M. Smith. 6 September 2016 / GSAC

Transcription

1 , Complexity, and Department of Mathematics University of Utah 6 September 2016 / GSAC

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4 Motivation The clock puzzle is an infamous part of the video game XIII-2 (2011). Most players rely on computer tools developed specifically to solve these puzzles. Main question: are these puzzles really that hard? Is there a simpler solution we just haven t found yet?

5 Live Demonstration Understanding the puzzle is easiest with a demonstration. Here is a video of some puzzles being solved: < Here is a link to a puzzle solver: <

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7 Formalizing the Puzzle A clock puzzle is 0,...N 1 points in a circle. Each point n has an associated integer k(n). From n we can move to either n + k(n) or n k(n) (mod N). This is data for a directed graph.

8 Formalizing the Solution A solution to the clock puzzle is a valid path that hits each integer n exactly once. This is called a Hamiltonian path if it exists.

9 A Familiar The Hamiltonian Path asks for a Hamiltonian path on a finite directed graph, or a proof that no such path exists. Clock puzzles are a special case of this well-studied problem in graph theory!

10 Brute Force Here s one way to find all Hamiltonian paths for a graph: k = 0. While (k < N!): P[k] = k-th permutation of 0,...,N-1. If P[k] gives an allowed path: Print P[k]. Increase k by 1. For a graph on N vertices, there are N! permutations. As N grows, the time needed to find all solutions grows rapidly!

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12 Motivation We have an algorithm which can find Hamiltonian paths (and hence solve clock puzzles), but it is very slow. Question: Is there a faster algorithm? To discuss what we know about this question, we need to do some rigorous math.

13 Turing Machines Informally, a Turing machine is a finite program attached to a read/write head which moves along an infinite tape. The tape is divided into squares, each of which contains one symbol or is blank. Initially the tape contains a finite non-blank input, and the Turing machine starts running at one end of the input.

14 Turing Machines A Turing machine is defined by M = (Σ, Γ, Q, δ). Σ, Γ, Q are finite nonempty sets. Q is disjoint from Γ. Σ Γ are alphabets. b Γ \ Σ is the blank character. The allowed inputs are finite strings over Σ. Q is the state of the machine. It contains three special states: the initial state q 0, q accept, and q reject. δ is the transition function.

15 Transition Function The transition function is defined by: δ : ( Q {q accept, q reject } ) Γ Q Γ {+1, 1} Suppose δ(q, s) = (q, s, h). The interpretation is: M is in state q scanning symbol s. Application of δ: Changes M to state q. Replaces symbol s on the tape with symbol s. Moves M to the left on the tape if h = 1, or to the right if h = +1.

16 Configurations A configuration of M is a string xqy, where x and y are finite strings in Γ and q Q. We interpret the configuration xqy to mean that the string xy is printed on the tape and M is in state q reading the leftmost character in y. If C, C are configurations of M, we write C M C if C arises from C after an application of δ.

17 Computation Given an input string w, the computation of M on w is a sequence of configurations C 0, C 1,... so that: C 0 = q 0 w (or C 0 = q 0 b with empty input). C i M C i+1 for each i with C i+1 in the computation. If q i = q accept or q reject, then the computation halts at step i. There are i + 1 configurations, and i applications of δ. Note that it is possible for q i to never be q accept or q reject, so the machine runs forever.

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19 Computation Time We say a Turing machine M accepts w as an input if the computation of M given w halts with M in state q accept. If M accepts w, the running time R(w) for the computation is the number of applications of the transition function δ. The worst case running time for M is a function T M : T M (n) = max{r(w) : length(w) n and M accepts w}

20 Big O Notation This notation is commonly used to describe running times. Given F, G : N R, we say that F(n) is O(G(n)) if there are constants C, N > 0 so that: F(n) C G(n) whenever n N. Note: this is asymmetric, and is not an equivalence relation!

21 Big O Example Let F(n) = 2n 2 1. For n 1 we compute: 2n 2 1 2n n 2 + n 2 3 n 2 so F(n) is O(n 2 ), with C = 3 and N = 1. By similar reasoning, a polynomial in n of degree d is O(n d ).

