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Algorithmic Combinatorics Based on Slicing Posets [chapter]

Vijay K. Garg
2002 Lecture Notes in Computer Science
This paper describes a technique based on a generalization of Birkhoff's theorem of representation of finite distributive lattices that can be used for solving such problems mechanically and efficiently  ...  Specifically, we give an efficient (polynomial time) algorithm to enumerate, count or detect structures that satisfy B when the total set of structures is large but the set of structures satisfying B is  ...  Every distributive lattice is a ranked poset [17] . Predicates on ideals Our technique is crucially based on the notion of predicates on the set of ideals.  ...

Algorithmic combinatorics based on slicing posets

Vijay K. Garg
2006 Theoretical Computer Science
This paper describes a technique based on a generalization of Birkhoff's theorem of representation of finite distributive lattices that can be used for solving such problems mechanically and efficiently  ...  Specifically, we give an efficient (polynomial time) algorithm to enumerate, count or detect structures that satisfy B when the total set of structures is large but the set of structures satisfying B is  ...  Every distributive lattice is a ranked poset [17] . Predicates on ideals Our technique is crucially based on the notion of predicates on the set of ideals.  ...

Thomas Lam
2010 Electronic Journal of Combinatorics
We construct examples based upon: the Fibonacci differential poset, permutations, standard Young tableau, and plane binary trees.  ...  algorithm.  ...  These examples are based on various combinatorial objects: the Fibonacci differential poset (also called the Young-Fibonacci lattice), permutations, standard Young tableau, and plane binary trees.  ...

Page 9207 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews
[Garg, Vijay Kumar] (1-TX-ELC; Austin, TX) Algorithmic combinatorics based on slicing posets.  ...  The slicing results are based on a generalization of Birkhoff’s The- orem of representation of finite distributive lattices.  ...

Venn Diagrams and Symmetric Chain Decompositions in the Boolean Lattice

Jerrold Griggs, Charles E. Killian, Carla D. Savage
2004 Electronic Journal of Combinatorics
A consequence of the approach is a constructive proof that the quotient poset of ${\cal B}_n$, under the relation "equivalence under rotation", has a symmetric chain decomposition whenever $n$ is prime  ...  We would like to thank Frank Ruskey for many helpful discussions and for his advice on drawing the figures.  ...  For comments and suggestions on an earlier version of this paper we are grateful to several people including Branko Grünbaum, Frank Ruskey, Donald Knuth, and the referees.  ...

Quantitative and Algorithmic aspects of Barrier Synchronization in Concurrency

OLivier Bodini, Matthieu Dien, Antoine Genitrini, Frédéric Peschanski
2021 Discrete Mathematics & Theoretical Computer Science
Based on this procedure, we develop a generic algorithm to generate process executions uniformly at random.  ...  This paper is essentially focusing on the the notion of synchronization from the point of view of combinatorics.  ...  ISSN 1365-8050 Acknowledgements The authors would like to thank the anonymous referees who provided useful and detailed comments on previous versions of that article.  ...

Techniques and applications of computation slicing

Neeraj Mittal, Vijay K. Garg
2005 Distributed computing
Nonetheless, for such a predicate, we develop an efficient heuristic algorithm for computing an approximate slice.  ...  We prove that the slice of a computation is uniquely defined for all predicates. We also present efficient algorithms for computing the slice for several useful classes of predicates.  ...  However, the algorithms based on this approach typically have exponential running time.  ...

Techniques and Applications of Computation Slicing [article]

Neeraj Mittal, Vijay K. Garg
2003 arXiv   pre-print
We prove that the slice exists and is uniquely defined for all predicates. We present efficient slicing algorithms for several useful classes of predicates.  ...  We develop efficient heuristic algorithms for computing an approximate slice for predicates for which computing the slice is otherwise provably intractable.  ...  However, the algorithms based on this approach may have exponential running time.  ...

Page 8149 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews
Garg [Vijay Kumar Garg], Algorithmic combinatorics based on slicing posets (169- 181); Jens Gramm, Jiong Guo and Rolf Niedermeier, Pattern matching for arc-annotated sequences (182-193); Bernhard Heine  ...  Prasad Sistla, Formal languages and algorithms for similarity based re- trieval from sequence databases (324-335); Walter Vogler and Ralf Wollowski, Decomposition in asynchronous circuit design (336-347  ...

Cyclage, catabolism, and the affine Hecke algebra

Jonah Blasiak
This turns out to be closely related to the combinatorics of the cells of H n worked out by Shi, Lusztig, and Xi, and we state explicit conjectures along these lines.  ...  Multiplying canonical basis elements by a certain element π ∈ H + n corresponds to rotations of words, and on cells corresponds to cocyclage.  ...  We now relate combinatorics of the cellular subquotient R 1 n to the cocyclage poset on standard tableaux.  ...

Cyclage, catabolism, and the affine Hecke algebra [article]

Jonah Blasiak
2010 arXiv   pre-print
We conjecture how this filtration relates to the combinatorics of the cells of _n worked out by Shi, Lusztig, and Xi.  ...  Multiplying canonical basis elements by a certain element π∈_n corresponds to rotations of words, and on cells corresponds to cocyclage.  ...  We now relate combinatorics of the cellular subquotient R 1 n to the cocyclage poset on standard tableaux.  ...

Metric Combinatorics of Convex Polyhedra: Cut Loci and Nonoverlapping Unfoldings

Ezra Miller, Igor Pak
2008 Discrete & Computational Geometry
Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding  ...  We characterize the cut locus (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets.  ...  Topological and Geometric Combinatorics (April, 2003) .  ...

Metric Combinatorics of Convex Polyhedra: Cut Loci and Nonoverlapping Unfoldings

Ezra Miller, Igor Pak
2006 Discrete & Computational Geometry
Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding  ...  We characterize the cut locus (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets.  ...  Topological and Geometric Combinatorics (April, 2003) .  ...

Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings [article]

Ezra Miller, Igor Pak
2003 arXiv   pre-print
Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding  ...  This paper is a study of the interaction between the combinatorics of boundaries of convex polytopes in arbitrary dimension and their metric geometry.  ...  Thus as time progresses, wavefront expansion builds the source poset by adding one event at a time. The algorithm.  ...

Combinatorial Reciprocity Theorems [article]

Matthias Beck
2012 arXiv   pre-print
A common theme of enumerative combinatorics is formed by counting functions that are polynomials evaluated at positive integers.  ...  In this expository paper, we focus on four families of such counting functions connected to hyperplane arrangements, lattice points in polyhedra, proper colorings of graphs, and P-partitions.  ...  For more on the combinatorics of hyperplane arrangements, we recommend the survey article [29] .  ...
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