Mike Paterson - Academia.edu (original) (raw)

Papers by Mike Paterson

Lecture Notes in Computer Science, 1977

A variety of theorems bounding the formula size of rather simple Boolean functions are described ... more A variety of theorems bounding the formula size of rather simple Boolean functions are described here for the first time. The p~incipal results are improved lower and upper bounds for symmetric functions. i. Introduction. In preparing my presentation to this 1977 GI-Conference I have borne in mind the high level of knowledge and expertise in the audience at such a meeting. I resolved that it would be most appropriate to talk about some of the most recent research upon which I have been engaged. The results to be described have been obtained by myself usually in collaboration with others, notably Mike Fischer, Albert Meyer and Bill McColl, during the pa~t year or so. One was completed only a few days ago and none has yet appeared in published form. Boolean function complexity is a key area of theoretical computer science. Questions of actual and potential efficiency in computation appear here in their ultimately refined form. It is the sticking-place for many problems arising from circuit design, algorithmic analysis and automata theory. The principal measures of complexity for Boolean functions are circuit size, formula size and depth. While the first must be regarded as the most fundamental measure, it is an unhappy historical fact that no lower bound non-linear in the number of arguments has yet been proved for the circuit size of an explicitly described function. This can be juxtaposed with the classical result that all but a vanishing fraction of Boolean functions have exponential circuit size. In contradistinction several lower bound theorems have liberated formula size from this "linear strait-jacket", disclosing a richer structure of function complexities.

Journal of Artificial Intelligence Research, 2011

Weighted voting is a classic model of cooperation among agents in decision-making domains. In suc... more Weighted voting is a classic model of cooperation among agents in decision-making domains. In such games, each player has a weight, and a coalition of players wins the game if its total weight meets or exceeds a given quota. A player's power in such games is usually not directly proportional to his weight, and is measured by a power index, the most prominent among which are the Shapley-Shubik index and the Banzhaf index.In this paper, we investigate by how much a player can change his power, as measured by the Shapley-Shubik index or the Banzhaf index, by means of a false-name manipulation, i.e., splitting his weight among two or more identities. For both indices, we provide upper and lower bounds on the effect of weight-splitting. We then show that checking whether a beneficial split exists is NP-hard, and discuss efficient algorithms for restricted cases of this problem, as well as randomized algorithms for the general case. We also provide an experimental evaluation of these ...

Theoretical Computer Science, 1976

Two fundamental complexity measures for a Boolean function f are its c'ircuit depth d(f) and its ... more Two fundamental complexity measures for a Boolean function f are its c'ircuit depth d(f) and its circu"it size c(f). It is shown that c '. * d"1o9rd for all f.

Parallel Processing Letters, 2002

We study the wavelength problem and arc (edge) congestion problem for communicating permutation i... more We study the wavelength problem and arc (edge) congestion problem for communicating permutation instances on a ring. We prove the best possible upper bounds on the number of wavelengths and arc (edge) congestion in both directed and undirected cases.

Lecture Notes in Computer Science, 1994

The length of a longest common subsequence (LLCS) of two or more strings is a useful measure of t... more The length of a longest common subsequence (LLCS) of two or more strings is a useful measure of their similarity. The LLCS of a pair of strings is related to the ‘edit distance’, or number of mutations/errors/editing steps required in passing from one string to the other.

45th Annual IEEE Symposium on Foundations of Computer Science

Recursively-constructed couplings have been used in the past for mixing on trees. We show for the... more Recursively-constructed couplings have been used in the past for mixing on trees. We show for the first time how to extend this technique to non-tree-like graphs such as the integer lattice. Using this method, we obtain the following general result. Suppose that G is a triangle-free graph and that for some ∆ ≥ 3, the maximum degree of G is at most ∆. We show that the spin system consisting of q-colourings of G has strong spatial mixing, provided q > α∆ − γ, where α ≈ 1.76322 is the solution to α α = e, and γ = 4α 3 −6α 2 −3α+4 2(α 2 −1) * This work was partially supported by the EPSRC grant "Discontinuous Behaviour in the Complexity of Randomized Algorithms".

