Observability of Lattice Graphs

We consider a graph observability problem: how many edge colors are needed for an unlabeled graph so that an agent, walking from node to node, can uniquely determine its location from just the observed color sequence of the walk? Specifically, let G(n,d) be an edge-colored subgraph of d-dimensional (directed or undirected) lattice of size n^d = n * n * ... * n. We say that G(n,d) is t-observable if an agent can uniquely determine its current position in the graph from the color sequence of any t-dimensional walk, where the dimension is the number of different directions spanned by the edges of the walk. A walk in an undirected lattice G(n,d) has dimension between 1 and d, but a directed walk can have dimension between 1 and 2d because of two different orientations for each axis. We derive bounds on the number of colors needed for t-observability. Our main result is that Theta(n^(d/t)) colors are both necessary and sufficient for t-observability of G(n,d), where d is considered a constant. This shows an interesting dependence of graph observability on the ratio between the dimension of the lattice and that of the walk. In particular, the number of colors for full-dimensional walks is Theta(n^(1/2)) in the directed case, and Theta(n) in the undirected case, independent of the lattice dimension. All of our results extend easily to non-square lattices: given a lattice graph of size N = n_1 * n_2 * ... * n_d, the number of colors for t-observability is Theta (N^(1/t))

Approximate Quantile Computation over Sensor Networks

Sensor networks have been deployed in various environments, from battle field surveillance to weather monitoring. The amount of data generated by the sensors can be large. One way to analyze such large data set is to capture the essential statistics of the data. Thus the quantile computation in the large scale sensor network becomes an important but challenging problem. The data may be widely distributed, e.g., there may be thousands of sensors. In addition, the memory and bandwidth among sensors could be quite limited. Most previous quantile computation methods assume that the data is either stored or streaming in a centralized site, which could not be directly applied in the sensor environment. In this paper, we propose a novel algorithm to compute the quantile for sensor network data, which dynamically adapts to the memory limitations. Moreover, since sensors may update their values at any time, an incremental maintenance algorithm is developed to reduce the number of times that a global recomputation is needed upon updates. The performance and complexity of our algorithms are analyzed both theoretically and empirically on various large data sets, which demonstrate the high promise of our method