63,906 research outputs found
Pattern Avoidance in k-ary Heaps
In this paper, we consider pattern avoidance in k-ary heaps, where the permutation associated with the heap is found by recording the nodes as they are encountered in a breadth-first search. We enumerate heaps that avoid patterns of length 3 and collections of patterns of length 3, first with binary heaps and then more generally with k-ary heaps
Zipf's Law Leads to Heaps' Law: Analyzing Their Relation in Finite-Size Systems
Background: Zipf's law and Heaps' law are observed in disparate complex
systems. Of particular interests, these two laws often appear together. Many
theoretical models and analyses are performed to understand their co-occurrence
in real systems, but it still lacks a clear picture about their relation.
Methodology/Principal Findings: We show that the Heaps' law can be considered
as a derivative phenomenon if the system obeys the Zipf's law. Furthermore, we
refine the known approximate solution of the Heaps' exponent provided the
Zipf's exponent. We show that the approximate solution is indeed an asymptotic
solution for infinite systems, while in the finite-size system the Heaps'
exponent is sensitive to the system size. Extensive empirical analysis on tens
of disparate systems demonstrates that our refined results can better capture
the relation between the Zipf's and Heaps' exponents. Conclusions/Significance:
The present analysis provides a clear picture about the relation between the
Zipf's law and Heaps' law without the help of any specific stochastic model,
namely the Heaps' law is indeed a derivative phenomenon from Zipf's law. The
presented numerical method gives considerably better estimation of the Heaps'
exponent given the Zipf's exponent and the system size. Our analysis provides
some insights and implications of real complex systems, for example, one can
naturally obtained a better explanation of the accelerated growth of scale-free
networks.Comment: 15 pages, 6 figures, 1 Tabl
Hollow Heaps
We introduce the hollow heap, a very simple data structure with the same
amortized efficiency as the classical Fibonacci heap. All heap operations
except delete and delete-min take time, worst case as well as amortized;
delete and delete-min take amortized time on a heap of items.
Hollow heaps are by far the simplest structure to achieve this. Hollow heaps
combine two novel ideas: the use of lazy deletion and re-insertion to do
decrease-key operations, and the use of a dag (directed acyclic graph) instead
of a tree or set of trees to represent a heap. Lazy deletion produces hollow
nodes (nodes without items), giving the data structure its name.Comment: 27 pages, 7 figures, preliminary version appeared in ICALP 201
Quantum heaps, cops and heapy categories
A heap is a structure with a ternary operation which is intuitively a group
with forgotten unit element. Quantum heaps are associative algebras with a
ternary cooperation which are to the Hopf algebras what heaps are to groups,
and, in particular, the category of copointed quantum heaps is isomorphic to
the category of Hopf algebras. There is an intermediate structure of a cop in
monoidal category which is in the case of vector spaces to a quantum heap about
what is a coalgebra to a Hopf algebra. The representations of Hopf algebras
make a rigid monoidal category. Similarly the representations of quantum heaps
make a kind of category with ternary products, which we call a heapy category.Comment: 10 pages, an adaptation of an old 2001 preprin
A new heap game
Given heaps of tokens. The moves of the 2-player game introduced
here are to either take a positive number of tokens from at most heaps,
or to remove the {\sl same} positive number of tokens from all the heaps.
We analyse this extension of Wythoff's game and provide a polynomial-time
strategy for it.Comment: To appear in Computer Games 199
On rank functions for heaps
Motivated by work of Stembridge, we study rank functions for Viennot's heaps
of pieces. We produce a simple and sufficient criterion for a heap to be a
ranked poset and apply the results to the heaps arising from fully commutative
words in Coxeter groups.Comment: 18 pages AMSTeX, 3 figure
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