Week 12: Running Time and Performance

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Lorena Chandler
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1 Week 12: Running Time and Performance 1

Most of the problems you have written in this class run in a few seconds or less Some kinds of programs can take much longer: Chess algorithms (Deep Blue) Routing and

2 Most of the problems you have written in this class run in a few seconds or less Some kinds of programs can take much longer: Chess algorithms (Deep Blue) Routing and scheduling algorithms (Traveling Sales Person, Purdue classroom scheduling) Medical imaging (massive amounts of data) Graphics processing (Pixar: Toy Story, Incredibles) 2

3 You can time the program What if it takes 1000 years to run? You can look at the program What are you looking for? Does the input matter? Sorting 100,000 numbers must take longer than sorting 10 numbers, right? 3

A sequence of steps used to solve a problem In CS, we take an algorithm and convert it into code -- Java in this class The study of algorithms is intended to: Make

4 A sequence of steps used to solve a problem In CS, we take an algorithm and convert it into code -- Java in this class The study of algorithms is intended to: Make sure that the code you write gets the right answer Make sure that the code runs as quickly as possible Make sure that the code uses a reasonable amount of memory 4

You want to find the definition of the word rheostat in the dictionary The Oxford English Dictionary has 301,100 main entries What if you search for the word rheostat by starting at the beginning and

5 You want to find the definition of the word rheostat in the dictionary The Oxford English Dictionary has 301,100 main entries What if you search for the word rheostat by starting at the beginning and checking each word to see if it is the one we are interested in Given a random word, you ll have to check about 150,550 entries before you find it Alternatively, you could jump to the correct letter of the alphabet, saving tons of time Searching for zither takes forever otherwise 5

6 Write a program to sum up the numbers between 1 and n Easy, right? Let us recall that a faster way might be to use the summation equation: n i = n( n+ 2 i= 1 1) 6

7 Memory usage is a problem too If you run out of memory, your program can crash Memory usage can have serious performance consequences too 7

the order of 100 times larger and 100 times

CPU Fast and expensive RAM Primary memory

relatively expensive Hard Drive Secondary

8 Remember, there are multiple levels of memory on a computer Each next level is on the order of 100 times larger and 100 times slower Size 100X 100X Cache Actually on the CPU Fast and expensive RAM Primary memory for a desktop computer Pretty fast and relatively expensive Hard Drive Secondary memory for a desktop computer Slow and cheap Speed 100X 100X 8

9 If you can do a lot of number crunching without leaving cache, that will be very fast If you have to fetch data from RAM, that will slow things down If you have to read and write data to the hard drive (unavoidable with large pieces of data like digital movies), you will slow things down a lot 9

10 Memory can be easier to estimate than running time Depending on your input, you will allocate a certain number of objects, arrays, and primitive data types It is possible to count the storage for each item allocated A reference to an object or an array costs an additional 4 bytes on top of the size of the object 10

11 A graph is a set of nodes and edges It s a really simple way to represent almost any kind of relationship B D A 8 C 3 4 The numbers on the edges could be distances, costs, time, whatever 16 E 11

12 There s a classic CS problem called the Traveling Salesperson Problem (TSP) Given a graph, find the shortest path that visits every node and comes back to the starting point Like a lazy traveling salesperson who wants to visit all possible clients and then return home as soon as possible 12

13 Strategies: Always visit the nearest neighbor next Randomized approaches Try every possible combination How can you tell what s the best solution? 13

14 We are tempted to always take the closest neighbor, but there are pitfalls 14

15 In a completely connected graph, we can try any sequence of nodes If there are n nodes, there are (n 1)! tours For 30 cities, 29! = If we can check 1,000,000,000 tours in one second, it will only take about 20,000 times the age of the universe to check them all We will (eventually) get the best answer! 15

16 This and similar problems are useful problems to solve for people who need to plan paths UPS Military commanders Pizza delivery people Internet routing algorithms No one knows a solution that always works in a reasonable amount of time We have been working on it for decades 16

17 The problem is so hard that you can win money if you solve it Clay Mathematics Institute has offered a $1,000,000 prize You can do one of two things to collect it: Find an efficient solution to the general problem Prove that it is impossible to find an efficient solution in some cases 17

18 How do we measure algorithm running time and memory usage in general? We want to take into account the size of the problem (TSP with 4 cities is certainly easier than TSP with 4,000 cities) Some algorithms might work well in certain cases and badly in others How do we come up with a measuring scheme? 18

19 Ideally, we want to classify both problems and algorithms We want to figure out which problems take a really long time to solve We want to figure out how we can compare one algorithm to another We want to prove that some algorithm is the best possible algorithm to solve a particular problem 19

20 Computer scientists use Big-Oh notation to give an indication of how much time (or memory) a program will take Big-Oh notation is: Asymptotic Focuses on the behavior as the size of input gets large Worst-Case Focuses on the absolute longest the program can take 20

