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1 Centre for Applied Internet Research Centre for Applied Internet Research www.cair-uk.org

2 Centre for Applied Internet Research How hard can it be? (A gentle overview of complexity theory and the limitations of computers) Professor Vic Grout Director of the Centre for Applied Internet Research (CAIR) Glyndŵr University, North Wales v.grout@glyndwr.ac.uk Inaugural Professorial Lecture, Glyndŵr University, 25 th June 2009

3 Centre for Applied Internet Research Two Apologies! This talk is aimed primarily at a non-specialist audience … … but there is some maths/computing in it … feel free to ignore it! For the mathematics/computer science purists … … some liberties have been taken with notation, terminology, simplification, etc. … please ignore it!

4 Centre for Applied Internet Research What is Complexity? Basically, some things we might want to do with a computer are a bit more awkward than they might seem! They can take ‘a long time’ We generally term anything we might want to do with a computer as a ‘problem’ (to be solved). In fact, the complexity of a problem is just one way in which it might be awkward. Some things can’t be done at all! We’ll come back to this, but first … … some ‘problems’ … … then, an argument … … then an example …

5 Centre for Applied Internet Research What is a Problem? Valid problems: Calculate 2 x 4 + 9 – 3 If 5 – a = 2 what’s a? Find the largest from 5,7,1,4,8,5,2,4,8,5,2,5,6,2,4,3,6,7,7,6,5,4 Sort 25,44,66,72,12,45,56,90,45,69,11,10,12,42,88 into order Arrange 1,2,3,4,5,6,7,8,9 into a ‘magic square’ What’s the quickest way to get to Paris? Invalid problems: What’s the meaning of life, the universe and everything? What will this week’s lottery numbers be?

6 Centre for Applied Internet Research It doesn’t really matter, does it? ‘Moore’s Law’ suggests (broadly speaking) that computing ‘power’ approximately doubles approximately every two years Two objections: It may not be true any more It doesn’t help anyway! Source: www.indybay.org

7 Centre for Applied Internet Research The ‘Travelling Salesman Problem’ (TSP) Imagine a sales rep moving from town to town, trying to sell their wares … Presumably, some ways are better than others?

8 Centre for Applied Internet Research The ‘Travelling Salesman Problem’ (TSP) The pedlar has to decide where to start …... then which town to visit in each turn until they get back to the start

9 Centre for Applied Internet Research Clearly, there are lots of choices … … but which is the best? The ‘Travelling Salesman Problem’ (TSP)

10 Centre for Applied Internet Research Surprisingly, no-one has found a much better way than … … trying out each possible route in turn! The ‘Travelling Salesman Problem’ (TSP)

11 Centre for Applied Internet Research So how complicated (complex) is that? (How many routes are there?) Well, consider these 16 towns … The ‘Travelling Salesman Problem’ (TSP)

12 Centre for Applied Internet Research There are 16 choices of the starting town That leaves 15 choices for the second town So there are 16 x 15 = 240 choices for the first two towns 14 choices for the third So there are 16 x 15 x 14 = 3,360 choices for the first three towns 13 choices for the fourth : 2 choices for the last but one 1 choice for the very last So there are 16 x 15 x 14 x 13 x … x 2 x 1 = 20,922,789,888,000 choices for visiting all 16 towns ( = 16! ‘sixteen factorial’ ) The ‘Travelling Salesman Problem’ (TSP)

13 Centre for Applied Internet Research 1!= 1Imagine 1 metre from here 2!= 22m 3!= 66m 4!= 24Fellows Bar 5!= 120Wrexham FC 6!= 720Wrexham 7!= 5,040Rhos 8!= 40,320Shrewsbury 9!= 362,880Southampton 10!= 3,628,800Istanbul 11!= 39,916,800Anywhere on Earth (and back) 12!= 479,001,600The Moon (and beyond) Factorials

