. analysis of algorithms involves analyzing their effectiveness need way of "guessing" how fast the algorithm will run without actually programming it and running it

leontyne
Download Presentation

RUNNING TIME

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. RUNNING TIME 10.4 – 10.5 (P. 551 – 555)

  2. RUNNING TIME • analysis of algorithms involves analyzing their effectiveness • need way of "guessing" how fast the algorithm will run without actually programming it and running it • to determine the speed, you must consider running time regardless of size of data set used for input • this is known as time complexity, and we often want to consider the worst case

  3. What is Big-O Notation • goal is to identify the dominant factor in determining the execution time of our algorithm • in most cases, this is some function of the size of the input data, denoted by n • goal then is to see how the execution time varies with size of data set • we consider the steps of an algorithm may be: • constant (will not change regardless of n) example: x[0] = initvalue; x[n-1] = initvalue; • a function of n (changes with input size) example: for (int i = 1; i < n; i++) x[i] = x[i-1] * factor; (number of steps executed changes with size of n)

  4. BIG-O Notation So, How Does this Relate to Big-O? • Big-O is the notation used for analyzing the upper bound of an algorithm • in the above examples, you can easily count the steps, but this is not always the case (sometimes it is almost impossible to tell how many steps will be taken) • thus, we usually talk about best case, average case or worst cast; • Big-O most often refers to worst case

  5. BIG-O Notation (Definition): • f(n) = O(g(n)) if and only if there exists positive constants C and n0 such that: f(n) <= Cg(n) for all n >= n0 • in other words: for all sufficiently large n, g(n) is an upper-bound for f • Explanation: • we are trying to predict the "time" of the algorithm using a function of the input size (n); • in other words, trying to determine f(n) • Big-O tries to determine the upper-bound for f(n) using another function, g(n) • bottom line: Big-O is used for stating the upper bound of the running time of an algorithm; the upper-bound ignores constants since for presumable large n, constants get lost anyway

  6. BIG-O NOTATION • O(1): constant time; doesn't change with problem size O(logn): logarithmic; very slow growing with problem size O(n): linear; increases at same rate as the problem size O(nlogn): More than linear, but not by much O(n2): Quadratic; when size doubles, time quadruples O(n3): Cubic; when size doubles, time increases eightfold O(2n): Exponential O(n!): factorial; HUGE increase in time! • Steps for determining complexity: • 1) Break the algorithm down into steps and analyze the complexity of each • Example: with a loop, analyze the body first and see how many times it executes • 2) Look for for-loops. They are easiest to analyze - give a clear upperbound • 3) Look for loops that operate over an entire data structure

  7. EXAMPLES OF BIG-O • Example 1: for (int i = 0; i < n; i++) x[i] = 0; • Time Complexity: O(n) • running time is directly related to the size of n loop will run n times, • Example 2: Linear Search: int LinearSearch(int a[], int listSize, int item) { for (int pos=0; pos < listSize; pos++) { if (a[pos] == item) return pos; } return -1; } - Time Complexity: O(n)

  8. EXAMPLES OF BIG-O • ) Example for Binary Search: int BinarySearch(int a[], int first, int last, int item) { if (last < First) return -1; else { int middle = (last + first) / 2; if (item == a[middle]) return middle; else (if a[middle] > item) return BinarySearch(a,first,middle-1,item); else return BinarySearch(a,middle+1,last,item); } }

  9. BIG-O OF BINARY SEARCH • list is halved each time BS is called • maximum number of comparisons: • 1) if n = 1, algorithm invoked 2 times • 2) if n > 1, algorithm invoked 2m times • where m is size of sequence being searched • 3) thus, total number of invocations: an = 1 + an/2 which is called the recurrance relation • 4) Thus, by solving the recurrance relation, we get: • a2k = 1 + a2k-1 • 2k-1 < n <= 2k • k-1 < log n <= k • so, Algorithm complexity: O(log n) • [see discrete textbook for more details]

  10. QUESTIONS? • Read HASHING Section (482-486)

More Related