3,529 research outputs found
Why some heaps support constant-amortized-time decrease-key operations, and others do not
A lower bound is presented which shows that a class of heap algorithms in the
pointer model with only heap pointers must spend Omega(log log n / log log log
n) amortized time on the decrease-key operation (given O(log n) amortized-time
extract-min). Intuitively, this bound shows the key to having O(1)-time
decrease-key is the ability to sort O(log n) items in O(log n) time; Fibonacci
heaps [M.L. Fredman and R. E. Tarjan. J. ACM 34(3):596-615 (1987)] do this
through the use of bucket sort. Our lower bound also holds no matter how much
data is augmented; this is in contrast to the lower bound of Fredman [J. ACM
46(4):473-501 (1999)] who showed a tradeoff between the number of augmented
bits and the amortized cost of decrease-key. A new heap data structure, the
sort heap, is presented. This heap is a simplification of the heap of Elmasry
[SODA 2009: 471-476] and shares with it a O(log log n) amortized-time
decrease-key, but with a straightforward implementation such that our lower
bound holds. Thus a natural model is presented for a pointer-based heap such
that the amortized runtime of a self-adjusting structure and amortized lower
asymptotic bounds for decrease-key differ by but a O(log log log n) factor
Strongly-cyclic branched coverings of (1,1)-knots and cyclic presentations of groups
We study the connections among the mapping class group of the twice punctured
torus, the cyclic branched coverings of (1,1)-knots and the cyclic
presentations of groups. We give the necessary and sufficient conditions for
the existence and uniqueness of the n-fold strongly-cyclic branched coverings
of (1,1)-knots, through the elements of the mapping class group. We prove that
every n-fold strongly-cyclic branched covering of a (1,1)-knot admits a cyclic
presentation for the fundamental group, arising from a Heegaard splitting of
genus n. Moreover, we give an algorithm to produce the cyclic presentation and
illustrate it in the case of cyclic branched coverings of torus knots of type
(k,hk+1) and (k,hk-1).Comment: 16 pages, 2 figures. to appear in the Mathematical Proceedings of the
Cambridge Philosophical Societ
Finite Countermodel Based Verification for Program Transformation (A Case Study)
Both automatic program verification and program transformation are based on
program analysis. In the past decade a number of approaches using various
automatic general-purpose program transformation techniques (partial deduction,
specialization, supercompilation) for verification of unreachability properties
of computing systems were introduced and demonstrated. On the other hand, the
semantics based unfold-fold program transformation methods pose themselves
diverse kinds of reachability tasks and try to solve them, aiming at improving
the semantics tree of the program being transformed. That means some
general-purpose verification methods may be used for strengthening program
transformation techniques. This paper considers the question how finite
countermodels for safety verification method might be used in Turchin's
supercompilation method. We extract a number of supercompilation sub-algorithms
trying to solve reachability problems and demonstrate use of an external
countermodel finder for solving some of the problems.Comment: In Proceedings VPT 2015, arXiv:1512.0221
Geometrical approach to SU(2) navigation with Fibonacci anyons
Topological quantum computation with Fibonacci anyons relies on the
possibility of efficiently generating unitary transformations upon
pseudoparticles braiding. The crucial fact that such set of braids has a dense
image in the unitary operations space is well known; in addition, the
Solovay-Kitaev algorithm allows to approach a given unitary operation to any
desired accuracy. In this paper, the latter task is fulfilled with an
alternative method, in the SU(2) case, based on a generalization of the
geodesic dome construction to higher dimension.Comment: 12 pages, 5 figure
A Tight Lower Bound for Decrease-Key in the Pure Heap Model
We improve the lower bound on the amortized cost of the decrease-key
operation in the pure heap model and show that any pure-heap-model heap (that
has a \bigoh{\log n} amortized-time extract-min operation) must spend
\bigom{\log\log n} amortized time on the decrease-key operation. Our result
shows that sort heaps as well as pure-heap variants of numerous other heaps
have asymptotically optimal decrease-key operations in the pure heap model. In
addition, our improved lower bound matches the lower bound of Fredman [J. ACM
46(4):473-501 (1999)] for pairing heaps [M.L. Fredman, R. Sedgewick, D.D.
Sleator, and R.E. Tarjan. Algorithmica 1(1):111-129 (1986)] and surpasses it
for pure-heap variants of numerous other heaps with augmented data such as
pointer rank-pairing heaps.Comment: arXiv admin note: substantial text overlap with arXiv:1302.664
On periodic Takahashi manifolds
In this paper we show that periodic Takahashi 3-manifolds are cyclic
coverings of the connected sum of two lens spaces (possibly cyclic coverings of
the 3-sphere), branched over knots. When the base space is a 3-sphere, we prove
that the associated branching set is a two-bridge knot of genus one, and we
determine its type. Moreover, a geometric cyclic presentation for the
fundamental groups of these manifolds is obtained in several interesting cases,
including the ones corresponding to the branched cyclic coverings of the
3-sphere.Comment: 12 pages, 5 figures. To appear in Tsukuba Journal of Mathematic
Partial-indistinguishability obfuscation using braids
An obfuscator is an algorithm that translates circuits into
functionally-equivalent similarly-sized circuits that are hard to understand.
Efficient obfuscators would have many applications in cryptography. Until
recently, theoretical progress has mainly been limited to no-go results. Recent
works have proposed the first efficient obfuscation algorithms for classical
logic circuits, based on a notion of indistinguishability against
polynomial-time adversaries. In this work, we propose a new notion of
obfuscation, which we call partial-indistinguishability. This notion is based
on computationally universal groups with efficiently computable normal forms,
and appears to be incomparable with existing definitions. We describe universal
gate sets for both classical and quantum computation, in which our definition
of obfuscation can be met by polynomial-time algorithms. We also discuss some
potential applications to testing quantum computers. We stress that the
cryptographic security of these obfuscators, especially when composed with
translation from other gate sets, remains an open question.Comment: 21 pages,Proceedings of TQC 201
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