This work addresses the problem of balancing the spatial distribution of the routing-load among the nodes in a given sensor network and the tradeoff that can be achieved for providing certain level of quality of service (QoS) guarantees. For high-density density networks, several studies have proposed field field-based routing paradigms to uniformly distribute the traffic load throughout the network. However, as network density decreases, we observe major shortcomings of the current st state-of-the-art: (i) path-merging merging leads to a reduction of path diversity, and (ii) the paths directed towards the border of the network merge into a single path along the border. These path merging effects decrease significantly the energy balance, and as consequence, onsequence, the lifetime of the network. In this article, we propose a novel mechanism to enable better load balancing for single-source single and multiple-source source scenarios, while minimizing the cost of the tradeoff for bounding the end-to-end end packet delivery la latencies. tencies. Our evaluations demonstrate that by using the proposed methodology, the network lifetime can be significantly prolonged, when long-term point-to-point queries are considered.

Given three permutations on the integers 1 through n, consider the set system consisting of each interval in each of the three permutations. József Beck conjectured (c. 1982) that the discrepancy of this set system is O(1). We give a counterexample to this conjecture: for any positive integer n = 3 k, we exhibit three permutations whose corresponding set system has discrepancy Ω(log n). Our counterexample is based on a simple recursive construction, and our proof of the discrepancy lower bound is by induction. This example also disproves a generalization of Beck’s conjecture due to Spencer, Srinivasan and Tetali, who conjectured that a set system corresponding to ℓ permutations has discrepancy O ( √ ℓ) [SST01]. 1

We study the mergeability of data summaries. Informally speaking, mergeability requires that, given two summaries on two data sets, there is a way to merge the two summaries into a summary on the two data sets combined together, while preserving the error and size guarantees. This property means that the summary can be treated like other algebraic objects such as sum and max, which is especially useful for computing summaries on massive distributed data. Many data summaries are trivially mergeable by construction, most notably those based on linear transformations. But some other fundamental ones like those for heavy hitters and quantiles, are not (known to be) mergeable. In this paper, we demonstrate that these summaries are indeed mergeable or can be made mergeable after appropriate modifications. Specifically, we show that for ε-approximate heavy hitters, there is a deterministic mergeable summary log(εn)) that has a restricted form of mergeability, and a randomized one of size O ( 1 ε ε) with full mergeability. We also extend our results to geometric summaries such as ε-approximations and ε-kernels. of size O(1/ε); for ε-approximate quantiles, there is a deterministic summary of size O ( 1 ε 1

We study computing geometric problems on uncertain points. An uncertain point is a point that does not have a fixed location, but rather is described by a probability distribution. When these probability distributions are restricted to a finite number of locations, the points are called indecisive points. In particular, we focus on geometric shape-fitting problems and on building compact distributions to describe how the solutions to these problems vary with respect to the uncertainty in the points. Our main results are: (1) a simple and efficient randomized approximation algorithm for calculating the distribution of any statistic on uncertain data sets; (2) a polynomial, deterministic and exact algorithm for computing the distribution of answers for any LP-type problem on an indecisive point set; and (3) the development of shape inclusion probability (SIP) functions which captures the ambient distribution of shapes fit to uncertain or indecisive point sets and are admissible to the two algorithmic constructions. 1

We show that the hereditary discrepancy of a hypergraph F on n pointsincreases by a factor of at most O(log n) when one adds a new edge to F.

We use entropy numbers in combination with the polynomial method to derive a new general lower bound for the n-th minimal error in the quantum setting of information-based complexity. As an application, we improve some lower bounds on quantum approximation of embeddings between finite dimensional Lp spaces and of Sobolev embeddings. 1