Distributed Algorithms for Scheduling on Line and Tree Networks

We have a set of processors (or agents) and a set of graph networks defined over some vertex set. Each processor can access a subset of the graph networks. Each processor has a demand specified as a pair of vertices $$, along with a profit; the processor wishes to send data between $u$ and $v$. Towards that goal, the processor needs to select a graph network accessible to it and a path connecting $u$ and $v$ within the selected network. The processor requires exclusive access to the chosen path, in order to route the data. Thus, the processors are competing for routes/channels. A feasible solution selects a subset of demands and schedules each selected demand on a graph network accessible to the processor owning the demand; the solution also specifies the paths to use for this purpose. The requirement is that for any two demands scheduled on the same graph network, their chosen paths must be edge disjoint. The goal is to output a solution having the maximum aggregate profit. Prior work has addressed the above problem in a distibuted setting for the special case where all the graph networks are simply paths (i.e, line-networks). Distributed constant factor approximation algorithms are known for this case. The main contributions of this paper are twofold. First we design a distributed constant factor approximation algorithm for the more general case of tree-networks. The core component of our algorithm is a tree-decomposition technique, which may be of independent interest. Secondly, for the case of line-networks, we improve the known approximation guarantees by a factor of 5. Our algorithms can also handle the capacitated scenario, wherein the demands and edges have bandwidth requirements and capacities, respectively.Comment: Accepted to PODC 2012, full versio

Subgraph Counting: Color Coding Beyond Trees

The problem of counting occurrences of query graphs in a large data graph, known as subgraph counting, is fundamental to several domains such as genomics and social network analysis. Many important special cases (e.g. triangle counting) have received significant attention. Color coding is a very general and powerful algorithmic technique for subgraph counting. Color coding has been shown to be effective in several applications, but scalable implementations are only known for the special case of {\em tree queries} (i.e. queries of treewidth one). In this paper we present the first efficient distributed implementation for color coding that goes beyond tree queries: our algorithm applies to any query graph of treewidth $2$. Since tree queries can be solved in time linear in the size of the data graph, our contribution is the first step into the realm of colour coding for queries that require superlinear running time in the worst case. This superlinear complexity leads to significant load balancing problems on graphs with heavy tailed degree distributions. Our algorithm structures the computation to work around high degree nodes in the data graph, and achieves very good runtime and scalability on a diverse collection of data and query graph pairs as a result. We also provide theoretical analysis of our algorithmic techniques, showing asymptotic improvements in runtime on random graphs with power law degree distributions, a popular model for real world graphs

Chylous Leak During Posterior Approach to Juvenile Scoliosis Surgery: A Case Report.

CaseWe report the first documented case of chylous leak recognized intraoperatively during posterior spinal instrumentation and fusion for juvenile scoliosis in a female patient with a history of thoracotomy and decortication for an empyema.ConclusionsThoracic duct injury can lead to severe morbidity and mortality because of chylothorax formation. Although chylous leaks are a well-documented complication of the anterior approach to spine surgery, leaks during the posterior approach are rarely reported. When these chylous leaks are recognized intraoperatively, the likelihood of serious complications may be minimized by drain placement before closure

On Optimizing Distributed Tucker Decomposition for Dense Tensors

The Tucker decomposition expresses a given tensor as the product of a small core tensor and a set of factor matrices. Apart from providing data compression, the construction is useful in performing analysis such as principal component analysis (PCA)and finds applications in diverse domains such as signal processing, computer vision and text analytics. Our objective is to develop an efficient distributed implementation for the case of dense tensors. The implementation is based on the HOOI (Higher Order Orthogonal Iterator) procedure, wherein the tensor-times-matrix product forms the core routine. Prior work have proposed heuristics for reducing the computational load and communication volume incurred by the routine. We study the two metrics in a formal and systematic manner, and design strategies that are optimal under the two fundamental metrics. Our experimental evaluation on a large benchmark of tensors shows that the optimal strategies provide significant reduction in load and volume compared to prior heuristics, and provide up to 7x speed-up in the overall running time.Comment: Preliminary version of the paper appears in the proceedings of IPDPS'1

Finding Independent Sets in Unions of Perfect Graphs

The maximum independent set problem (MaxIS) on general graphs is known to be NP-hard to approximate within a factor of $n^{1-epsilon}$, for any $epsilon > 0$. However, there are many ``easy" classes of graphs on which the problem can be solved in polynomial time. In this context, an interesting question is that of computing the maximum independent set in a graph that can be expressed as the union of a small number of graphs from an easy class. The MaxIS problem has been studied on unions of interval graphs and chordal graphs. We study the MaxIS problem on unions of perfect graphs (which generalize the above two classes). We present an $O(sqrt{n})$-approximation algorithm when the input graph is the union of two perfect graphs. We also show that the MaxIS problem on unions of two comparability graphs (a subclass of perfect graphs) cannot be approximated within any constant factor