Costas Busch

Approximating Congestion + Dilation in Networks via "Quality of Routing” Games

A classic optimization problem in network routing is to minimize C + D, where C is the maximum edge congestion and D is the maximum path length (also known as dilation). The problem of computing the optimal C* + D* is NP-complete even when either C* or D* is a small constant. We study routing games in general networks where each player i selfishly selects a path that minimizes Ci + Di the sum of congestion and dilation of the players path. We first show that there are instances of this game without Nash equilibria. We then turn to the related quality of routing (QoR) games which always have Nash equilibria. QoR games represent networks with a small number of service classes where paths in different classes do not interfere with each other (with frequency or time division multiplexing). QoR games have O(log4 n) price of anarchy when either C* or D* is a constant. Thus, Nash equilibria of QoR games give poly-log approximations to hard optimization problems.

Contention-free MAC protocols for wireless sensor networks

A MAC protocol specifies how nodes in a sensor network access a shared communication channel. Desired properties of such MAC protocol are: it should be distributed and contention-free (avoid collisions); it should self-stabilize to changes in the network (such as arrival of new nodes), and these changes should be contained, i.e., affect only the nodes in the vicinity of the change; it should not assume that nodes have a global time reference, i.e., nodes may not be time-synchronized. We give the first MAC protocols that satisfy all of these requirements, i.e., we give distributed, contention-free, self-stabilizing MAC protocols which do not assume a global time reference. Our protocols self-stabilize from an arbitrary initial state, and if the network changes the changes are contained and the protocol adjusts to the local topology of the network. The communication complexity, number and size of messages, for the protocol to stabilize is small (logarithmic in network size).

Analysis of link reversal routing algorithms for mobile ad hoc networks

Link reversal algorithms provide a simple mechanism for routing in mobile ad hoc networks. These algorithms maintain routes to any particular destination in the network, even when the network topology changes frequently. In link reversal, a node reverses its incident links whenever it loses routes to the destination. Link reversal algorithms have been studied experimentally and have been used in practical routing algorithms, including [8].This paper presents the first formal performance analysis of link reversal algorithms. We study these algorithms in terms of work (number of node reversals) and the time needed until the network stabilizes to a state in which all the routes are reestablished. We focus on the full reversal algorithm and the partial reversal algorithm, both due to Gafni and Berstekas [5]; the first algorithm is simpler, while the latter has been found to be more efficient for typical cases. Our results are as follows:(1) The full reversal algorithm requires O(n2) work and time, where n is the number of nodes which have lost the routes to the destination.(2) The partial reversal algorithm requires O(n • a* + n2) work and time, where a* is a non-negative integer which depends on the state of the network. This bound is tight in the worst case, for any a*.(3) There are networks such that for every deterministic link reversal algorithm, there are initial states which require requires ω(n2) work and time to stabilize. Therefore, surprisingly, the full reversal algorithm is asymptotically optimal in the worst case, while the partial reversal algorithm is not, since a* can grow arbitrarily large.

Sketching asynchronous streams over a sliding window

We study the problem of maintaining sketches of recent elements of a data stream. Motivated by applications involving network data, we consider streams that are asynchronous, in which the observed order of data is not the same as the time order in which the data was generated. The notion of recent elements of a stream is modeled by the sliding timestamp window, which is the set of elements with timestamps that are close to the current time. We design algorithms for maintaining sketches of all elements within the sliding timestamp window that can give provably accurate estimates of two basic aggregates, the sum and the median, of a stream of numbers. The space taken by the sketches, the time needed for querying the sketch, and the time for inserting new elements into the sketch are all polylog with respect to the maximum window size and the values of the data items in the window. Our sketches can be easily combined in a lossless and compact way, making them useful for distributed computations over data streams. Previous works on sketching recent elements of a data stream have all considered the more restrictive scenario of synchronous streams, where the observed order of data is the same as the time order in which the data was generated. Our notion of recency of elements is more general than that studied in previous work, and thus our sketches are more robust to network delays and asynchrony.

Oblivious routing on geometric networks

We study oblivious routing in which the packet paths are constructed independently of each other. We give a simple oblivious routing algorithm for geometric networks in which the nodes are embedded in the Euclidean plane. In our algorithm, a packet path is constructed by first choosing a random intermediate node in the space between the source and destination, and then the packet is sent to its destination through the intermediate node. We analyze the performance of the algorithm in terms of the stretch and congestion of the resulting paths. We show that the stretch is constant, and the congestion is near optimal when the network paths can be chosen to be close to the geodesic lines that connect the end points of the paths. We give applications of our general result to the mesh topology and uniformly distributed disc graphs. Previous oblivious routing algorithms with near optimal congestion use many intermediate nodes and do not control the stretch.

A deterministic algorithm for summarizing asynchronous streams over a sliding window

We consider the problem of maintaining aggregates over recent elements of a massive data stream. Motivated by applications involving network data, we consider asynchronous data streams, where the observed order of data may be different from the order in which the data was generated. The set of recent elements is modeled as a sliding timestamp window of the stream, whose elements are changing continuously with time. We present the first deterministic algorithms for maintaining a small space summary of elements in a sliding timestamp window of an asynchronous data stream. The summary can return approximate answers for the following fundamental aggregates: basic count, the number of elements within the sliding window, and sum, the sum of all element values within the sliding window. For basic counting, the space taken by our summary is O(logW ċ log B ċ (log W + log B)/Ɛ) bits, where B is an upper bound on the value of the basic count, W is an upper bound on the width of the timestamp window, and Ɛ is the desired relative error. Our algorithms are based on a novel data structure called splittable histogram. Prior to this work, randomized algorithms were known for this problem, which provide weaker guarantees than those provided by our deterministic algorithms.

Analysis of Link Reversal Routing Algorithms

Link reversal algorithms provide a simple mechanism for routing in communication networks whose topology is frequently changing, such as in mobile ad hoc networks. A link reversal algorithm routes by imposing a direction on each network link such that the resulting graph is a destination oriented DAG. Whenever a node loses routes to the destination, it reacts by reversing some (or all) of its incident links. Link reversal algorithms have been studied experimentally and have been used in practical routing algorithms, including TORA [V. D. Park and M. S. Corson, A highly adaptive distributed routing algorithm for mobile wireless networks, in Proc. INFOCOM, IEEE, Los Alamitos, CA, 1997, pp. 1405--1413]. This paper presents the first formal performance analysis of link reversal algorithms. We study these algorithms in terms of work (number of node reversals) and the time needed until the network stabilizes to a state in which all the routes are reestablished. We focus on the full reversal algorithm and the partial reversal algorithm, both due to Gafni and Bertsekas [IEEE Trans. Comm.}, 29 (1981), pp. 11--18]; the first algorithm is simpler, while the latter has been found to be more efficient for typical cases. Our results are as follows: The full reversal algorithm requires O(n2) work and time, where n is the number of nodes that have lost routes to the destination. This bound is tight in the worst case. The partial reversal algorithm requires O(n

Routing without flow control

\cdot

Contention in counting networks

a* + n2) work and time, where a* is a nonnegative integral function of the initial state of the network. Further, for every nonnegative integer

Window-based greedy contention management for transactional memory

\alpha