In this paper we give a simple algorithm for constructing sparse spanners for arbitrary weighted graphs. We then apply this algorithm to obtain specific results for planar graphs and Euclidean graphs. We discuss the optimality of our results and present several nearly matching lower bounds.

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With ever growing competition in telecommunications markets, operators have to increasingly rely on business intelligence to offer the right incentives to their customers. Toward this end, existing approaches have almost solely focussed on the individual behaviour of customers. Call graphs, that is, graphs induced by people calling each other, can allow telecom operators to better understand the interaction behaviour of their customers, and potentially provide major insights for designing effective incentives.In this paper, we use the Call Detail Records of a mobile operator from four geographically disparate regions to construct call graphs, and analyse their structural properties. Our findings provide business insights and help devise strategies for Mobile Telecom operators. Another goal of this paper is to identify the shape of such graphs. In order to do so, we extend the well-known reachability analysis approach with some of our own techniques to reveal the shape of such massive graphs. Based on our analysis, we introduce the Treasure-Hunt model to describe the shape of mobile call graphs. The proposed techniques are general enough for analysing any large graph. Finally, how well the proposed model captures the shape of other mobile call graphs needs to be the subject of future studies.

Let G = (V, E) be an n-vertex connected graph with positive edge weights.A subgraph G' = (V, E') is a tspanner of G if for all u, v E V, the weighted distance between u and v in G' is at most t times the weighted distance between u and v in G. We consider the problem of constructing sparse spanners. Sparseness of spanners is measured by two criteria, the size, defined as the number of edges in the spanner, and the weight, defined as the sum of the edge weights in the spanner. In this paper, we concentrate on constructing spanners of small weight.For an arbitrary positive edge-weighted graph G, for any t > 1, and any c >0, we show that a t-spanner of G with weight O(n* ) . wt(MST) can be constructed in polynomial time. We also show that (logz n)-spanners of weight O(1) . wt(MST) can be constructed.We then consider spanners for complete graphs induced by a set of points in d-dimensional real normed space. The weight of an edge Zy is the norm of the~y vector. We show that for these graphs, t-spanners with total weight O(log n) . wt (MST) can be constructed in polynomial time.

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