Fundamentals of the Analysis of Algorithm Efficiency

Transcription

1 Fundamentals of the Analysis of Algorithm Efficiency DR. JIRABHORN CHAIWONGSAI ดร.จ ราพร ไชยวงศ สาย D E PA R T M E N T O F C O M P U T E R E N G I N E E R I N G S C H O O L O F I N F O R M AT I O N A N D C O M M U N I C AT I O N T E C H N O L O GY U N I V E R S I T Y O F P H AYA O

2 Analysis of algorithms Correctness Time efficiency o How fast an algorithm in question runs Space efficiency o Extra space the algorithm requires Speed can be achieve much more spectacular progress than space Focus on time efficiency 2

3 Analysis framework Measuring an Input s Size All algorithms run longer on larger inputs Ex. take longer to sort larger arrays Unit for measuring running time Time efficiency is analyzed by determining the number of repetitions of the basic operation as a function of input size basic operation is the most important operation of the algorithm Contribute the total running time Compute the number of times that the basic operation is executed The most consuming operation in the algorithm 3

4 Ex.1 Sorting algorithm Work by comparing elements (keys) of a list being sorted with each other Basic operation =? Input size =? 4

5 Input size and basic operation examples Problem Searching for key in a list of n items Input size Number of list s items, i.e. n Basic operation Key comparison Multiplication of two matrices Matrix dimensions or total number of elements Multiplication of two numbers Checking primality of a given integer n Typical graph problem n size = number of digits (in binary representation) #vertices and/or edges Division Visiting a vertex or traversing an edge 5

6 Unit for measuring running time Basic operation: the operation that contributes most towards the running time of the algorithm input size running time T(n) c op C(n) execution time for basic operation Number of times basic operation is executed Let c op be the execution time of an algorithm s basic operation C(n) be the number of times this operation needs to be executed for this algorithm 6

7 Unit for measuring running time (cont.) Why do we use? c op is not always easy to assess How much faster would this algorithm run on a machine that is ten times faster than the one we have? Assume that a addition operation requires a cycle to execute We need 99 addition operations Compare the clock rate of a CPU between running at 1.0 GHz and 2.0 GHz 7

8 Unit for measuring running time (cont.) What about C(n)? Homework1 Assume that How much longer will the algorithm run if we double its input size? 8

9 Order of Growth 9

10 Order of Growth (cont.) time 10

11 Order of Growth (cont.) 11

12 Best-case, Average-case, Worst-case For some algorithms efficiency depends on form of input: Worst case: C worst (n) = maximum over inputs of size n Best case: C best (n) = minimum over inputs of size n Average case: C avg (n) = average over inputs of size n Number of times the basic operation will be executed on typical input NOT the average of worst and best case Expected number of basic operations considered as a random variable under some assumption about the probability distribution of all possible inputs 12

13 Example2: Sequential search Worst case: Best case: Average case: 13

14 Example2: Sequential search (cont.) C worst (n) = the algorithm makes the largest number of comparisons among all possible inputs of size n For any instance of size n, the running time will not exceed C worst (n) C worst (n) =? 14

15 Example2: Sequential search (cont.) C best (n) = the algorithm runs the fastest among all possible inputs of size n The best-case efficiency is not nearly as important as the worst-case C best (n) =? 15

16 Example2: Sequential search (cont.) To analyze the average-case efficiency, we must make some assumption about possible inputs of size n 16

17 Example2: Sequential search (cont.) (a) The probability of successful search = (b) The probability of the first match occurring in the -th position of the list is the same for every In case of successful search, the probability of the first match occurring in the -th position of the list is for every C avg (n) =? In case of unsuccessful search, the number of the comparisons is with the probability 17

18 Example2: Sequential search (cont.) If If = 1 (the search must be successful), (the search must be unsuccessful), 18

19 Best-case, average-case, worst-case 19

20 Best-case, average-case, worst-case Average-case efficiency is considerably more difficult than the worst-case and best-case efficiencies The average-case efficiency can t be obtained by taking the average of the worst-case and the best-case efficiencies 20