80 research outputs found
Dynamic Planar Embeddings of Dynamic Graphs
We present an algorithm to support the dynamic embedding in the plane of a
dynamic graph. An edge can be inserted across a face between two vertices on
the face boundary (we call such a vertex pair linkable), and edges can be
deleted. The planar embedding can also be changed locally by flipping
components that are connected to the rest of the graph by at most two vertices.
Given vertices , linkable decides whether and are
linkable in the current embedding, and if so, returns a list of suggestions for
the placement of in the embedding. For non-linkable vertices , we
define a new query, one-flip-linkable providing a suggestion for a flip
that will make them linkable if one exists. We support all updates and queries
in O(log) time. Our time bounds match those of Italiano et al. for a
static (flipless) embedding of a dynamic graph.
Our new algorithm is simpler, exploiting that the complement of a spanning
tree of a connected plane graph is a spanning tree of the dual graph. The
primal and dual trees are interpreted as having the same Euler tour, and a main
idea of the new algorithm is an elegant interaction between top trees over the
two trees via their common Euler tour.Comment: Announced at STACS'1
Planar Reachability in Linear Space and Constant Time
We show how to represent a planar digraph in linear space so that distance
queries can be answered in constant time. The data structure can be constructed
in linear time. This representation of reachability is thus optimal in both
time and space, and has optimal construction time. The previous best solution
used space for constant query time [Thorup FOCS'01].Comment: 20 pages, 5 figures, submitted to FoC
Faster Fully-Dynamic Minimum Spanning Forest
We give a new data structure for the fully-dynamic minimum spanning forest
problem in simple graphs. Edge updates are supported in
amortized time per operation, improving the amortized bound of
Holm et al. (STOC'98, JACM'01). We assume the Word-RAM model with standard
instructions.Comment: 13 pages, 2 figure
Dynamic Planar Embeddings of Dynamic Graphs
We present an algorithm to support the dynamic embedding in the plane
of a dynamic graph. An edge can be inserted across a face between two vertices on the boundary (we call such a vertex pair linkable), and edges can be deleted. The planar embedding can also be changed locally by flipping components that are connected to the rest of the graph by at most two vertices. Given vertices u,v, linkable(u,v) decides whether u and v are linkable, and if so, returns a list of suggestions for the placement of (u,v) in the embedding. For non-linkable vertices u,v, we define a new query, one-flip-linkable(u,v) providing a suggestion for a flip that will make them linkable if one exists. We will support all updates and queries in O(log^2 n) time. Our time bounds match those of Italiano et al. for a static (flipless) embedding of a dynamic graph.
Our new algorithm is simpler, exploiting that the complement of a spanning tree of a connected plane graph is a spanning tree of the dual graph. The primal and dual trees are interpreted as having the same Euler tour, and a main idea of the new algorithm is an elegant
interaction between top trees over the two trees via their common Euler tour
Good r-Divisions Imply Optimal Amortized Decremental Biconnectivity
We present a data structure that, given a graph G of n vertices and m edges, and a suitable pair of nested r-divisions of G, preprocesses G in O(m+n) time and handles any series of edge-deletions in O(m) total time while answering queries to pairwise biconnectivity in worst-case O(1) time. In case the vertices are not biconnected, the data structure can return a cutvertex separating them in worst-case O(1) time.
As an immediate consequence, this gives optimal amortized decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs
The Bane of Low-Dimensionality Clustering
In this paper, we give a conditional lower bound of on
running time for the classic k-median and k-means clustering objectives (where
n is the size of the input), even in low-dimensional Euclidean space of
dimension four, assuming the Exponential Time Hypothesis (ETH). We also
consider k-median (and k-means) with penalties where each point need not be
assigned to a center, in which case it must pay a penalty, and extend our lower
bound to at least three-dimensional Euclidean space.
This stands in stark contrast to many other geometric problems such as the
traveling salesman problem, or computing an independent set of unit spheres.
While these problems benefit from the so-called (limited) blessing of
dimensionality, as they can be solved in time or
in d dimensions, our work shows that widely-used clustering
objectives have a lower bound of , even in dimension four.
We complete the picture by considering the two-dimensional case: we show that
there is no algorithm that solves the penalized version in time less than
, and provide a matching upper bound of .
The main tool we use to establish these lower bounds is the placement of
points on the moment curve, which takes its inspiration from constructions of
point sets yielding Delaunay complexes of high complexity
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