In order for metogivebackyour midterms, please form. a line and sort yourselves in alphabetical order, from A

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1 Parallel Bulesort In order for metogiveackyour midterms, please form a line and sort yourselves in alphaetical order, from A to Z.

2 Cominatorial Search We have seen how clever algorithms can reduce sorting from O(n 2 )too(n log n). However, the stakes are even higher for cominatorially explosive prolems: The Travelling Salesman Prolem Given a weighted graph, nd the shortest cycle which visits each vertex once. Applications include minimizing plotter movement, printedcircuit oard wiring, transportation prolems, etc. There is no known polynomial time algorithm (ie. O(n k ) for some xed k) for this prolem, so search ased algorithm are the only way togoifyou need an optional solution.

3 But I want to use a Supercomputer Moving to a faster computer can only uy you a relatively small improvement: Hardware clock rates on the fastest computers only improved y a factor of 6 from 1976 to 1989, from 12ns to 2ns. Moving to a machine with 100 processors can only give you a factor of 100 speedup, even if your jo can e perfectly parallelized (ut of course it can't). The fast Fourier algorithm (FFT) reduced computation from O(n 2 )too(n lg n). This is a speedup of 340 times on n =4096 and revolutionized the eld of image processing. The fast multipole method for n-particle interaction reduced the computation from O(n 2 )too(n). This is a speedup of 4000 times on n =4096.

4 Can Eight Pieces Cover a Chess Board? Consider the 8 main pieces in chess (king, queen, two rooks, two ishops, two knights). Can they e positioned on a chessoard so every square is threatened? N N B R R B Q K Only 63 square are threatened in this conguration. Since 1849, no one had een ale to nd an arrangement with ishops on dierent colors to cover all squares. Of course, this is not an important prolem, ut we will use it as an example of how to attack a cominatorial search prolem.

5 How many positions to test? Picking a square for each piece gives us the ound: 64!=(64 ; 8)! = Anything much larger than 10 8 is unreasonale to search on a modest computer in a modest amount of time. However, we can exploit symmetry to save work. With reections along horizontal, vertical, and diagonal axis, the queen can go in only 10 non-equivallent positions. Even etter, we can restrict the white ishop to 16 spots and the queen to 16, while eing certain that we get all distinct congurations. Q Q Q =

6 Backtracking Backtracking is a systematic way to go through all the possile congurations of a search space. In the general case, we assume our solution is a vector v = (a 1 a 2 ::: an) where each element a i is selected from a nite ordered set S i, We uild from a partial solution of length k. v = (a 1 a 2 ::: an) and try to extend it y adding another element. After extending it, we will test whether what we have so far is still possile as a partial solution. If it is still a candidate solution, great. If not, we delete a k and try the next element from S k : Compute S 1,thesetofcandidate rst elements of v. count =0 k =1 While k>0do While S k 6= do (*advance*) a k =anelement in S k S k S k ; a k count = count ; 1 if (a 1 a 2 ::: a k )issolution, print! k = k +1 compute S k,candidate kth elements given v. k = k ; 1 (*acktrack*)

7 Recursive Backtracking Recursion can e used for elegant and easy implementation of acktracking. Backtrack(a, k) if a is a solution, print(a) else f k = k +1 compute S k while S k 6= do a k =anelement in S k S k = S k ; a k Backtrack(a, k) g Backtracking can easily e used to iterate through all susets or permutations of a set. Backtracking ensures correctness y enumerating all possiilities. For acktracking to e ecient, we must prune the search space.

