8,344 research outputs found
A Linear Time Algorithm for Finding Minimum Spanning Tree Replacement Edges
Given an undirected, weighted graph, the minimum spanning tree (MST) is a
tree that connects all of the vertices of the graph with minimum sum of edge
weights. In real world applications, network designers often seek to quickly
find a replacement edge for each edge in the MST. For example, when a traffic
accident closes a road in a transportation network, or a line goes down in a
communication network, the replacement edge may reconnect the MST at lowest
cost. In the paper, we consider the case of finding the lowest cost replacement
edge for each edge of the MST. A previous algorithm by Tarjan takes time, where is the inverse Ackermann's function.
Given the MST and sorted non-tree edges, our algorithm is the first that runs
in time and space to find all replacement edges. Moreover, it
is easy to implement and our experimental study demonstrates fast performance
on several types of graphs. Additionally, since the most vital edge is the tree
edge whose removal causes the highest cost, our algorithm finds it in linear
time
Parallel Batch-Dynamic Graph Connectivity
In this paper, we study batch parallel algorithms for the dynamic
connectivity problem, a fundamental problem that has received considerable
attention in the sequential setting. The most well known sequential algorithm
for dynamic connectivity is the elegant level-set algorithm of Holm, de
Lichtenberg and Thorup (HDT), which achieves amortized time per
edge insertion or deletion, and time per query. We
design a parallel batch-dynamic connectivity algorithm that is work-efficient
with respect to the HDT algorithm for small batch sizes, and is asymptotically
faster when the average batch size is sufficiently large. Given a sequence of
batched updates, where is the average batch size of all deletions, our
algorithm achieves expected amortized work per
edge insertion and deletion and depth w.h.p. Our algorithm
answers a batch of connectivity queries in expected
work and depth w.h.p. To the best of our knowledge, our algorithm
is the first parallel batch-dynamic algorithm for connectivity.Comment: This is the full version of the paper appearing in the ACM Symposium
on Parallelism in Algorithms and Architectures (SPAA), 201
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
An O(1)-Approximation for Minimum Spanning Tree Interdiction
Network interdiction problems are a natural way to study the sensitivity of a
network optimization problem with respect to the removal of a limited set of
edges or vertices. One of the oldest and best-studied interdiction problems is
minimum spanning tree (MST) interdiction. Here, an undirected multigraph with
nonnegative edge weights and positive interdiction costs on its edges is given,
together with a positive budget B. The goal is to find a subset of edges R,
whose total interdiction cost does not exceed B, such that removing R leads to
a graph where the weight of an MST is as large as possible. Frederickson and
Solis-Oba (SODA 1996) presented an O(log m)-approximation for MST interdiction,
where m is the number of edges. Since then, no further progress has been made
regarding approximations, and the question whether MST interdiction admits an
O(1)-approximation remained open.
We answer this question in the affirmative, by presenting a 14-approximation
that overcomes two main hurdles that hindered further progress so far.
Moreover, based on a well-known 2-approximation for the metric traveling
salesman problem (TSP), we show that our O(1)-approximation for MST
interdiction implies an O(1)-approximation for a natural interdiction version
of metric TSP
Effective Edge-Fault-Tolerant Single-Source Spanners via Best (or Good) Swap Edges
Computing \emph{all best swap edges} (ABSE) of a spanning tree of a given
-vertex and -edge undirected and weighted graph means to select, for
each edge of , a corresponding non-tree edge , in such a way that the
tree obtained by replacing with enjoys some optimality criterion (which
is naturally defined according to some objective function originally addressed
by ). Solving efficiently an ABSE problem is by now a classic algorithmic
issue, since it conveys a very successful way of coping with a (transient)
\emph{edge failure} in tree-based communication networks: just replace the
failing edge with its respective swap edge, so as that the connectivity is
promptly reestablished by minimizing the rerouting and set-up costs. In this
paper, we solve the ABSE problem for the case in which is a
\emph{single-source shortest-path tree} of , and our two selected swap
criteria aim to minimize either the \emph{maximum} or the \emph{average
stretch} in the swap tree of all the paths emanating from the source. Having
these criteria in mind, the obtained structures can then be reviewed as
\emph{edge-fault-tolerant single-source spanners}. For them, we propose two
efficient algorithms running in and time, respectively, and we show that the guaranteed (either
maximum or average, respectively) stretch factor is equal to 3, and this is
tight. Moreover, for the maximum stretch, we also propose an almost linear time algorithm computing a set of \emph{good} swap edges,
each of which will guarantee a relative approximation factor on the maximum
stretch of (tight) as opposed to that provided by the corresponding BSE.
Surprisingly, no previous results were known for these two very natural swap
problems.Comment: 15 pages, 4 figures, SIROCCO 201
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