3,502 research outputs found
Reconstructing Biological and Digital Phylogenetic Trees in Parallel
In this paper, we study the parallel query complexity of reconstructing biological and digital phylogenetic trees from simple queries involving their nodes. This is motivated from computational biology, data protection, and computer security settings, which can be abstracted in terms of two parties, a responder, Alice, who must correctly answer queries of a given type regarding a degree-d tree, T, and a querier, Bob, who issues batches of queries, with each query in a batch being independent of the others, so as to eventually infer the structure of T. We show that a querier can efficiently reconstruct an n-node degree-d tree, T, with a logarithmic number of rounds and quasilinear number of queries, with high probability, for various types of queries, including relative-distance queries and path queries. Our results are all asymptotically optimal and improve the asymptotic (sequential) query complexity for one of the problems we study. Moreover, through an experimental analysis using both real-world and synthetic data, we provide empirical evidence that our algorithms provide significant parallel speedups while also improving the total query complexities for the problems we study
Optimal distance query reconstruction for graphs without long induced cycles
Let be an -vertex connected graph of maximum degree .
Given access to and an oracle that given two vertices , returns
the shortest path distance between and , how many queries are needed to
reconstruct ? We give a simple deterministic algorithm to reconstruct trees
using distance queries and show that even
randomised algorithms need to use at least
queries in expectation. The best previous lower bound was an
information-theoretic lower bound of . Our lower
bound also extends to related query models including distance queries for
phylogenetic trees, membership queries for learning partitions and path queries
in directed trees.
We extend our deterministic algorithm to reconstruct graphs without induced
cycles of length at least using queries, which
includes various graph classes of interest such as chordal graphs, permutation
graphs and AT-free graphs. Since the previously best known randomised algorithm
for chordal graphs uses queries in expectation, we both
get rid off the randomness and get the optimal dependency in for chordal
graphs and various other graph classes.
Finally, we build on an algorithm of Kannan, Mathieu, and Zhou [ICALP, 2015]
to give a randomised algorithm for reconstructing graphs of treelength
using queries in expectation.Comment: 35 page
Recommended from our members
Fully dynamic maintenance of Euclidean minimum spanning trees and maxima of decomposable functions
We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in time O(n^1/2 log^2 n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in time O(n^E) per update. Our algorithm uses a novel construction, the ordered nearest neighbors of a sequence of points. Any point set or bichromatic point set can be ordered so that this graph is a simple path. Our results generalize to higher dimensions, and to fully dynamic algorithms for maintaining maxima of decomposable functions, including the diameter of a point set and the bichromatic farthest pair
Distance Oracles for Time-Dependent Networks
We present the first approximate distance oracle for sparse directed networks
with time-dependent arc-travel-times determined by continuous, piecewise
linear, positive functions possessing the FIFO property.
Our approach precomputes approximate distance summaries from
selected landmark vertices to all other vertices in the network. Our oracle
uses subquadratic space and time preprocessing, and provides two sublinear-time
query algorithms that deliver constant and approximate
shortest-travel-times, respectively, for arbitrary origin-destination pairs in
the network, for any constant . Our oracle is based only on
the sparsity of the network, along with two quite natural assumptions about
travel-time functions which allow the smooth transition towards asymmetric and
time-dependent distance metrics.Comment: A preliminary version appeared as Technical Report ECOMPASS-TR-025 of
EU funded research project eCOMPASS (http://www.ecompass-project.eu/). An
extended abstract also appeared in the 41st International Colloquium on
Automata, Languages, and Programming (ICALP 2014, track-A
- …