Algorithm Analysis. Gunnar Gotshalks. AlgAnalysis 1

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1 Algorithm Analysis AlgAnalysis 1

2 How Fast is an Algorithm? 1 Measure the running time» Run the program for many data types > Use System.currentTimeMillis to record the time Worst Time Average Best» Usually take worst to be safe (air traffic control)» Average is difficult to measure probability distribution A B C D E F Input Instance AlgAnalysis 2

3 How fast is an Algorithm? 2 But worst depends upon the size of the input. The more input the longer an algorithm takes. Time Size of input AlgAnalysis 3

4 Measurement Limitations Need to implement and test the algorithm Can only run for limited number of data sets Depends upon hardware and software Depends upon how good the programmer is Need a more general methodology» Take into account all possible inputs» Use high level description of algorithm» Evaluate independent of hardware, software and programmer AlgAnalysis 4

5 Compute execution time 1 Each statement set has no loops > can compute its constant time statement set 1 // exec time is T1 units while (condition 1) { // Loop 1 statement set 2 // exec time is T2 units while (condition 2) { // Loop 2 statement set 3 } // exec time is T3 units statement set 4 } // exec time is T4 units statement set 5 // exec time is T5 units while (condition 3) { // Loop 3 statement set 6 } // exec time is T6 units statement set 7 // exec time is T7 units AlgAnalysis 5

6 Compute execution time 2 Suppose each loop executes Li times, then» Loop 2 executes for L2*T3 units» Loop 1 executes for (T2 + T3*L2 + T4)*L1 units» Loop 3 executes for T6*L3 units The entire program executes for» T1 + (T2 + T3*L2 + T4)*L1 + T5 + T6*L3 + T7 Rearranging we have» T3*L2*L1 + (T2 + T4)*L1 + T6*L3 + (T1 + T5 + T7) The Ti are constants so combine them as follows giving us expression 1» C3*L2*L1 + C2*L1 + C1*L3 + C0 AlgAnalysis 6

7 Compute execution time 3 Now determine L1, L2 and L3 as a function of the size of the input There are three basic choices» The loop does not depend upon the size of the input» The loop depends upon the size of the input in a linear or proportional way» The loop depends upon the size of the input in a logarithmic way AlgAnalysis 7

8 Constant loops The loop does not depend upon the size of the input i! 1 while ( i " 10) { some statements i++ } The loop body is executed a fixed number of times Replace the loop counter Li with a constant k AlgAnalysis 8

9 Linear Loops The loop depends upon the size of input in a linear or proportional way» The loop body executes for every input, every 2'nd, 3'rd or some other proportional amount In the following example it loops n times for a file of length n open file get acharacter while ( acharacter! endoffile) { some statements no get get acharacter } Replace the loop counter Li with n AlgAnalysis 9

10 Logarithmic Loops The loop depends upon the size of the input in a logarithmic way» Every time around the loop the size of the remaining data to be processed is divided by a constant amount» Log base p is the number of times one can divide by p to get to 1 > log base 2 means divide by 2 e.g. binary search > log base 3 means divide by 3, etc. i! 1 ; j! n while ( i < j ) { some statements do not change i or j j! (i + j) / 2 } Replace the loop counter Li with log n AlgAnalysis 10

11 Compute time variation 1 1 AlgAnalysis 11

12 Compute time variation 1 2 We are interested in the general behaviour as a function of n» Let n grow towards infinity» Get the asymptotic behaviour of the algorithm Under this assumption the terms other than C1*n eventually become as small as we like relative to C1*n» C3*k*log n + C2*log n + C1*n + C0 Asymptotically we are left with» C1*n We ignore the constant because we see the algorithm behaves linearly with n» Double the size of the input the algorithm takes twice as long» We say the big O timing of our algorithm is O(n) AlgAnalysis 12

13 Compute time variation 2 We had the following execution time expression for our algorithm» C3*L2*L1 + C2*L1 + C1*L3 + C0 Suppose» L1 = n L2 = n L3 = n Make the substitutions and we get» C3*n*n + C2*n + C1*n + C0 Let n grow towards infinity to get the asymptotic behaviour of the algorithm» C3*n 2 Ignoring the constant C3 we have an O(n 2 ) algorithm AlgAnalysis 13