Lecture Notes in Computer Science, 2000

We study contention-resolution protocols for multiple-access channels. We show that every backoff... more We study contention-resolution protocols for multiple-access channels. We show that every backoff protocol is transient if the arrival rate, λ, is at least 0.42 and that the capacity of every backoff protocol is at most 0.42. Thus, we show that backoff protocols have (provably) smaller capacity than full-sensing protocols. Finally, we show that the corresponding results, with the larger arrival bound of 0.531, also hold for every acknowledgment-based protocol.

Theoretical Computer Science, 1980

When selecting fnom, on sorting, a file stored on a reao-only tape and the irrternal storage is n... more When selecting fnom, on sorting, a file stored on a reao-only tape and the irrternal storage is nather linritedrseveral !'asses of the inpr-rt tape may be requir,ed. he strrdy the relation befween t}'re arnount of internal stor"age available and the number of passes r:equirecl tc select the Kti] h.ghest of N inputs. We show, :fc,.r'example, that to find the median irrtwc passes requires a.t 'l 1 ].east A(Nz) and at rnost O(N2 log N) in'i*rriai sto:-age. For 'i 1.;t:'r:babi1.i.stic nrr'tli,t,Js, O(il2) internal sloJ..rge is ne<.:essa:.y antJ. r;':i..{-'ir:i.er:t fcr'a sing'rl pdss nrethocl which finrls Ltre rnerlian with arbitr.ar,ily high pnoba):i Ii ty" 1" f ntr:r:ducti-on As a paladig'matic study of effects cf interrral stor'age limitations on J-arge-scale data-pr<;cessing tasks, we consider problens of searching and iorr.:ng in da'ta stored on a one-way r"eadrcnly tape when tlre amount of random*access wonking space is severeiy constrained. We shall <luantify nather'ciose.l-y the relatiorr between the number. of passes oven the input file which are requir.ed for these tasks anri the amo'unt r:f' storage available for a given size of the file" In sever.al cas€s the upper bounds are demonstr.ated by new sampling algorithms of some practical interest " 'r this author was partial)y suppcrted by a SeniortVisiting feJlowship frprn the Scrlgnqs Reseanch Council v,rhj-le vjsitine the U;riversitv of Wan*ick.

Theoretical Computer Science, 2003

We show that for all integers n ¿ 7 and , such that 5 6 6 2n − 2 and satisfying some coprimality ... more We show that for all integers n ¿ 7 and , such that 5 6 6 2n − 2 and satisfying some coprimality conditions, there exists a minimum n-state nondeterministic ÿnite automaton that is equivalent to a minimum deterministic ÿnite automaton with exactly 2 n − states.

SIAM Journal on Computing, 1998

We consider the problem of fitting an n x iz distance matrix D by a tree metric T. Let e be the d... more We consider the problem of fitting an n x iz distance matrix D by a tree metric T. Let e be the distance to the closest tree metric under the l,norm, that is, € = minr{ll f-Dll-}. First we present an O(n2) algorithm for finding a tree metric Z such that ll Z-D ll-< 3e. Second we show that it is NPhard to find a tree metric f such that ll T-D ll-< 9/8e. This paper presents the first algorithm for this problem with a performance guarantee.

SIAM Journal on Computing, 2004

SIAM Journal on Computing, 2008

The existence of polynomial time algorithms for the solution of parity games is a major open prob... more The existence of polynomial time algorithms for the solution of parity games is a major open problem. The fastest known algorithms for the problem are randomized algorithms that run in subexponential time. These algorithms are all ultimately based on the randomized subexponential simplex algorithms of Kalai and of Matoušek, Sharir and Welzl. Randomness seems to play an essential role in these algorithms. We use a completely different, and elementary, approach to obtain a deterministic subexponential algorithm for the solution of parity games. The new algorithm, like the existing randomized subexponential algorithms, uses only polynomial space, and it is almost as fast as the randomized subexponential algorithms mentioned above.