21 Add up the operations done by the following code: int sum = 0; for( int i = 0; i < n; i++ ) sum += i; System.out.println( Sum: + sum); Initialization: 1 operation Loop: 1 initialization + n checks + n increments + n additions to sum = 3n + 1 Output: 1 operation Total: 3n

22 We could express the time taken by the code on the previous slide as a function of n: f(n) = 3n + 3 This approach has a number of problems: We assumed that each line takes 1 time unit to accomplish, but the output line takes much longer than an integer operation This program is 4 lines long, a longer program is going to have a very messy running time function We can get nit picky about details: Does sum += i; take one operation or two if we count the addition and the store separately? 22

In short, this way of getting a running time function is almost useless because: It cannot be used to give us an idea of how long the program really runs in seconds It is complex and unwieldy The

23 In short, this way of getting a running time function is almost useless because: It cannot be used to give us an idea of how long the program really runs in seconds It is complex and unwieldy The most important thing about the analysis of the code that we did is learning that the growth of the function should be linear (3n+3) A general description of how the running time grows as the input grows would be useful 23

Enter Big Oh notation Big Oh simplifies a complicated running time function into a simple statement about its worst case growth rate All constant coefficients are

24 Enter Big Oh notation Big Oh simplifies a complicated running time function into a simple statement about its worst case growth rate All constant coefficients are ignored All low order terms are ignored 3n + 3 is O(n) Big Oh is a statement that a particular running time is no worse than some function, with appropriate constants 24

25 147n 3 + 2n 2 + 5n is O(n 3 ) 15n 2 + 6n + 7log n is O(n 2 ) log n is O(log n) Note: In CS, we use log 2 unless stated otherwise 25

26 How long does it take to do multiplication by hand? 123 x Let s assume that the length of the numbers is n digits (n multiplications + n carries) x n digits + (n + 1 digits) x n additions = 3n 2 + n Running time: O(n 2 ) 26

27 How do we find the largest element in an array? int largest = array[0]; for( int i = 1; i < array.length; i++ ) if( array[i] > largest ) largest = array[i]; System.out.println( Largest: + largest); Running time: O(n) if n is the length of the array What if the array is sorted in ascending order? System.out.println( Largest: + array[array.length-1]); Running time: O(1) 27

28 Given two n x n matrices A and B, the code to multiply them is: double[][] c = new double[n][n]; for( int i = 0; i < n; i++ ) for( int j = 0; j < n; j++ ) for( int k = 0; k < n; k++ ) c[i][j] += a[i][k]*b[k][j]; Running time: O(n 3 ) Is there a faster way to multiply matrices? Yes, but it s complicated and has other problems 28

29 Here is some code that sorts an array in ascending order What is its running time? for( int i = 0; i < array.length; i++ ) for( int j = 0; j < array.length - 1; j++ ) if( array[j] > array[j + 1] ) { int temp = array[j]; array[j] = array[j + 1]; array[j + 1] = temp; } Running time: O(n 2 ) 29

30 There is nothing better than constant time (O(1)) Logarithmic time (O(log n)means that the problem can become much larger and only take a little longer Linear time (O(n))means that time grows with the problem Quadratic time (O(n 2 )) means that expanding the problem size significantly could make it impractical Cubic time (O(n 3 )) is about the reasonable maximum if we expect the problem to grow Exponential and factorial time mean that we cannot solve anything but the most trivial problem instances 30

31 Looking at the code and determining the order of complexity is one technique Sometimes this only captures worst-case behavior The code can be complicated and difficult to understand Another possibility is running experiments to see how the running time changes as the problem size increases 31

32 The doubling hypothesis states that it is often possible to determine the order of the running time of a program by progressively doubling the input size and measuring the changes in time 32

33 Let s use the doubling hypothesis to test the running time of the program that finds the largest element in an unsorted array Size Time Ratio ms ms ms ms 1.48 Looks like a decent indication of O(n) 33

34 Let s check matrix multiplication Size Time Ratio ms ms ms ms 8.81 A factor of 8 is expected for O(n 3 ) Not too bad 34

Let s try bubble sort Size Time Ratio 1024 8.648306 ms - 2048 28.331604 ms 3.28 4096 108.931582 ms 3.84 8192 404.

35 Let s try bubble sort Size Time Ratio ms ms ms ms 3.71 Success! O(n 2 ) is supposed to give a ratio of 4 We should have done more trials to do this scientifically 35

What did we talk about last time? Finished hunters and prey Class variables Constants Class constants Started Big Oh notation

Week 12 - Friday What did we talk about last time? Finished hunters and prey Class variables Constants Class constants Started Big Oh notation Here is some code that sorts an array in ascending order