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48
Designing Programs That Check Their Work
, 1989
"... A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It d ..."
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Cited by 307 (17 self)
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A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It designs program checkers for a few specific and carefully chosen problems in the class FP of functions computable in polynomial time. Problems in FP for which checkers are presented in this paper include Sorting, Matrix Rank and GCD. It also applies methods of modern cryptography, especially the idea of a probabilistic interactive proof, to the design of program checkers for group theoretic computations. Two strucural theorems are proven here. One is a characterization of problems that can be checked. The other theorem establishes equivalence classes of problems such that whenever one problem in a class is checkable, all problems in the class are checkable.
The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory
 SIAM J. Comput
, 1998
"... ..."
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Exact and Approximation Algorithms for Sorting By Reversals, With Application to Genome Rearrangement
, 1995
"... Motivated by the problem in computational biology of reconstructing the series of chromosome inversions by which one organism evolved from another, we consider the problem of computing the shortest series of reversals that transform one permutation to another. The permutations describe the order of ..."
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Cited by 78 (4 self)
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Motivated by the problem in computational biology of reconstructing the series of chromosome inversions by which one organism evolved from another, we consider the problem of computing the shortest series of reversals that transform one permutation to another. The permutations describe the order of genes on corresponding chromosomes, and a reversal takes an arbitrary substring of elements and reverses their order. For this problem we develop two algorithms: a greedy approximation algorithm that finds a solution provably close to optimal in O(n 2 ) time and O(n) space for an n element permutation, and a branch and bound exact algorithm that finds an optimal solution in O(mL(n;n)) time and O(n 2 ) space, where m is the size of the branch and bound search tree and L(n; n) is the time to solve a linear program of n variables and n constraints. The greedy algorithm is the first to come within a constant factor of the optimum; it guarantees a solution that uses no more than twice the min...
Coordinating Pebble Motion on Graphs, The Diameter Of Permutation Groups, And Applications
"... The problem of memory management in totally distributed computing systems leads to the following movers' problem on graphs: Let G be a graph with n vertices with k < n pebbles numbered 1, ... ,k on distinct vertices. A move consists of transferring a pebble to an adjacent unoccupied vertex.
The prob ..."
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Cited by 25 (0 self)
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The problem of memory management in totally distributed computing systems leads to the following movers' problem on graphs: Let G be a graph with n vertices with k < n pebbles numbered 1, ... ,k on distinct vertices. A move consists of transferring a pebble to an adjacent unoccupied vertex.
The problem is to decide whether one· arrangement of the pebbles is reachable from another, and to find the shortest sequence of moves to find the rearrangement when it is possible.
In the case that G is biconnected and k=n1, Wilson (1974) gave an efficient decision procedure. However, he did not determine whether solutions require at most polynomially many moves. We generalize by giving a Ptime decision procedure for all graphs and any number of pebbles. Further, we prove matching O(n^3) upper and lower bounds on the number. of moves required, and show how to efficiently plan solutions.
It is hoped that the algebraic methods introduced for the graph puzzle will be applicable to special cases of the general geometric movers' problem, which is PSPACEhard (Reif (1979)).
We consider the related question of permutation group diameter. Driscoll and Furst (1983) obtained a polynomial upper bound on the diameter of permutation groups generated by cycles of bounded length. By making effective some standard results in permutation group theory, we obtain the following partial extension of their result to unbounded vcycles:
If G (on n letters) is generated by cycles, one of which has prime length p < 2n/3, and G is primitive, then G = A_n or S_n and has diameter less than 2^((p^1/6)+4) n^8. This is a moderately exponential bound.
Sylow's Theorem in Polynomial Time
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1985
"... Given a set r of permutations of an nset, let G be the group of permutations generated by f. If p is a prime, a Sylow psubgroup of G is a subgroup whose order is the largest power of p dividing IGI. For more than 100 years it has been known that a Sylow psubgroup exists, and that for any two Sylo ..."
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Cited by 24 (8 self)
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Given a set r of permutations of an nset, let G be the group of permutations generated by f. If p is a prime, a Sylow psubgroup of G is a subgroup whose order is the largest power of p dividing IGI. For more than 100 years it has been known that a Sylow psubgroup exists, and that for any two Sylow psubgroups PI, P, of G there is an element go G such that Pz = g‘PI g. We present polynomialtime algorithms that find (generators for) a Sylow psubgroup of G, and that find ge G such that P, = g‘P, g whenever (generators for) two Sylow psubgroups PI, Pz are given. These algorithms involve the classification of all tinite simple groups. 0 1985 Academic Press. Inc. PART I 1.
