Debmalya Panigrahi

A general framework for graph sparsification

We present a general framework for constructing cut sparsifiers in undirected graphs --- weighted subgraphs for which every cut has the same weight as the original graph, up to a multiplicative factor of (1 ε). Using this framework, we simplify, unify and improve upon previous sparsification results. As simple instantiations of this framework, we show that sparsifiers can be constructed by sampling edges according to their strength (a result of Benczur and Karger), effective resistance (a result of Spielman and Srivastava), edge connectivity, or by sampling random spanning trees. Sampling according to edge connectivity is the most aggressive method, and the most challenging to analyze. Our proof that this method produces sparsifiers resolves an open question of Benczur and Karger. While the above results are interesting from a combinatorial standpoint, we also prove new algorithmic results. In particular, we develop techniques that give the first (optimal) O(m)-time sparsification algorithm for unweighted graphs. Our algorithm has a running time of O(m) + ~O(n/ε2) for weighted graphs, which is also linear unless the input graph is very sparse itself. In both cases, this improves upon the previous best running times (due to Benczur and Karger) of O(m log2 n) (for the unweighted case) and O(m log3 n) (for the weighted case) respectively. Our algorithm constructs sparsifiers that contain O(n log n/ε2) edges in expectation; the only known construction of sparsifiers with fewer edges is by a substantially slower algorithm running in O(n3 m / ε2) time. A key ingredient of our proofs is a natural generalization of Kargers bound on the number of small cuts in an undirected graph. Given the numerous applications of Kargers bound, we suspect that our generalization will also be of independent interest.

A New Channel Assignment Mechanism for Rural Wireless Mesh Networks

In this paper we present a new channel allocation scheme for IEEE 802.11 based mesh networks with point-to- point links, designed for rural areas. Our channel allocation scheme allows continuous full-duplex data transfer on every link in the network. Moreover, we do not require any synchronization across the links as the channel assignment prevents cross link interference. Our approach is simple. We consider any link in the network as made up of two directed edges. To each directed edge at a node, we assign a non-interfering IEEE 802.11 channel so that the set of channels assigned to the outgoing edges is disjoint from channels assigned to the incoming edges. Evaluation of this scheme in a testbed demonstrate throughput gains of between 50 - 100%, and significantly less end-to-end delays, over existing link scheduling/channel allocation protocols (such as 2P [11]) designed for point-to-point mesh networks. Formally speaking, this channel allocation scheme is equivalent to an edge-coloring problem, that we call the directed edge coloring (DEC) problem. We establish a relationship between this coloring problem and the classical vertex coloring problem, and thus, show that this problem is NP-hard. More precisely, we give an algorithm that, given k vertex coloring of a graph can directed edge color it using xi(k) colors, where xi(k) is the smallest integer n such that (lfloorn/2rfloor/n ) ges k.

Minimum Cost Topology Construction for Rural Wireless Mesh Networks

IEEE 802.11 WiFi equipment based wireless mesh networks have recently been proposed as an inexpensive approach to connect far-flung rural areas. Such networks are built using high-gain directional antennas that can establish long-distance wireless point-to-point links. Some nodes in the network (called gateway nodes) are directly connected to the wired internet, and the remaining nodes connect to the gateway(s) using one or more hops. The dominant cost of constructing such a mesh network is the cost of constructing antenna towers at nodes. The cost of a tower depends on its height, which in turn depends on the length of its links and the physical obstructions along those links. We investigate the problem of selecting which links should be established such that all nodes are connected, while the cost of constructing the antenna towers required to establish the selected links is minimized. We show that this problem is NP-hard and that a better than O(log n) approximation cannot be expected, where n is the number of vertices in the graph. We then present the first algorithm in the literature, for this problem, with provable performance bounds. More precisely, we present a greedy algorithm that is an O(log n) approximation algorithm for this problem. Finally, through simulations, we compare our approximation algorithm with both the optimal solution, and a naive heuristic.

Online Node-Weighted Steiner Tree and Related Problems

We obtain the first online algorithms for the node-weighted Steiner tree, Steiner forest and group Steiner tree problems that achieve a poly-logarithmic competitive ratio. Our algorithm for the Steiner tree problem runs in polynomial time, while those for the other two problems take quasi-polynomial time. Our algorithms can be viewed as online LP rounding algorithms in the framework of Buchbinder and Naor (Foundations and Trends in Theoretical Computer Science, 2009); however, while the natural LP formulation of these problems do lead to fractional algorithms with a poly-logarithmic competitive ratio, we are unable to round these LPs online without losing a polynomial factor. Therefore, we design new LP formulations for these problems drawing on a combination of paradigms such as spider decompositions, low-depth Steiner trees, generalized group Steiner problems, etc. and use the additional structure provided by these to round the more sophisticated LPs losing only a poly-logarithmic factor in the competitive ratio. As further applications of our techniques, we also design polynomial-time online algorithms with poly-logarithmic competitive ratios for two fundamental network design problems in edge-weighted graphs: the group Steiner forest problem (thereby resolving an open question raised by Chekuri et. al. (SODA 2008)) and the single source ℓ-vertex connectivity problem (which complements similar results for the corresponding edge-connectivity problem due to Gupta et. al. (STOC 2009)).