14 Centre for Applied Internet Research There are 16 choices of the starting town That leaves 15 choices for the second town So there are 16 x 15 = 240 choices for the first two towns 14 choices for the third So there are 16 x 15 x 14 = 3360 choices for the first three towns 13 choices for the fourth : 2 choices for the last but one 1 choice for the very last So there are 16 x 15 x 14 x 13 x … x 2 x 1 = 20,922,789,888,000 choices for visiting all 16 towns ( = 16! ‘sixteen factorial’ ) OK, but does this really matter? The ‘Travelling Salesman Problem’ (TSP)

15 Centre for Applied Internet Research Suppose we have a computer that can (just) solve the TSP for 20 towns That is, one that can search through the 20 x 19 x 18 x … x 2 x 1 = 2,432,902,008,176,640,000 possible choices (in ‘reasonable’ time) How much extra effort is involved in solving the TSP for 21 towns? That would be 21 x 20 x 19 x 18 x … 2 x 1, which is 21 times as much as for 20 towns According to Moore’s Law, it will take about another 8 years to develop a computer this powerful To solve the TSP for 30 towns would take a computer 30 x 29 x 28 x … x 22 x 21 = 109,027,350,432,000 times as powerful About 95 years by Moore’s Law! … but this will fail long before that! How far could we ever get? The ‘Travelling Salesman Problem’ (TSP)

16 Centre for Applied Internet Research Suppose we turn the Universe into a computer and run it as efficiently as theoretically possible Suppose every sub-atomic particle in the Universe is a component of the computer. Suppose we manage to perform a calculation every time a sub-atomic particle changes quantum state, for as many such particles as there are, for as many states as they have, as often as such changes are possible and we run the whole thing for the lifetime of the Universe What’s the biggest TSP we can solve? Approximately … 80 (towns) The problem is that the complexity of the TSP increases exponentially We’ll come back to that later as well! The ‘Perfect Computer’!

17 Centre for Applied Internet Research Finding the Limitations of Computers Trying to work out (theoretically) what computers can and can’t do and how difficult some things might be is, in itself, tricky because computers are themselves … well complex! So we tend to start with something simpler An idealised model of a computer There are lots of these around But probably the most well-known, and certainly the first, is the … Turing Machine (Alan Turing, 1936) A ‘computer’ is modelled as being something that can exist in a number of ‘states’, can respond to simple input (i.e. change state) and produce simple output For some initial input, it does this over and over again until it produces some final output (hopefully).

18 Centre for Applied Internet Research q … r … s … t … y … z … The Turing Machine ‘State’ s Read/Write Head 0 1 & $ % 1 0 $ Data ‘tape’ Input/output data 0 1 % & $ * Input ‘Action Grid’ (program) State : : : : : : Write ‘0’ Change to state ‘t’ Move left

19 Centre for Applied Internet Research Problem ‘Solvability’ A problem is ‘solvable’ if its input can be written to the data tape of an appropriate Turing machine and, through the application of a suitable action grid (program), its output eventually read from the data tape If a problem is unsolvable, either it will not be possible to construct such a machine or any such attempt will fail to terminate (run indefinitely) It is possible to show that some very complex problems can be solved by a Turing machine However, Turing himself proved that there are some problems that are simply not solvable For example, the ‘Halting Problem’ Given a program and its input (as a Turing machine, say), can we produce another program that will determine if the first will finish? NO!

20 Centre for Applied Internet Research ‘Decision’ Functions Most ‘natural’ or ‘general’ problems have a variety of answers … number, name, colour, shape, size, direction, etc. A ‘decision function’ instead always evaluates to ‘yes’ or ‘no’ However … Every general problem has an equivalent decision function! For example, in complexity terms, ‘who is the tallest person in the room’ is the same as ‘is there anyone in the room taller than #.## metres?’ For some values of #.##, it will be equally difficult to determine either A problem is solvable if its equivalent decision function is ‘decidable’ In 1931, Kurt Gödel proved that not all functions are decidable In 1936, Alan Turing proved that not all problems are solvable Same thing!

