Sunil M. Shende

Static Frequency Assignment in Cellular Networks

Abstract. A cellular network is generally modeled as a subgraph of the triangular lattice. In the static frequency assignment problem, each vertex of the graph is a base station in the network, and has associated with it an integer weight that represents the number of calls that must be served at the vertex by assigning distinct frequencies per call. The edges of the graph model interference constraints for frequencies assigned to neighboring stations. The static frequency assignment problem can be abstracted as a graph multicoloring problem. We describe an efficient algorithm to multicolor optimally any weighted even or odd length cycle representing a cellular network. This result is further extended to any outerplanar graph. For the problem of multicoloring an arbitrary connected subgraph of the triangular lattice, we demonstrate an approximation algorithm which guarantees that no more than 4/3 times the minimum number of required colors are used. Further, we show that this algorithm can be implemented in a distributed manner, where each station needs to have knowledge only of the weights at a small neighborhood.

Distributed Online Frequency Assignment in Cellular Networks

A cellular network is generally modeled as a subgraph of the triangular lattice. The distributed online frequency assignment problem can be abstracted as a multicoloring problem on a weighted graph, where the weight vector associated with the vertices models the number of calls to be served at the vertices and is assumed to change over time. In this paper, we develop a framework for studying distributed online frequency assignment in cellular networks. We present the first distributed online algorithms for this problem with proven bounds on their competitive ratios. We show a series of algorithms that use at each vertex information about increasingly larger neighborhoods of the vertex, and that achieve better competitive ratios. In contrast, we show lower bounds on the competitive ratios of some natural classes of online algorithms.

Approximate Hotlink Assignment

Consider a directed rooted tree T = (V,E) of maximal degree d representing a collection V of web pages connected via a set E of links all reachable from a source home page, represented by the root of T. Each leaf web page carries a weight representative of the frequency with which it is visited. By adding hotlinks, shortcuts from a node to one of its descendents, we are interested in minimizing the expected number of steps needed to visit the leaf pages from the home page. We give an O(N2) time algorithm for assigning hotlinks so that the expected number of steps to reach the leaves from the root of the tree is at most H(p)/log(d+1)-(d/(d+1)) log d + d+1/d, where H(p) is the entropy of the probability (frequency) distribution p = on the N leaves of the given tree, i.e., pi is the weight on the ith leaf. The best known lower bound for this problem is H(p)/log(d+1). Thus our algorithm approximates the optimal hotlink assignment to within a constant for any fixed d.

Packet routing on square meshes with row and column buses

General point-to-point communication among processors in the classical two-dimensional n*n square mesh architecture necessarily takes at least 2n-2 time steps. The authors investigate the problem of routing arbitrary permutations on an enhanced square mesh with separate broadcast buses along each of its rows and columns. They prove that any packet routing algorithm on this mesh takes Theta (2n/3) time steps. Further, they demonstrate a simple algorithm which, for any chosen 2/n<or= in <or=1/2, routes packets in (7n/6+O( in n)) time steps with local queue-size at most (3/ in +5).<<ETX>>

Complexity of barrier coverage with relocatable sensors in the plane

We consider several variations of the problems of covering a set of barriers (modeled as line segments) using sensors that can detect any intruder crossing any of the barriers. Sensors are initially located in the plane and they can relocate to the barriers. We assume that each sensor can detect any intruder in a circular area of fixed range centered at the sensor. Given a set of barriers and a set of sensors located in the plane, we study three problems: (i) the feasibility of barrier coverage, (ii) the problem of minimizing the largest relocation distance of a sensor (MinMax), and (iii) the problem of minimizing the sum of relocation distances of sensors (MinSum). When sensors are permitted to move to arbitrary positions on the barrier, the MinMax problem is shown to be strongly NP-complete for sensors with arbitrary ranges. We also study the case when sensors are restricted to use perpendicular movement to one of the barriers. We show that when the barriers are parallel, both the MinMax and MinSum problems can be solved in polynomial time. In contrast, we show that even the feasibility problem is strongly NP-complete if two perpendicular barriers are to be covered, even if the sensors are located at integer positions, and have only two possible sensing ranges. On the other hand, we give an O ( n 3 / 2 ) algorithm for a natural special case of this last problem.

Encoding 2D range maximum queries

We consider the two-dimensional range maximum query (2D-RMQ) problem: given an array containing elements from an ordered set, encode the array so that the position of the maximum element in any specified range of rows and range of columns can be found efficiently. We focus on determining the effective entropy of 2D-RMQ, i.e., how many bits are needed to encode an array so that 2D-RMQ queries can be answered without accessing the array. We give tight upper and lower bounds on the expected effective entropy for the case when A contains independent identically-distributed random values, and give new upper and lower bounds for the case when the array contains few rows. The latter results improve upon the upper and lower bounds by Brodal et al. 4. We also give some efficient data structures for 2D-RMQ whose space usage is close to the effective entropy.

Online compression of video sequences using adaptive VQ codebooks

Proposes a novel approach that combines the space covering property of high rate lattice VQ with the pattern matching ability of clustering VQ. The proposed scheme encompasses a broad range of online algorithms that use suitable VQ encodings and fixed-size, adaptive codebooks. The generic baseline algorithm for the scheme has the following desirable characteristics: the distortion per individual vector is guaranteed to be less than a user specified threshold. Secondly, the algorithm is amenable to fast realtime implementation and requires minimal statistical assumptions for analysis. Finally, with careful analysis, the coding rate can be bounded with respect to some theoretical benchmark.<<ETX>>

Upper bounds on recognition of a hierarchy of non-context-free languages

Control grammars, a generalization of context-free grammars recently introduced for use in natural language recognition, are investigated. In particular, it is shown that a hierarchy of non-context-free languages, called control language hierarchy (CLH), generated by control grammars can be recognized in polynomial time. Previously, the best-known upper bound was exponential time. It is also shown that CLH is in NC(2), the class of languages recognizable by uniform boolean circuits of polynomial size and O(log2 n) depth.

Pumping lemmas for the control language hierarchy

We investigate a progression of grammatically defined language families, thecontrol language hierarchy. This hierarchy has been studied recently from the perspective of providing a linguistic framework for natural language syntax. We exhibit a progression of pumping lemmas, one for each family in the hierarchy, thereby showing that the hierarchy is strictly separable.

On Multidimensional Packet Routing for Meshes with Buses

Routing problems on the r-dimensional n × n × · · · × n mesh architecture, enhanced with independent buses along each dimension, are investigated. We analyze several different kinds of routing problems, among them off-line routing, and provide routing algorithms for the one-dimensional, two-dimensional, and r-dimensional meshes. Asymptotically, we show that, queuesize being equal, the bus-equipped mesh architecture outperforms the classical mesh by speeding up the routing time by a factor of 127.