VillageNet: A low-cost, 802.11-based mesh network for rural regions

VillageNet is a wireless mesh network that aims to provide low-cost broadband Internet access for rural regions. The cost of building the network is kept low by using off-the-shelf IEEE 802.11 equipment and optimizing the network topology to minimize cost. In this paper we describe the over-all operation of VillageNet and discuss two fundamental problems in building such a network. Nodes in VillageNet communicate using long-distance point-to-point wireless links that are established using high-gain directional antenna. VillageNet uses the 2P MAC protocol [?], that is suited for the interference pattern within such a network. However, the 2P protocol requires the underlying mesh graph (for each 802.11 channel) to be bi-partite. Thus, if K channels are available, then an important consideration is how to select K bi-partite subgraphs to activate, such that the demands of the nodes are best met. We formally pose this problem and present some initial results. Second, we observe that the dominant cost of constructing such a mesh network is the cost of constructing antenna towers at nodes. The cost of a tower depends on its height, which in turn depends on the length of its links, and the physical obstructions along those links. Thus to minimize cost, we pose the problem of deciding which links should be established, such that all villages are connected and the cost of constructing antenna towers to establish the selected links is minimized.

Online selection of diverse results

The phenomenal growth in the volume of easily accessible information via various web-based services has made it essential for service providers to provide users with personalized representative summaries of such information. Further, online commercial services including social networking and micro-blogging websites, e-commerce portals, leisure and entertainment websites, etc. recommend interesting content to users that is simultaneously diverse on many different axes such as topic, geographic specificity, etc. The key algorithmic question in all these applications is the generation of a succinct, representative, and relevant summary from a large stream of data coming from a variety of sources. In this paper, we formally model this optimization problem, identify its key structural characteristics, and use these observations to design an extremely scalable and efficient algorithm. We analyze the algorithm using theoretical techniques to show that it always produces a nearly optimal solution. In addition, we perform large-scale experiments on both real-world and synthetically generated datasets, which confirm that our algorithm performs even better than its analytical guarantees in practice, and also outperforms other candidate algorithms for the problem by a wide margin.

Tight Bounds for Online Vector Scheduling

Modern data centers face a key challenge of effectively serving user requests that arrive online. Such requests are inherently multi-dimensional and characterized by demand vectors over multiple resources such as processor cycles, storage space, and network bandwidth. Typically, different resources require different objectives to be optimized, and Lr norms of loads are among the most popular objectives considered. Furthermore, the server clusters are also often heterogeneous making the scheduling problem more challenging. To address these problems, we consider the online vector scheduling problem in this paper. Introduced by Chekuri and Khanna (SIAM J. of Comp. 2006), vector scheduling is a generalization of classical load balancing, where every job has a vector load instead of a scalar load. The scalar problem, introduced by Graham in 1966, and its many variants (identical and unrelated machines, makespan and Lr-norm optimization, offline and online jobs, etc.) have been extensively studied over the last 50 years. In this paper, we resolve the online complexity of the vector scheduling problem and its important generalizations - for all Lr norms and in both the identical and unrelated machines settings. Our main results are: · For identical machines, we show that the optimal competitive ratio is Θ(log d/ log log d) by giving an online lower bound and an algorithm with an asymptotically matching competitive ratio. The lower bound is technically challenging, and is obtained via an online lower bound for the minimum mono-chromatic clique problem using a novel online coloring game and randomized coding scheme. Our techniques also extend to asymptotically tight upper and lower bounds for general Lr norms. · For unrelated machines, we show that the optimal competitive ratio is Θ(log m + log d) by giving an online lower bound that matches a previously known upper bound. Unlike identical machines, however, extending these results, particularly the upper bound, to general Lr norms requires new ideas. In particular, we use a carefully constructed potential function that balances the individual Lr objectives with the overall (convexified) min-max objective to guide the online algorithm and track the changes in potential to bound the competitive ratio.

TDMA Scheduling in Long-Distance WiFi Networks

In the last few years, long-distance WiFi networks have been used to provide Internet connectivity in rural areas. The strong requirement to support real-time applications in these settings leads us to consider TDMA link scheduling. In this paper, we consider the FRACTEL architecture for long-distance mesh networks. We propose a novel angular interference model, which is not only practical, but also makes the problem of TDMA scheduling tractable. We then consider delay-bounded scheduling and present an algorithm which uses at most 1/3rd more time-slots than the optimal number of slots required without the delay bound. Our evaluation on various network topologies shows that the algorithm is practical, and more efficient in practice than its worst-case bound.

Near-Optimal Online Algorithms for Prize-Collecting Steiner Problems

In this paper, we give the first online algorithms with a poly-logarithmic competitive ratio for the node-weighted prize-collecting Steiner tree and Steiner forest problems. The competitive ratios are optimal up to logarithmic factors. In fact, we give a generic technique for reducing online prize-collecting Steiner problems to the fractional version of their non-prize-collecting counterparts losing only a logarithmic factor in the competitive ratio. This reduction is agnostic to the cost model (edge-weighted or node-weighted) of the input graph and applies to a wide class of network design problems including Steiner tree, Steiner forest, group Steiner tree, and group Steiner forest. Consequently, we also give the first online algorithms for the edge-weighted prize-collecting group Steiner tree and group Steiner forest problems with a poly-logarithmic competitive ratio, since corresponding fractional guarantees for the non-prize-collecting variants of these problems were previously known.

Online Node-Weighted Steiner Forest and Extensions via Disk Paintings

We give the first polynomial-time online algorithm for the node-weighted Steiner forest problem with a poly-logarithmic competitive ratio. The competitive ratio of our algorithm is optimal up to a logarithmic factor. For the special case of graphs with an excluded fixed minor (e.g., planar graphs), we obtain a logarithmic competitive ratio, which is optimal up to a constant, using a different online algorithm. Both these results are obtained as special cases of generic results for a large class of problems that can be encoded as online 0, 1-proper functions. Our results are obtained by using a new framework for online network design problems that we call disk paintings. The central idea in this technique is to amortize the cost of primal updates to a set of carefully selected mutually disjoint fixed-radius dual disks centered at a subset of terminals. We hope that this framework will be useful for other online network design problems.