1. Week 13 - Monday

2.  What did we talk about last time?  B-trees  Hamiltonian tour  Traveling Salesman Problem

7.  Given a graph, find the shortest tour that visits every node and comes back to the starting point  Like a lazy traveling salesman who wants to visit all possible clients and then return to his home

8.  Find the shortest TSP tour:  Starts anywhere  Visits all nodes  Has the shortest length of such a tour E E A A D D B B F F C C 5 6 4 4 3 5 2 7 3 3

9.  Strategies:  Always add the nearest neighbor  Randomized approaches  Try every possible combination  Why are we only listing strategies?

10.  So far we haven’t given a foolproof solution to TSP  Let’s look at two possibilities:  Greedy Solution: Pick the closest neighbor  Brute Force: Try all possibilities

11.  We are tempted to always take the closest neighbor, but there are pitfalls GreedyOptimal

12.  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! = 8841761993739701954543616000000  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!

13.  No one knows how to find the best solution to TSP in an efficient amount of time  For a general graph, no good approximation even exists  For a graph with the triangle inequality, there is an approximation that yields a tour no more than 3/2 the optimal  Some variations on the problem are easier than others

14.  Is TSP the hardest problem there is?  Are there other problems equally hard?  How do we compare the difficulty of one problem to another?

15.  TSP is one of many problems which are called NP-complete  All of these problems share the characteristic that the only way we know to find best solution uses brute force  All NP-complete problems are reducible to all other NP-complete problems  An efficient solution to one would guarantee an efficient solution to all

16.  Traveling Salesman Problem  Hamiltonian Cycle (and Path)  Let’s see just a couple more…

17.  Find the smallest number of colors for coloring nodes such that no two adjacent nodes have the same color E E A A D D B B F F C C E A D B F C

18.  What about this graph?

19.  Like TSP and Hamiltonian Cycle, large graphs get really hard to find the smallest coloring for  It might not be obvious, but graph coloring has practical applications  Perhaps the most common example is register allocation inside of a compiler

20.  Find the largest complete subgraph in a given graph  This graph has a clique of size 3 E E A A D D B B F F C C

21.  What about this graph?  This graph has a clique of size 4  As before, a large graph can be very difficult E E A A D D B B F F C C

22.  Not all NP-complete problems are graph problems  The knapsack problem is the following:  Imagine that you are Indiana Jones  You are the first to open the tomb of some long-lost pharaoh  You have a knapsack that can hold m pounds of loot, but there's way more than that in the tomb  Because you're Indiana Jones, you can instantly tell how much everything weighs and how valuable it is  You want to find the most valuable loot that weighs less than or equal to m pounds

23.  This one is a little bit mathematical  Say you have a set of numbers  Somebody gives you a number k  Is there any subset of the numbers in your set that add up to exactly k?  Example:  Set: { 3, 9, 15, 22, 35, 52, 78, 141}  Is there a subset that adds up to exactly 100?  What about 101?

24.  These NP-complete problems are very hard  Many of them are really useful  Especially if you are a lazy traveling salesman  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 any of the problems  Prove that one cannot have an efficient solution

26.  A Turing machine is a mathematical model for computation  It consists of a head, an infinitely long tape, a set of possible states, and an alphabet of characters that can be written on the tape  A list of rules saying what it should write and should it move left or right given the current symbol and state 1011110000 A A

27.  3 state, 2 symbol “busy beaver” Turing machine:  Starting state A Tape Symbol State AState BState C WriteMoveNextWriteMoveNextWriteMoveNext 01RB0RC1LC 11RHALT1RB1LA

28.  If an algorithm exists, a Turing machine can perform that algorithm  In essence, a Turing machine is the most powerful model we have of computation  Power, in this sense, means the ability to compute some function, not the speed associated with its computation

29.  NP is actually a class of problem  Problems in NP can be solved in polynomial time on a non-deterministic computer  A deterministic computer is the kind you know:  First it has to consider possibility A, then, it can consider possibility B

30.  A non-deterministic computer (which, as far as we know, only exists in our imagination) can consider both possibility A and possibility B at the same time Deterministic A B CDEF A B C D E F Non- Deterministic

31.  P is the class of decision problems that can be solved in polynomial time by a deterministic computer  Lots of great problems are in P:  Is this list sorted?  Is this number prime?  Is the largest number in this B-tree equal to 38?  Many problems are unknown:  Does this number have exactly two factors?  Is this graph equivalent to this other graph?

32.  Everything in P is also in NP  Some problems are the “hardest” problems in NP  This means that any problem in NP can be converted into one of these problem in polynomial time  These problems make up the class NP- complete

33.  Notice that P, NP, and NP-complete are all decision problems  So, the TSP we stated is not technically NP- complete  The NP-complete version is:  Is there a tour of length less than or equal to 24 in this graph?  The optimization versions of NP-complete problems are called NP-hard

34.  EXPSPACE  Exponential space  EXP  Exponential time on a deterministic Turing machine  PSPACE  Polynomial space  NP  Polynomial time on a non- deterministic Turing machine PP  Polynomial time on a deterministic Turing machine LL  Logarithmic space on a deterministic Turing machine EXPSPACE EXP PSPACE NP P L

35. RE Co-RE  There are also problems that cannot be solved by any computer (or algorithm) in a finite amount of time  Problems in R can be solved by an algorithm in a finite amount of time  Problems in RE can reach a "yes" answer in a finite amount of time, but no algorithm is guaranteed to reach completion if the answer is "no"  Problems in Co-RE can reach a "no" answer in a finite amount of time, but no algorithm is guaranteed to reach completion if the answer is "yes" R EXPSPACE  R is the set of recursive decision problems  RE is the set of recursively enumerable decision problems  Co-RE is the complement of set RE

36.  Given a Turing machine and input x, does it reach the halt state?  It turns out that this problem is undecidable  That means that there is no algorithm that can be to determine if any Turing machine will go into an infinite loop  Consequently, there is no algorithm that can take any program and check to see if it goes into an infinite loop

37.  Given two finite lists of words A and B, can you pick k words (repetitions allowed) from A and k words (repetitions allowed) from B so that the words from A concatenated are exactly the same string as the words from B concatenated  Example:  Solution:  aa + bb + bb = aabbbb = a + a + bbbb  The Post correspondence problem (PCP) is undecidable (there is no algorithm that can solve all instances of it) AB aabbbb abaa bbabba

39.  Elementary sorts  Merge sort  Quicksort  Priority queues  Heapsort

40.  Work on Project 4  Work on Assignment 6  Read sections 2.1 through 2.4