21 Centre for Applied Internet Research q … r … s … t … y … z … Extended Turing Machines ‘State’ s Read/Write Head 0 1 & $ % 1 0 $ Data ‘tape’ Input/output data 0 1 % & $ * Input ‘Action Grid’ (program) State : : : : : : Write ‘0’ Change to state ‘t’ Move left

22 Centre for Applied Internet Research q … r … s … t … y … z … Extended Turing Machines s 0 1 & $ % 1 0 $ 0 1 % & $ *

23 Centre for Applied Internet Research Extended Turing Machines Multiple machines/tapes Multiple Action grids Write ‘0 1 0 …’ Change to state ‘t’ Move left 3 places Extended functionality There’s nothing an extended Turing machine can do that a basic Turing machine can’t (It might just take a longer program (tape) or more time)

24 Centre for Applied Internet Research Equivalent Models So an extended Turing machine is (essentially) no different to a standard Turing machine It can’t really do any more … or less In fact there are many models of the computational process … … and they’re all the same! We can start with something very ‘simple’ and show what we can and can’t do with it … … then gradually get more ‘complex’, showing that we haven’t gained or lost anything at each step … … until we’ve effectively got a ‘real’ computer So we can say what a computer can and can’t do Unfortunately, the results aren’t always … ‘comfortable’!

25 Centre for Applied Internet Research A decision function may be … Enough of that! (For now …) Levels of Solvability YESNOUNDECIDABLE CAN’T TELL!

26 Centre for Applied Internet Research Measuring Complexity Some things are simple! For example ‘find x if x+3 = 5’ is a trivial calculation x = 5 - 3 This has ‘no complexity at all’ or ‘constant’ complexity However, some things take longer! What really matters is how complexity grows ‘Who’s the tallest person in the room?’ involves a ‘search’            ------------------------------------------------------------------------------  Searching 10 people involves 10 comparisons searching 20 people involves 20 comparisons In general, searching ‘n’ people involves ‘n’ comparisons We say the (time) complexity is ‘of the order of n’ … ‘O(n)’ for short

27 Centre for Applied Internet Research Measuring Complexity Searching a map (n x n grid) for something … … involves n x n checks, so has O(n x n) or O(n 2 ) complexity 1 2 3 … n 123:n123:n

28 Centre for Applied Internet Research Measuring Complexity Searching a volume (n x n x n) … … has O(n x n x n) = O(n 3 ) complexity We call this O(n), O(n 2 ), O(n 3 ), etc. complexity ‘polynomial complexity’ … or ‘EASY’ … or ‘P’ n n

29 Centre for Applied Internet Research Measuring Complexity But not all problems have polynomial complexity (Well, probably not!) A basic TSP search, for example has O(n x n-1 x n-2 x … x 2 x 1) = O(n!) complexity The ‘best’ known TSP solution has O(2 n ) complexity (2 n = 2 x 2 x 2 x …….. x 2 ) This O(2 n ), O(n!), O(n n ), etc. complexity is known as ‘exponential complexity’ or ‘HARD’ We like EASY (polynomial) We don’t like HARD (exponential) Why?

30 Centre for Applied Internet Research time (logarithmic) 10 000 1000 100 10 0 time 400 300 200 100 0 Polynomial vs. Exponential Time n n n n2n2 n3n3 2n2n n! n n n2n2 n3n3 2n2n n

31 Centre for Applied Internet Research All problems Complexity Classes It should look like this … … but does it? ‘NP’ problems ‘P’ problems Solvable problems Realistic problems (Hard) (Easy) * Halting problem * TSP

32 Centre for Applied Internet Research An algorithm is a method of solution for a problem … the steps to be followed … an abstract version of a program The complexity of a problem is really the complexity of the (best) algorithm that solves it But there’s a big issue here … How do we know we’ve got the best algorithm for the job? A problem could be simpler than we realise … … there might be an easy algorithm for it out there … … we just haven’t found it yet! Let’s take an example … Algorithms

