13 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
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
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
Algorithms for Fundamental Problems in Computer Networks.
Traditional studies of algorithms consider the sequential setting, where the whole input data is fed into a single device that computes the solution. Today, the network, such as the Internet, contains of a vast amount of information. The overhead of aggregating all the information into a single device is too expensive, so a distributed approach to solve the problem is often preferable. In this thesis, we aim to develop efficient algorithms for the following fundamental graph problems that arise in networks, in both sequential and distributed settings.
Graph coloring is a basic symmetry breaking problem in distributed computing. Each node is to be assigned a color such that adjacent nodes are assigned different colors. Both the efficiency and the quality of coloring are important measures of an algorithm. One of our main contributions is providing tools for obtaining colorings of good quality whose existence are non-trivial. We also consider other optimization problems in the distributed setting. For example, we investigate efficient methods for identifying the connectivity as well as the bottleneck edges in a distributed network. Our approximation algorithm is almost-tight in the sense that the running time matches the known lower bound up to a poly-logarithmic factor. For another example, we model how the task allocation can be done in ant colonies, when the ants may have different capabilities in doing different tasks.
The matching problems are one of the classic combinatorial optimization problems. We study the weighted matching problems in the sequential setting. We give a new scaling algorithm for finding the maximum weight perfect matching in general graphs, which improves the long-standing Gabow-Tarjan's algorithm (1991) and matches the running time of the best weighted bipartite perfect matching algorithm (Gabow and Tarjan, 1989). Furthermore, for the maximum weight matching problem in bipartite graphs, we give a faster scaling algorithm whose running time is faster than Gabow and Tarjan's weighted bipartite {it perfect} matching algorithm.PhDComputer Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113540/1/hsinhao_1.pd