Analyzing Massive Graphs in the Semi-streaming Model

Massive graphs arise in a many scenarios, for example, traffic data analysis in large networks, large scale scientific experiments, and clustering of large data sets. The semi-streaming model was proposed for processing massive graphs. In the semi-streaming model, we have a random accessible memory which is near-linear in the number of vertices. The input graph (or equivalently, edges in the graph) is presented as a sequential list of edges (insertion-only model) or edge insertions and deletions (dynamic model). The list is read-only but we may make multiple passes over the list. There has been a few results in the insertion-only model such as computing distance spanners and approximating the maximum matching. In this thesis, we present some algorithms and techniques for (i) solving more complex problems in the semi-streaming model, (for example, problems in the dynamic model) and (ii) having better solutions for the problems which have been studied (for example, the maximum matching problem). In course of both of these, we develop new techniques with broad applications and explore the rich trade-offs between the complexity of models (insertion-only streams vs. dynamic streams), the number of passes, space, accuracy, and running time. 1. We initiate the study of dynamic graph streams. We start with basic problems such as the connectivity problem and computing the minimum spanning tree. These problems are trivial in the insertion-only model. However, they require non-trivial (and multiple passes for computing the exact minimum spanning tree) algorithms in the dynamic model. 2. Second, we present a graph sparsification algorithm in the semi-streaming model. A graph sparsification is a sparse graph that approximately preserves all the cut values of a graph. Such a graph acts as an oracle for solving cut-related problems, for example, the minimum cut problem and the multicut problem. Our algorithm produce a graph sparsification with high probability in one pass. 3. Third, we use the primal-dual algorithms to develop the semi-streaming algorithms. The primal-dual algorithms have been widely accepted as a framework for solving linear programs and semidefinite programs faster. In contrast, we apply the method for reducing space and number of passes in addition to reducing the running time. We also present some examples that arise in applications and show how to apply the techniques: the multicut problem, the correlation clustering problem, and the maximum matching problem. As a consequence, we also develop near-linear time algorithms for the $b$-matching problems which were not known before

Correlation Clustering in Data Streams

Clustering is a fundamental tool for analyzing large data sets. A rich body of work has been devoted to designing data-stream algorithms for the relevant optimization problems such as k-center, k-median, and k-means. Such algorithms need to be both time and and space efcient. In this paper, we address the problem of correlation clustering in the dynamic data stream model. The stream consists of updates to the edge weights of a graph on n nodes and the goal is to find a node-partition such that the end-points of negative-weight edges are typically in diferent clusters whereas the end-points of positive-weight edges are typically in the same cluster. We present polynomial-time, O(n ⋅ polylog n)-space approximation algorithms for natural problems that arise. We frst develop data structures based on linear sketches that allow the “quality” of a given node-partition to be measured. We then combine these data structures with convex programming and sampling techniques to solve the relevant approximation problem. Unfortunately, the standard LP and SDP formulations are not obviously solvable in O(n ⋅ polylog n)-space. Our work presents space-efcient algorithms for the convex programming required, as well as approaches to reduce the adaptivity of the sampling

Correlation clustering in data streams

In this paper, we address the problem of correlation clustering in the dynamic data stream model. The stream consists of updates to the edge weights of a graph on n nodes and the goal is to find a node-partition such that the end-points of negative-weight edges are typically in different clusters whereas the end-points of positive-weight edges are typically in the same cluster. We present polynomial-time, O(n·polylog n)-space approximation algorithms for natural problems that arise. We first develop data structures based on linear sketches that allow the “quality” of a given node-partition to be measured. We then combine these data structures with convex programming and sampling techniques to solve the relevant approximation problem. However the standard LP and SDP formulations are not obviously solvable in O(n·polylog n)-space. Our work presents space-efficient algorithms for the convex programming required, as well as approaches to reduce the adaptivity of the sampling. Note that the improved space and running-time bounds achieved from streaming algorithms are also useful for offline settings such as MapReduce models

Cardioprotective Effect of the SDF-1α/CXCR4 Axis in Ischemic Postconditioning in Isolated Rat Hearts

Background and Objectives: Information about the role of the stromal cell-derived factor1a (SDF-1 alpha)/chemokine receptor type 4 (CXCR4) axis in ischemic postconditioning (IPOC) is currently limited. We hypothesized that the SDF-1 alpha/CXCR4 signaling pathway is directly involved in the cardioprotective effect of IPOC. Methods: Isolated rat hearts were divided into four groups. The control group was subjected to 30-min of regional ischemia and 2-hour of reperfusion (n=12). The IPOC group was induced with 6 cycles of 10-second reperfusion and 10-second global ischemia (n=8) in each cycle. The CXCR4 antagonist, AMD3100, was applied before reperfusion in the IPOC group (AMD+IPOC group, n=11) and control group (AMD group, n=9). Hemodynamic changes with electrocardiography were monitored and infarct size was measured. The SDF-1 alpha, lactate dehydrogenase (LDH) and creatine kinase (CK) concentrations in perfusate were measured. We also analyzed extracellular signal-regulated kinase 1/2 (ERK1/2) and Akt phosphorylation state expression. Results: IPOC significantly reduced infarct size, but AMD3100 attenuated the infarct reducing effect of IPOC. IPOC significantly decreased LDH and CK, but these effects were reversed by AMD3100. ERK1/2 and Akt phosphorylation increased with IPOC and these effects were blocked by AMD3100. Conclusion: Based on the results of this study, SDF-1 alpha/CXCR4 signaling may be involved in IPOC cardioprotection and this signaling pathway couples to the ERK1/2 and Akt pathways.111Ysciescopuskc