Computing the Composition Factors of a Permutation Group in Polynomial Time
 Combinatorica
, 1987
"... Givengenerators for a group of permutations, it is shown that generators for the subgroups in a composition series can be found in polynomial time. The procedure also yields permutation representations of the composition factors. ..."
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Cited by 21 (2 self)
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Givengenerators for a group of permutations, it is shown that generators for the subgroups in a composition series can be found in polynomial time. The procedure also yields permutation representations of the composition factors.
Recognition Of Linear ContextFree Rewriting Systems
, 1992
"... has been introduced as a generalization of a class of grammar formalisms known as mildly contextsensitive. The recognition problem for linear contextfree rewriting languages is studied at length here, presenting evidence that, even in some restricted cases, it cannot be solved efficiently. This en ..."
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Cited by 20 (7 self)
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has been introduced as a generalization of a class of grammar formalisms known as mildly contextsensitive. The recognition problem for linear contextfree rewriting languages is studied at length here, presenting evidence that, even in some restricted cases, it cannot be solved efficiently. This entails the existence of a gap between, for example, tree adjoining languages and the subclass of linear contextfree rewriting languages that generalizes the former class; such a gap is attributed to "crossing configurations". A few other interesting consequences of the main result are discussed, that concern the recognition problem for linear contextfree rewriting languages.
New Approaches to Designing Public Key Cryptosystems Using OneWay Functions and TrapDoors in Finite Groups
 Journal of Cryptology
"... A symmetric key cryptosystem based on logarithmic signatures for nite permutation groups was described by the rst author in [6], and its algebraic properties were studied in [7]. In this paper we describe two possible approaches to the construction of new public key cryptosystems with message spa ..."
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Cited by 18 (1 self)
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A symmetric key cryptosystem based on logarithmic signatures for nite permutation groups was described by the rst author in [6], and its algebraic properties were studied in [7]. In this paper we describe two possible approaches to the construction of new public key cryptosystems with message space a large nite group G, using logarithmic signatures and their generalizations. The rst approach relies on the fact that permutations of the message space G induced by transversal logarithmic signatures almost always generate the full symmetric group SG on the message space. The second approach could potentially lead to new ElGamal  like systems based on trapdoor, oneway functions induced Research supported in part by National Science Foundation grant CCR9610138 y Research supported in part by NSERC grants IRC #21643196 and RGPIN # 20311498. 1 by logarithmic signaturelike objects we call meshes, which are uniform covers for G. Key words. Trapdoor oneway functions...
The Mathematics of Perfect Shuffles
 ADVANCES IN APPLIED MATHEMATICS
, 1983
"... There are two ways to perfectly shuffle a deck of 2n cards. Both methods cut the deck in half and interlace perfectly. The out shuffle 0 leaves the original top card on top. The in shuffle I leaves the original top card second from the top. Applications to the design of computer networks and card tr ..."
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Cited by 16 (3 self)
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There are two ways to perfectly shuffle a deck of 2n cards. Both methods cut the deck in half and interlace perfectly. The out shuffle 0 leaves the original top card on top. The in shuffle I leaves the original top card second from the top. Applications to the design of computer networks and card tricks are reviewed. The main result is the determination of the group (I, 0) generated by the two shuffles, for all n. If 2 n is not a power of 2, and if 2n * 12,24, then (I, 0) has index 1,2, or 4 in the Weyl group B, (the group of all 2”n! signed n x n permutation matrices). If 2n = 2“, then (I, 0) is isomorphic to a semidirect product of Zi and Z,. When 2 n = 24, (I, 0) is isomorphic to a semidirect product of 2j ’ and M,,, the Mathieu group of degree 12. When 2n = 12, (I, 0) is isomorphic to a semidirect product of Zi and the group PGL(2,5) of all linear fractional transformations over GF(5). 1. I~RODUCTI~N There are two ways to perfectly shuffle a deck of 2n cards. Both cut the deck in half and interlace perfectly. The in shuffle I leaves the original top card second from the top. The out shuffle 0 leaves the original top card on top. Let the deck be labeled (0, 1,..., n 1, n,..., 2n 1). After an in shuffle the order is (n, 0, n + 1,..., 2n 1, n 1). After an out shuffle, the order is (O,n,l,n+ l,..., n 1,2n 1). These shuffles have been used by gamblers and magicians to manipulate cards. A historical review,