14 Polynomial Algorithms Big O Big O is a measure of the nesting structure of loops and the work done as a function of the size of the input Here is a list, in increasing complexity, the most common big O time complexities» O(1) no loops, or constant time loops > Typical algorithm: Hashing into a table to find an entry» O(log N) dominant single loop is a divide by 2 on each iteration > Typical algorithm: binary search» O(N) dominant single loop dependent linearly on n > Typical algorithm: sequential search AlgAnalysis 14

15 Polynomial Algorithms Big O 2» O(N log N) dominant looping structure is a nested loop one loop dependent on N the other log N. > Typical algorithm is an efficient sort» O(N P ) dominant looping structure is a sequence of loops nested p deep > Typical algorithm: inefficient sort N 2, matrix multiplication N 3 Polynomial time algorithms are considered to be feasible» Although as P grows, we still can only deal with smaller and smaller problems AlgAnalysis 15

16 Exponential algorithms Big O The following time complexities are exponential due to the explosive nature of the growth of execution time with the size of the input.» O(P N ) dominant looping structure is a sequence of loops nested N deep. Each loop is constant time P.» O(N N ) dominant looping structure is a sequence of loops nested N deep. Each loop is time N. AlgAnalysis 16

17 Exponential algorithms Big O - 2 Algorithms with exponential time complexity are considered to be unfeasible Typical algorithms depend upon permutations and combinations of the input.» Essential have to try all possible orderings of the input, or, at least, a substantial subset > Typical algorithms Travelling salesman shortest path to visit N cities Knapsack packing how to best pack a knapsack with the most objects of varying size. AlgAnalysis 17

18 Typical complexity curves O(n n ) O(n p ) Time O(n) O(log n) O(1) Size of input AlgAnalysis 18

19 Growth table for big O A subset of the table on page 171 of Gooderich & Tomassia, 4 th edition Note 1 about 5 years worth of instructions on a super computer Note 2 About 600,000 times greater than the age of the universe in nanoseconds AlgAnalysis 19

20 Maximum Size Problem The following table shows how much input can be processed in fixed time intervals, with each input being processed in 1 microsecond Taken from Gooderich & Tomassia, p53, 1 st edition A superset of that on page 171, 4 th edition AlgAnalysis 20

21 New Maximum Size Problem The following table shows how much more input can be processed if we use a computer 256 times faster > m is the size of the input for the slower computer Gooderich & Tomassia, p53, 1 st edtion A superset of that on page 171, 4 th edition AlgAnalysis 21

22 When "Inefficient" Algorithms are Better An n**2 algorithm (insert sort) can have a lower setup time than an "n log n" algorithm (quicksort average case).» For low n n**2 is faster than n log n when n < n 0 O(n 2 ) Time O(n log n) Set up n log n Set up n 2 n 0 Size of input AlgAnalysis 22

23 Formal definition of Big O Given functions f(n) and g(n), we say that» f(n) is O(g(n)) if and only if» f(n)! c*g(n) for n > n 0 > c and n 0 are constants > f and g are functions over the non negative integers g(n) f(n) Time f(n)! g(n) most of the time i.e. f(n) is smaller than g(n) within a constant factor n 0 Size of input AlgAnalysis 23

24 Formal definition of Big Omega Given functions f(n) and g(n), we say that» f(n) is! (g(n)) if and only if g(n) is O(f(n))» f(n) " c*g(n) for n > n 0 > c and n 0 are constants > f and g are functions over the non negative integers f(n) g(n) Time f(n)! g(n) most of the time i.e. f(n) is larger than g(n) within a constant factor n 0 Size of input AlgAnalysis 24

25 Formal definition of Big Theta Given functions f(n) and g(n), we say that» f(n) is! (g(n)) if and only if» f(n) is O(g(n)) and f(n) is " (g(n))» c' * g(n) # f(n) # c'' * g(n) for n > n 0 and constants c', c'' c'' * g(n) f(n) c' * g(n) Time f(n)! g(n) most of the time i.e. f(n) = g(n) within a constant factor n 0 Size of input AlgAnalysis 25

26 Space complexity Definition The amount of space an algorithm requires as a function of the size of the input» The amount of storage we need to store the data» Plus the amount of scratch space we need for an algorithm to run > The space needed to keep track of the intermediate results before they can be combined > Similar to your needing some paper on which to write your computations before you can state a result AlgAnalysis 26

27 Space complexity Examples Insert sort and quicksort require O(1) extra space Merge sort requires O(n) extra space Storing a dictionary with an index requires O(n log n) space where n is the number of words in the dictionary AlgAnalysis 27

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