Random Structures & Algorithms, 2004

We consider the problem of uniformly sampling proper 3‐colorings of an m × n rectangular region o... more We consider the problem of uniformly sampling proper 3‐colorings of an m × n rectangular region of ℤ2. We show that the single‐site “Glauber‐dynamics” Markov chain is rapidly mixing. Our result complements an earlier result of Luby, Randall, and Sinclair, which demonstrates rapid mixing when there is a fixed boundary (whose color cannot be changed). © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004

Journal of the ACM, 1996

Let M(m,n) be the minimum number of comparators needed in a comparator network that merges m elem... more Let M(m,n) be the minimum number of comparators needed in a comparator network that merges m elements x 1 ≤ x 2 ≤ … ≤ x m and n elements y 1 ≤ y 2 ≤ … ≤ y m , where n ≥ m. Batcher's odd-even merge yields the following upper bound: M(m,n) ≤ ½(m + n)log 2 m + O(n); in particular, M(n,n) ≤ n log 2 n + o(n) We prove the following lower bound that matches the upper bound above asymptotically as n ≥ m →∞; M(m,n) ≥ ½(m+n)log 2 m - O(m) in particular, M(n,n) ≥ n log 2 - O(n). Our proof technique extends to give similarily tight lower bounds for the size of monotone Boolean circuits for merging, and for the size of switching networks capable of realizing the set of permutations that arise from merging.

Journal of the ACM, 2007

It is known that if P and NP are different then there is an infinite hierarchy of different compl... more It is known that if P and NP are different then there is an infinite hierarchy of different complexity classes that lie strictly between them. Thus, if P ≠ NP, it is not possible to classify NP using any finite collection of complexity classes. This situation has led to attempts to identify smaller classes of problems within NP where dichotomy results may hold: every problem is either in P or is NP-complete. A similar situation exists for counting problems. If P ≠#P, there is an infinite hierarchy in between and it is important to identify subclasses of #P where dichotomy results hold. Graph homomorphism problems are a fertile setting in which to explore dichotomy theorems. Indeed, Feder and Vardi have shown that a dichotomy theorem for the problem of deciding whether there is a homomorphism to a fixed directed acyclic graph would resolve their long-standing dichotomy conjecture for all constraint satisfaction problems. In this article, we give a dichotomy theorem for the problem of...

Information and Computation, 1991

Discrete Applied Mathematics, 1996

A fold of a nite string S over a given alphabet is an embedding of S in some xed in nite grid, su... more A fold of a nite string S over a given alphabet is an embedding of S in some xed in nite grid, such as the square or cubic mesh. The score of a fold is the number of pairs of matching string symbols which are embedded at adjacent grid vertices. Folds of strings in two-and threedimensional meshes are considered, and the corresponding problems of optimizing the score or achieving a given target score are shown to be NP-hard.

Computational Complexity, 1993

Combinatorics, Probability and Computing, 1998

Five natural relations for sets, such as inclusion, disjointness, intersection, etc., are introdu... more Five natural relations for sets, such as inclusion, disjointness, intersection, etc., are introduced in terms of the emptiness of the subsets de ned by Boolean combinations of the sets. Let N denote f1; 2; : : :; ng and N 2 denote f(i; j) j i; j 2 N and i < jg. A function on N 2 speci es one of these relations for each pair of indices. Then is said to be consistent on M N if and only if there exists a collection of sets corresponding to indices in M such that the relations speci ed by hold between each associated pair of the sets. In this paper it is proved that if is consistent on all subsets of N of size three then is consistent on N. Furthermore, conditions that make consistent on a subset of size three are given explicitly.