4 research outputs found
Adaptive Out-Orientations with Applications
We give simple algorithms for maintaining edge-orientations of a
fully-dynamic graph, such that the out-degree of each vertex is bounded. On one
hand, we show how to orient the edges such that the out-degree of each vertex
is proportional to the arboricity of the graph, in a worst-case update
time of . On the other hand, motivated by applications
in dynamic maximal matching, we obtain a different trade-off, namely the
improved worst case update time of for the problem of
maintaining an edge-orientation with at most out-edges per
vertex. Since our algorithms have update times with worst-case guarantees, the
number of changes to the solution (i.e. the recourse) is naturally limited.
Our algorithms make choices based entirely on local information, which makes
them automatically adaptive to the current arboricity of the graph. In other
words, they are arboricity-oblivious, while they are arboricity-sensitive. This
both simplifies and improves upon previous work, by having fewer assumptions or
better asymptotic guarantees.
As a consequence, one obtains an algorithm with improved efficiency for
maintaining a approximation of the maximum subgraph density,
and an algorithm for dynamic maximal matching whose worst-case update time is
guaranteed to be upper bounded by , where
is the arboricity at the time of the update
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