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137
Fourier meets Möbius: fast subset convolution
 Proceedings of the 39th Annual ACM Symposium on Theory of Computing
, 2007
"... We present a fast algorithm for the subset convolution problem: given functions f and g defined on the lattice of subsets of an nelement set N, compute their subset convolution f ∗g, defined for all S ⊆ N by (f ∗ g)(S) = X f(T)g(S \ T), T ⊆S where addition and multiplication is carried out in an a ..."
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Cited by 48 (7 self)
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We present a fast algorithm for the subset convolution problem: given functions f and g defined on the lattice of subsets of an nelement set N, compute their subset convolution f ∗g, defined for all S ⊆ N by (f ∗ g)(S) = X f(T)g(S \ T), T ⊆S where addition and multiplication is carried out in an arbitrary ring. Via Möbius transform and inversion, our algorithm evaluates the subset convolution in O(n 2 2 n) additions and multiplications, substantially improving upon the straightforward O(3 n) algorithm. Specifically, if the input functions have an integer range {−M, −M+1,..., M}, their subset convolution over the ordinary sum–product ring can be computed in Õ(2 n log M) time; the notation Õ suppresses polylogarithmic factors. Furthermore, using a standard embedding technique we can compute the subset convolution over the max–sum or min–sum semiring in Õ(2n M) time. To demonstrate the applicability of fast subset convolution, we present the first Õ(2k n 2 + nm) algorithm for the Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights, improving upon the Õ(3k n+2 k n 2 +nm) time bound of the classical Dreyfus– Wagner algorithm. We also discuss extensions to recent Õ(2 n)time algorithms for covering and partitioning problems
Generating Linear Extensions Fast
"... One of the most important sets associated with a poset P is its set of linear extensions, E(P) . "ExtensionFast.html" 87 lines, 2635 characters One of the most important sets associated with a poset P is its set of linear extensions, E(P) . In this paper, we present an algorithm to generate all of t ..."
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Cited by 36 (6 self)
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One of the most important sets associated with a poset P is its set of linear extensions, E(P) . "ExtensionFast.html" 87 lines, 2635 characters One of the most important sets associated with a poset P is its set of linear extensions, E(P) . In this paper, we present an algorithm to generate all of the linear extensions of a poset in constant amortized time; that is, in time O(e(P)) , where e ( P ) =  E(P) . The fastest previously known algorithm for generating the linear extensions of a poset runs in time O(n e(P)) , where n is the number of elements of the poset. Our algorithm is the first constant amortized time algorithm for generating a ``naturally defined'' class of combinatorial objects for which the corresponding counting problem is #Pcomplete. Furthermore, we show that linear extensions can be generated in constant amortized time where each extension differs from its predecessor by one or two adjacent transpositions. The algorithm is practical and can be modified to efficiently count linear extensions, and to compute P(x < y) , for all pairs x,y , in time O( n^2 + e ( P )).
Cycles and paths in semicomplete multipartite digraphs, theorems and algorithms: a survey
 J. Graph Theory
, 1995
"... A digraph obtained by replacing each edge of a complete mpartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete mpartite digraph. We describe results (theorems and algorithms) on directed walks in semicomplete m partite digraphs including s ..."
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Cited by 34 (18 self)
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A digraph obtained by replacing each edge of a complete mpartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete mpartite digraph. We describe results (theorems and algorithms) on directed walks in semicomplete m partite digraphs including some recent results concerning tournaments. 1
Degenerations Of Flag And Schubert Varieties To Toric Varieties
 Transformation Groups
, 1996
"... . In this paper, we prove the degenerations of Schubert varieties in a minuscule G=P , as well as the class of Kempf varieties in the flag variety SL(n)=B, to (normal) toric varieties. As a consequence, we obtain that determinantal varieties degenerate to (normal) toric varieties. Introduction In ..."
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Cited by 32 (4 self)
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. In this paper, we prove the degenerations of Schubert varieties in a minuscule G=P , as well as the class of Kempf varieties in the flag variety SL(n)=B, to (normal) toric varieties. As a consequence, we obtain that determinantal varieties degenerate to (normal) toric varieties. Introduction In this paper, we carry out the proof of the results announced in [21]. Let G be a semisimple, simply connected algebraic group defined over an algebraically closed field k. Fix a maximal torus T in G, a Borel subgroup B oe T . Let W be the Weyl group of G relative to T . Let Q ' B be a parabolic subgroup of classical type, say Q = T r i=1 P k i , where P k i , 1 i r, is a maximal parabolic subgroup of classical type (see [26] for the definition of a parabolic subgroup of classical type). Let W (Q) be the Weyl group of Q. For w 2 W=W (Q), let X(w)(= BwQ (mod Q) with the canonical reduced structure of a scheme) denote the Schubert variety in G=Q, associated to w. Given m = (m 1 ; : : : ; m ...
On Automatic Class Insertion with Overloading
, 1996
"... Several algorithms [Cas92, MS89, Run92, DDHL94a, DDHL95, GMM95] have been proposed to automatically insert a class into an inheritance hierarchy. But actual hierarchies all include overriden and overloaded properties that these algorithms handle either very partially or not at all. Partially handled ..."
