4,079 research outputs found
A primal Barvinok algorithm based on irrational decompositions
We introduce variants of Barvinok's algorithm for counting lattice points in
polyhedra. The new algorithms are based on irrational signed decomposition in
the primal space and the construction of rational generating functions for
cones with low index. We give computational results that show that the new
algorithms are faster than the existing algorithms by a large factor.Comment: v3: New all-primal algorithm. v4: Extended introduction, updated
computational results. To appear in SIAM Journal on Discrete Mathematic
Counting Points and Hilbert Series in String Theory
The problem of counting points is revisited from the perspective of reflexive
4-dimensional polytopes. As an application, the Hilbert series of the
473,800,776 reflexive polytopes (equivalently, their Calabi-Yau hypersurfaces)
are computed.Comment: 13 pages, 4 figures. Contribution to the Max Kreuzer memorial volum
Computing parametric rational generating functions with a primal Barvinok algorithm
Computations with Barvinok's short rational generating functions are
traditionally being performed in the dual space, to avoid the combinatorial
complexity of inclusion--exclusion formulas for the intersecting proper faces
of cones. We prove that, on the level of indicator functions of polyhedra,
there is no need for using inclusion--exclusion formulas to account for
boundary effects: All linear identities in the space of indicator functions can
be purely expressed using half-open variants of the full-dimensional polyhedra
in the identity. This gives rise to a practically efficient, parametric
Barvinok algorithm in the primal space.Comment: 16 pages, 1 figure; v2: Minor corrections, new example and summary of
algorithm; submitted to journa
Integer polyhedra for program analysis
Polyhedra are widely used in model checking and abstract interpretation. Polyhedral analysis is effective when the relationships between variables are linear, but suffers from imprecision when it is necessary to take into account the integrality of the represented space. Imprecision also arises when non-linear constraints occur. Moreover, in terms of tractability, even a space defined by linear constraints can become unmanageable owing to the excessive number of inequalities. Thus it is useful to identify those inequalities whose omission has least impact on the represented space. This paper shows how these issues can be addressed in a novel way by growing the integer hull of the space and approximating the number of integral points within a bounded polyhedron
Symbolic and analytic techniques for resource analysis of Java bytecode
Recent work in resource analysis has translated the idea of amortised resource analysis to imperative languages using a program logic that allows mixing of assertions about heap shapes, in the tradition of separation logic, and assertions about consumable resources. Separately, polyhedral methods have been used to calculate bounds on numbers of iterations in loop-based programs. We are attempting to combine these ideas to deal with Java programs involving both data structures and loops, focusing on the bytecode level rather than on source code
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