14,940 research outputs found
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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
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Fully dynamic maintenance of euclidean minimum spanning trees
We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in time O(n^5/6 log1^2/2 n) per update operation. No nontrivial dynamic geometric minimum spanning tree algorithm was previously known. We reduce the problem to maintaining bichromatic closest pairs, which we also solve in the same time bounds. 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
The Power of Dynamic Distance Oracles: Efficient Dynamic Algorithms for the Steiner Tree
In this paper we study the Steiner tree problem over a dynamic set of
terminals. We consider the model where we are given an -vertex graph
with positive real edge weights, and our goal is to maintain a tree
which is a good approximation of the minimum Steiner tree spanning a terminal
set , which changes over time. The changes applied to the
terminal set are either terminal additions (incremental scenario), terminal
removals (decremental scenario), or both (fully dynamic scenario). Our task
here is twofold. We want to support updates in sublinear time, and keep
the approximation factor of the algorithm as small as possible. We show that we
can maintain a -approximate Steiner tree of a general graph in
time per terminal addition or removal. Here,
denotes the stretch of the metric induced by . For planar graphs we achieve
the same running time and the approximation ratio of .
Moreover, we show faster algorithms for incremental and decremental scenarios.
Finally, we show that if we allow higher approximation ratio, even more
efficient algorithms are possible. In particular we show a polylogarithmic time
-approximate algorithm for planar graphs.
One of the main building blocks of our algorithms are dynamic distance
oracles for vertex-labeled graphs, which are of independent interest. We also
improve and use the online algorithms for the Steiner tree problem.Comment: Full version of the paper accepted to STOC'1
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Offline algorithms for dynamic minimum spanning tree problems
We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications. The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known. The algorithm performs (log n) work per modification, where n is the number of vertices in the graph. We use our techniques to solve the offline geometric MST problem for a planar point set subject to insertions and deletions; our algorithm for this problem performs O(log^2 n) work per modification. No previous dynamic geometric MST algorithm was known
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
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