8 Constructing all Susets How many susets are there of an n-element set? To construct all 2 n susets, set up an array/vector of n cells, where the value of a i is either true or false, signifying whether the ith item is or is not in the suset. Touse the notation of the general acktrack algorithm, S k =(true false), and v is a solution whenever k n. What order will this generate the susets of f1 2 3g? (1)! (1 2)! (1 2 3)! (1 2 ;)!(1 ;)! (1 ; 3)! (1 ; ;)!(1 ;)! (1)! (;)! (; 2)! (; 2 3)! (; 2 ;)!(; ;)! (; ; 3)! (; ; ;)!(; ;)! (;)! ()

9 Constructing all Permutations How many permutations are there of an n-element set? Toconstruct all n! permutations, set up an array/vector of n cells, where the value of a i is an integer from 1 to n which has not appeared thus far in the vector, corresponding to the ith element of the permutation. Touse the notation of the general acktrack algorithm, S k =(1 ::: n) ; v, and v is a solution whenever k n.

10 The n-queens Prolem INCLUDE TWO PAGES OF PICTURES ON THE EIGHT QUUENS The rst use of pruning to deal with the cominatorial explosion was y the king who rewarded the fellow who discovered chess! In the eight Queens, we prune whenever one queen threatens another.

11 Covering the Chess Board In covering the chess oard, we prune whenever we nd there is a square which we cannot cover given the initial conguration! Specically, each piece can threaten a certain maximum numer of squares (queen 27, king 8, rook 14, etc.) Whenever the numer of unthreated squares exceeds the sum of the maximum numer of coverage remaining in unplaced squares, we can prune. As implemented y a graduate student project, this acktrack search eliminates 95% of the search space, when the pieces are ordered y decreasing moility. With precomputing the list of possile moves, this program could search 1,000 positions per second. But this is too slow! =10 3 =10 9 seconds > 1000 days Although we might further speed the program y an order of magnitude, we need to prune more nodes! By using a more clever algorithm, we eventually were ale to prove no solution existed, in less than one day's worth of computing. You too can ght the cominatorial explosion!

12 The Backtracking Contest: Bandwidth The andwidth prolem takes as input a graph G, with n vertices and m edges (ie. pairs of vertices). The goal is to nd a permutation of the vertices on the line which minimizes the maximum length of any edge The andwidth prolem has a variety of applications, including circuit layout, linear algera, and optimizing memory usage in hypertext documents. The prolem is NP-complete, meaning that it is exceedingly unlikely that you will e ale to nd an algorithm with polynomial worst-case running time. It remains NP-complete even for restricted classes of trees. Since the goal of the prolem is to nd a permutation, a acktracking program which iterates through all the n! possile permutations and computes the length of the longest edge for each gives an easy O(n! m) algorithm. But the goal of this assignment is to nd as practically good an algorithm as possile.

13 Rules of the Game 1. Everyone must do this assignment separately. Just this once, you are not allowed to work with your partner. The idea is to think aout the prolem from scratch. 2. If you do not completely understand what the andwidth of a graph is, you don't have the slightest chance of producing a working program. Don't e afraid to ask for a clarication or explanation!!!!! 3. There will e a variety of dierent data les of dierent sizes. Test on the smaller les rst. Do not e afraid to create your own test les to help deug your program. 4. The data les are availale via the course WWW page. 5. You will e graded on how fast and clever your program is, not on style. No credit will e given for incorrect programs. 6. The programs are to run on the whatever computer you have access to, although it must e vanilla enough that I can run the program on something I have access to. 7. You are to turn in a listing of your program, along with a rief description of your algorithm and any

14 interesting optimizations, sample runs, and the time it takes on sample data les. Report the largest test le your program could handle in one minute or less of wall clock time. 8. The top ve self-reported times / largest sizes will e collected and tested y me to determine the winner.

15 Producing Ecient Programs 1. Don't optimize prematurely: Worrying aout recursion vs. iteration is counter-productive until you have worked out the est way to prune the tree. That is where the money is. 2. Choose your data structures for areason: What operations will you e doing? Is case of insertion/deletion more crucial than fast retrieval? When in dout, keep it simple, stupid (KISS). 3. Let the proler determine where to do nal tuning: Your program is proaly spending time where you don't expect.

Lecture 20: Combinatorial Search (1997) Steven Skiena. skiena

Lecture 20: Combinatorial Search (1997) Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Give an O(n lg k)-time algorithm