33 Centre for Applied Internet Research Climbing stairs … … taking one or two steps at a time How many different ways are there of taking 6 steps? 1 6 5 4 3 2 The Stairs Problem

34 Centre for Applied Internet Research Climbing stairs … … taking one or two steps at a time How many different ways are there of taking 6 steps? 1 6 5 4 3 2 The Stairs Problem

35 Centre for Applied Internet Research Climbing stairs … … taking one or two steps at a time How many different ways are there of taking 6 steps? 1 6 5 4 3 2 The Stairs Problem

36 Centre for Applied Internet Research Climbing stairs … … taking one or two steps at a time How many different ways are there of taking 6 steps? How many different ways are there of taking n steps? Call this f n Then f 1 = 1 f 2 = 2 f 3 = 3 f 4 = 5 ?? The Stairs Problem 1 n n-1 : : 2

37 Centre for Applied Internet Research How about the general case, f n ? Well, consider the first step … A single step leaves n-1 A double step leaves n-2 There are f n-1 ways of taking n-1 steps … … and f n-1 ways of taking n-1 steps It has to be one or the other so … … the number of ways of taking n steps is made up of the number of ways of taking n-1 steps and the number of ways of taking n-2 steps In other ‘words’ … … a ‘recursive definition’ 1 2 n n-1 : : f n-1 f n-2 f n = f n-1 + f n-2 The Stairs Problem

38 Centre for Applied Internet Research We can write a program to do this … Recursion f3f3 f2f2 f4f4 f2f2 f1f1 f3f3 f2f2 f1f1 f3f3 f2f2 f1f1 f2f2 f4f4 f5f5 f6f6 (2) (1)

39 Centre for Applied Internet Research Let’s try running this … [PROGRAM][PROGRAM] Recursion // C++ fn by recursion #include int n; float f(int n); main() { for (n = 3; n <= 100; ++n) cout << “n = “ << n << “: f(n) = “ << f(n) << “\n”; } float f(int n) { if (n == 1 || n == 2) return n; else return f(n-1) + f(n-2); }

40 Centre for Applied Internet Research Recursion is elegant, not efficient! Recursion f3f3 f2f2 f4f4 f2f2 f1f1 f3f3 f2f2 f1f1 f3f3 f2f2 f1f1 f2f2 f4f4 f5f5 f6f6 (2) (1)

41 Centre for Applied Internet Research How about the general case, f n ? Well, consider the first step … A single step leaves n-1 A double step leaves n-2 There are f n-1 ways of taking n-1 steps … … and f n-1 ways of taking n-1 steps It has to be one or the other so … … the number of ways of taking n steps is made up of the number of ways of taking n-1 steps and the number of ways of taking n-2 steps In other ‘words’ … … a ‘recursive definition’ 1 2 n n-1 : : f n-1 f n-2 f n = f n-1 + f n-2 The Stairs Problem

42 Centre for Applied Internet Research Start with the values we know, not the ones we don’t! f 1 = 1 f 2 = 2 f 3 = f 2 + f 1 = 2 + 1 = 3 f 4 = f 3 + f 2 = 3 + 2 = 5 f 5 = f 4 + f 3 = 5 + 3 = 8 This is the ‘Fibonacci Sequence’ n1234567... f n 123581321 … Iteration

43 Centre for Applied Internet Research Much simpler! We can write a program to do this … Iteration f1f1 f2f2 f3f3 f4f4 f5f5 f6f6 (1) (2)

44 Centre for Applied Internet Research Let’s try running this … [PROGRAM][PROGRAM] Iteration // C++ fn by iteration #include int n; float fn_2 = 1; fn_1 = 2, fn; main() { for (n = 3; n <= 100; ++n) { fn = fn_1 + fn_2; fn_2 = fn_1; fn_1 = fn; cout << “n = “ << n << “: f(n) = “ << fn << “\n”; }

