9,133 research outputs found
Near-linear Time Algorithm for Approximate Minimum Degree Spanning Trees
Given a graph , we wish to compute a spanning tree whose maximum
vertex degree, i.e. tree degree, is as small as possible. Computing the exact
optimal solution is known to be NP-hard, since it generalizes the Hamiltonian
path problem. For the approximation version of this problem, a
time algorithm that computes a spanning tree of degree at most is
previously known [F\"urer \& Raghavachari 1994]; here denotes the
minimum tree degree of all the spanning trees. In this paper we give the first
near-linear time approximation algorithm for this problem. Specifically
speaking, we propose an time algorithm that
computes a spanning tree with tree degree for any constant .
Thus, when , we can achieve approximate solutions with
constant approximate ratio arbitrarily close to 1 in near-linear time.Comment: 17 page
Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees
Let be an -node and -edge positively real-weighted undirected
graph. For any given integer , we study the problem of designing a
sparse \emph{f-edge-fault-tolerant} (-EFT) {\em -approximate
single-source shortest-path tree} (-ASPT), namely a subgraph of
having as few edges as possible and which, following the failure of a set
of at most edges in , contains paths from a fixed source that are
stretched at most by a factor of . To this respect, we provide an
algorithm that efficiently computes an -EFT -ASPT of size . Our structure improves on a previous related construction designed for
\emph{unweighted} graphs, having the same size but guaranteeing a larger
stretch factor of , plus an additive term of .
Then, we show how to convert our structure into an efficient -EFT
\emph{single-source distance oracle} (SSDO), that can be built in
time, has size , and is able to report,
after the failure of the edge set , in time a
-approximate distance from the source to any node, and a
corresponding approximate path in the same amount of time plus the path's size.
Such an oracle is obtained by handling another fundamental problem, namely that
of updating a \emph{minimum spanning forest} (MSF) of after that a
\emph{batch} of simultaneous edge modifications (i.e., edge insertions,
deletions and weight changes) is performed. For this problem, we build in time a \emph{sensitivity} oracle of size , that
reports in time the (at most ) edges either exiting from
or entering into the MSF. [...]Comment: 16 pages, 4 figure
Decomposing highly edge-connected graphs into homomorphic copies of a fixed tree
The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for
every tree there exists a natural number such that the following
holds: If is a -edge-connected simple graph with size divisible by
the size of , then can be edge-decomposed into subgraphs isomorphic to
. So far this conjecture has only been verified for paths, stars, and a
family of bistars. We prove a weaker version of the Tree Decomposition
Conjecture, where we require the subgraphs in the decomposition to be
isomorphic to graphs that can be obtained from by vertex-identifications.
We call such a subgraph a homomorphic copy of . This implies the Tree
Decomposition Conjecture under the additional constraint that the girth of
is greater than the diameter of . As an application, we verify the Tree
Decomposition Conjecture for all trees of diameter at most 4.Comment: 18 page
A nearly-mlogn time solver for SDD linear systems
We present an improved algorithm for solving symmetrically diagonally
dominant linear systems. On input of an symmetric diagonally
dominant matrix with non-zero entries and a vector such that
for some (unknown) vector , our algorithm computes a
vector such that
{ denotes the A-norm} in time
The solver utilizes in a standard way a `preconditioning' chain of
progressively sparser graphs. To claim the faster running time we make a
two-fold improvement in the algorithm for constructing the chain. The new chain
exploits previously unknown properties of the graph sparsification algorithm
given in [Koutis,Miller,Peng, FOCS 2010], allowing for stronger preconditioning
properties. We also present an algorithm of independent interest that
constructs nearly-tight low-stretch spanning trees in time
, a factor of faster than the algorithm in
[Abraham,Bartal,Neiman, FOCS 2008]. This speedup directly reflects on the
construction time of the preconditioning chain.Comment: to appear in FOCS1
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