79,373 research outputs found
Optimal Binary Search Trees with Near Minimal Height
Suppose we have n keys, n access probabilities for the keys, and n+1 access
probabilities for the gaps between the keys. Let h_min(n) be the minimal height
of a binary search tree for n keys. We consider the problem to construct an
optimal binary search tree with near minimal height, i.e.\ with height h <=
h_min(n) + Delta for some fixed Delta. It is shown, that for any fixed Delta
optimal binary search trees with near minimal height can be constructed in time
O(n^2). This is as fast as in the unrestricted case.
So far, the best known algorithms for the construction of height-restricted
optimal binary search trees have running time O(L n^2), whereby L is the
maximal permitted height. Compared to these algorithms our algorithm is at
least faster by a factor of log n, because L is lower bounded by log n
Counting smaller elements in the Tamari and m-Tamari lattices
We introduce new combinatorial objects, the interval- posets, that encode
intervals of the Tamari lattice. We then find a combinatorial interpretation of
the bilinear operator that appears in the functional equation of Tamari
intervals described by Chapoton. Thus, we retrieve this functional equation and
prove that the polynomial recursively computed from the bilinear operator on
each tree T counts the number of trees smaller than T in the Tamari order. Then
we show that a similar m + 1-linear operator is also used in the functionnal
equation of m-Tamari intervals. We explain how the m-Tamari lattices can be
interpreted in terms of m+1-ary trees or a certain class of binary trees. We
then use the interval-posets to recover the functional equation of m-Tamari
intervals and to prove a generalized formula that counts the number of elements
smaller than or equal to a given tree in the m-Tamari lattice.Comment: 46 pages + 3 pages of code appendix, 27 figures. Long version of
arXiv:1212.0751. To appear in Journal of Combinatorial Theory, Series
Recommended from our members
Generation of optimal binary split trees
A binary split tree is a search structure combining features of heaps and binary search trees. Building an optimal binary split tree was originally conjectured to be intractable due to difficulties in applying dynamic programming techniques to the problem. However, two algorithms have recently been published which purportedly generate optimal trees in O(n^5) time, for records with distinct access probabilities. An extension allowing non-distinct access probabilities required exponential time. These algorithms consider a range of values when only a single value is possible, and may select an infeasible value which leads to an incorrect result. A dynamic programming method for determining the correct value is given, resulting in an algorithm which builds an optimal binary split tree in O(n^5) time for non-distinct access probabilities and Θ(n^4) time for distinct access probabilities
Prefix Codes: Equiprobable Words, Unequal Letter Costs
Describes a near-linear-time algorithm for a variant of Huffman coding, in
which the letters may have non-uniform lengths (as in Morse code), but with the
restriction that each word to be encoded has equal probability. [See also
``Huffman Coding with Unequal Letter Costs'' (2002).]Comment: proceedings version in ICALP (1994
Extracting Hierarchies of Search Tasks & Subtasks via a Bayesian Nonparametric Approach
A significant amount of search queries originate from some real world
information need or tasks. In order to improve the search experience of the end
users, it is important to have accurate representations of tasks. As a result,
significant amount of research has been devoted to extracting proper
representations of tasks in order to enable search systems to help users
complete their tasks, as well as providing the end user with better query
suggestions, for better recommendations, for satisfaction prediction, and for
improved personalization in terms of tasks. Most existing task extraction
methodologies focus on representing tasks as flat structures. However, tasks
often tend to have multiple subtasks associated with them and a more
naturalistic representation of tasks would be in terms of a hierarchy, where
each task can be composed of multiple (sub)tasks. To this end, we propose an
efficient Bayesian nonparametric model for extracting hierarchies of such tasks
\& subtasks. We evaluate our method based on real world query log data both
through quantitative and crowdsourced experiments and highlight the importance
of considering task/subtask hierarchies.Comment: 10 pages. Accepted at SIGIR 2017 as a full pape
A simple fixed parameter tractable algorithm for computing the hybridization number of two (not necessarily binary) trees
Here we present a new fixed parameter tractable algorithm to compute the
hybridization number r of two rooted, not necessarily binary phylogenetic trees
on taxon set X in time (6^r.r!).poly(n)$, where n=|X|. The novelty of this
approach is its use of terminals, which are maximal elements of a natural
partial order on X, and several insights from the softwired clusters
literature. This yields a surprisingly simple and practical bounded-search
algorithm and offers an alternative perspective on the underlying combinatorial
structure of the hybridization number problem
- …