45 Centre for Applied Internet Research In fact the ‘equation’ f n = f n-1 + f n-2 can be ‘solved’ to give … (an interesting expression for a whole number?) This is a single step surely? Calculation

46 Centre for Applied Internet Research Calculation // C++ fn by calculation #include int n; float f(int n); main() { for (n = 3; n <= 100; ++n) cout << “n = “ << n << “: f(n) = “ << f(n) << “\n”; } float f(int n) { return (pow((1 + sqrt(5)) / 2, n+1) - pow((1 – sqrt(5)) / 2, n+1)) / sqrt(5); }

47 Centre for Applied Internet Research In fact the ‘equation’ f n = f n-1 + f n-2 can be ‘solved’ to give … (an interesting expression for a whole number?) This is a single step surely? Not for a computer! (‘powers are complex’) The key point is that we can’t be sure whether we’ve got the best algorithm so how can we say whether a problem is easy or hard? Calculation

48 Centre for Applied Internet Research Examples But the equivalent problem for wireless networks (MCDS) … … appears to be HARD The shortest path problem (SPP) … … is EASY The shortest tour problem (TSP) … … appears to be HARD The minimum connecting network problem (MST) … … is EASY

49 Centre for Applied Internet Research Some problems are easy polynomial (P) Some problems don’t appear to be but we can’t be sure Some of these ‘harder’ problems are ‘equivalent’ to each other a polynomial (easy) algorithm for any one of them could be easily adapted to all of the others OR a proof that no polynomial (easy) algorithm exists for any one of them would prove that none exists for any of the others We call this class of equivalent ‘harder’ problems … NP-complete What DO We Know …

50 Centre for Applied Internet Research How it might look But only if P  NP (otherwise there are no hard problems) Complexity Classes All problems ‘NP’ problems ‘P’ problems ‘NP-complete problems

51 Centre for Applied Internet Research How it might look if P = NP Complexity Classes All problems ‘P’ problems ‘NP’ problems ‘NP-complete problems

52 Centre for Applied Internet Research It comes down to this … This has exercised the minds of some of our greatest thinkers over the last few decades … The Big Question … P NP = ?

53 Centre for Applied Internet Research The Big Question …

54 Centre for Applied Internet Research The Million Dollar Question … In 1998, the Clay Mathematics Institute offered prizes of $1,000,000 for solving each of seven Millennium Prize Problems ‘P vs. NP’ is one of them So far, none have been solved (Well, maybe one?) But, before you dig out the pencil and paper, consider this …

55 Centre for Applied Internet Research is a decision function! Will We Ever Know? YESNOUNDECIDABLE CAN’T TELL! P = NP ?

56 Centre for Applied Internet Research is a decision function! Where does it live? Will We Ever Know? UNDECIDABLE CAN’T TELL! P = NP ? NP P NP-Comp

57 Centre for Applied Internet Research This ‘P easy – NP hard’ idea may not be as clear as we might think! How large does n have to get before x n becomes larger than n y ? ( eg. 1.0001 n vs. n 9999 ) For x small enough and/or y large enough, n has to be as large as you like! So what’s the distinction? If there is one, it’s purely theoretical rather than practical! Will We Ever Know? n x n n x

58 Centre for Applied Internet Research Conclusions Probably no serious mathematician/computer scientist thinks that P = NP In other words, yes, there are some genuinely hard problems But this P ≠ NP may be very difficult to prove However, we can make a problem harder than it needs to be through choosing a poor algorithm Ultimately, computers really do seem to have simple, but real, limits Good luck to anyone who can prove this for sure!

59 Centre for Applied Internet Research Thank-you … … Any questions? Professor Vic Grout Director of the Centre for Applied Internet Research (CAIR) Glyndŵr University, North Wales v.grout@glyndwr.ac.uk Inaugural Professorial Lecture, Glyndŵr University, 25 th June 2009

60 Centre for Applied Internet Research Centre for Applied Internet Research www.cair-uk.org


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