22 Common and their Orders Using the worst case running time, we can compute how complicated certain tasks are based on their input size. This determines the time complexity of the problem. O(1) : Printing "Billy Crystal". O(log n): Binary search in a list of size n. O(n): Addition of two n-digit integers. O(n 2 ): Simple multiplication of two n-digit integers. O(c n ), c > 1: Best known Hamiltonian path finder. O(n!): Brute force Hamiltonian path finder, n vertices.

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24 Slow Fast Consider the Deep Thought algorithm: wait 7.5 million years. print "42". This algorithm is O(1), but we would not call it "fast". Since the constants in Big O notation can be arbitrarily large, we need to be careful!

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26 Motivation Despite the caveat, we can consider polynomial time algorithms to be "fast" or "easy" and algorithms which run in more than polynomial time to be "slow" or "hard". However, just because there is no known "fast" algorithm for a problem does not mean that there isn t one! So, we need further work to show that clock puzzles are "hard".

27 Class P P stands for deterministic polynomial time. A problem is in class P if there exists a Turing machine which can solve it in polynomial time or faster. Binary search, addition, and multiplication are all problems in class P: we have O(log n), O(n) and O(n 2 ) algorithms (respectively).

28 Class NP NP stands for nondeterministic polynomial time. Informally, a problem is in class NP if there exists a "verification algorithm" for the problem which runs in polynomial time. The Hamiltonian path problem is in class NP, since it takes O(n) steps to verify that a path on n vertices is a Hamiltonian path.

29 P NP It is easy to show that P NP, as a polynomial time algorithm gives a polynomial time verification for a problem. It is unknown whether P = NP! This gives us some hope that "hard" problems might be easy.

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31 Historical Background One of the seven Clay Millenium s, announced in 2000, is to determine whether P = NP or not. In the 1960 s, the notion of polynomial time computation was formalized by Cobham and Edmonds. In 1971, the notion of NP-completeness was introduced by Cook. In 1972 Karp demonstrated that many natural problems are NP-complete. Karp also introduced the notation P and NP.

32 NP-completeness Informally, a fixed problem in NP is NP-complete if it is possible to "reduce" any NP problem to the fixed problem in polynomial time. There are hundreds of NP-complete problems, including the Hamiltonian path problem! This means we can say XIII-2 is "NP-complete" (in some sense).

33 What happens if P = NP? If P = NP, then there is a polynomial time algorithm solving an NP-complete problem. By the reduction definition of NP-completeness, this algorithm is universal: it can solve all problems in NP. The existence of a fast algorithm for NP-complete problems would change the world!

34 What happens if P = NP? Modern cryptography is based on integer factorization, which is in NP, and believed to be "hard". But if P = NP, then many aspects of the internet, including financial transactions, could no longer be secure! On the other hand, many important optimization problems like finding Hamiltonian paths would become easy to solve.

35 Alternative Scenarios Most complexity theorists believe that P NP because of the stunning consequences of equality. However, polynomial time does not mean "fast". We could have P = NP, but the best universal algorithm has coefficients on the order of Graham s number. Another option is that P = NP, but the proof is not constructive. There would be a universal algorithm for NP problems, but no-one would be able to write it down.

36 What About P NP? To prove P NP, it suffices to find a superpolynomial lower bound for the running time of an NP problem. However, complexity theory has failed in many cases to provide interesting lower bounds in many cases, including for P and NP.

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38 Summary We don t know whether s clock puzzles are actually hard! A solution to the problem could take many different forms, some with dramatic consequences. Many unexpected things can lead to beautiful mathematics.

39 References and Further Reading XIII-2 Nathaniel Johnston s blog < counting-and-solving-final-fantasy-xiii-2s- > Pictures on slides 7 and 8 are taken from the above post.

40 References and Further Reading Big O https: //en.wikipedia.org/wiki/big_o_notation Has a more exhaustive list of problems and orders of solution algorithms. < millennium-problems/p-vs-np-problem> The official problem description by Cook is the major reference for this talk.