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Cited by 30 (13 self)
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Several algorithms [Cas92, MS89, Run92, DDHL94a, DDHL95, GMM95] have been proposed to automatically insert a class into an inheritance hierarchy. But actual hierarchies all include overriden and overloaded properties that these algorithms handle either very partially or not at all. Partially handled means handled provided there is a separate given function f able to compare overloaded properties [DDHL95, GMM95]. In this paper, we describe a new version of our algorithm (named Ares) which handles automatic class insertion more efficiently using such a function f . Although impossible to fully define, this function can be computed for a number of well defined cases of overloading and overriding. We give a classification of such cases and describe the computation process for a welldefined set of nontrivial cases. The algorithm preserves these important properties:  preservation of the maximal factorization of properties  preservation of the underlying structure (Galois lattice) of t...
Peakword condensation and submodule lattices: an application of the MEATAXE
 J. Symb. Comp
, 1994
"... Abstract. We describe a new condensation method for computing the submodule lattice of a module for a finite dimensional algebra over a finite field, which exploits the idea of condensation and extends it to the case of primitive idempotents. The method has been implemented in the new C version of t ..."
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Cited by 25 (14 self)
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Abstract. We describe a new condensation method for computing the submodule lattice of a module for a finite dimensional algebra over a finite field, which exploits the idea of condensation and extends it to the case of primitive idempotents. The method has been implemented in the new C version of the MeatAxe developed at Aachen, and we give several examples which have been analysed with our method.
A Combinatorial Treatment of Balancing Networks
, 1999
"... Balancing networks, originally introduced by Aspnes et al. (Proc. of the 23rd Annual ACM Symposium on Theory of Computing, pp. 348358, May 1991), represent a new class of distributed, lowcontention data structures suitable for solving many fundamental multiprocessor coordination problems that can ..."
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Cited by 23 (11 self)
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Balancing networks, originally introduced by Aspnes et al. (Proc. of the 23rd Annual ACM Symposium on Theory of Computing, pp. 348358, May 1991), represent a new class of distributed, lowcontention data structures suitable for solving many fundamental multiprocessor coordination problems that can be expressed as balancing problems. In this work, we present a mathematical study of the combinatorial structure of balancing networks, andavariety of its applications. Our study identies important combinatorial transfer parameters of balancing networks. In turn, necessary and sucient combinatorial conditions are established, expressed in terms of transfer parameters, which precisely characterize many important and well studied classes of balancing networks suchascounting networks and smoothing networks.We propose these combinatorial conditions to be \balancing analogs" of the well known ZeroOne principle holding for sorting networks.
Functional Dependencies in Relational Databases: A Lattice Point of View
, 1992
"... A lattice theoretic approach is developed to study the properties of functional dependencies in relational databases. The particular attention is paid to the analysis of the semilattice of closed sets, the lattice of all closure operations on a given set and to a new characterization of normal form ..."
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Cited by 23 (2 self)
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A lattice theoretic approach is developed to study the properties of functional dependencies in relational databases. The particular attention is paid to the analysis of the semilattice of closed sets, the lattice of all closure operations on a given set and to a new characterization of normal form relation schemes. Relation schemes with restrictions on functional dependencies are also studied. 1. Introduction The relational datamodel was defined by E.F. Codd [14] in 1970, and it is still one of the most powerful database models. In this model a relation is a matrix (table) every row of which corresponds to a record and every column to an attribute. This model has been widely studied. One of the most important branches in the theory of relational databases is that dealing with the design of database schemes. This branch is based on the theory of dependencies and constraints. In this paper we study the functional dependencies. Informally, functional dependency means that some attribu...
Algebraic Combinatorics in Mathematical Chemistry. Methods and Algorithms. I. Permutation Groups and Coherent (Cellular) Algebras.
, 1997
"... ..."
The price of anarchy for polynomial social cost
 In Proceedings of the 29th International Symposium of Mathematical Foundations of Computer Science (MFCS
, 2004
"... Abstract. In this work, we consider an interesting variant of the wellstudied KP model [18] for selfish routing that reflects some influence from the much older Wardrop model [31]. In the new model, user traffics are still unsplittable, while social cost is now the expectation of the sum, over all l ..."
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Cited by 18 (4 self)
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Abstract. In this work, we consider an interesting variant of the wellstudied KP model [18] for selfish routing that reflects some influence from the much older Wardrop model [31]. In the new model, user traffics are still unsplittable, while social cost is now the expectation of the sum, over all links, of a certain polynomial evaluated at the total latency incurred by all users choosing the link; we call it polynomial social cost. The polynomials that we consider have nonnegative coefficients. We are interested in evaluating Nash equilibria in this model, and we use the Price of Anarchy as our evaluation measure. We prove the Fully Mixed Nash Equilibrium Conjecture for identical users and two links, and establish an approximate version of the conjecture for arbitrary many links. Moreover, we give upper bounds on the Price of Anarchy. 1