Oblivious routing is a well-studied paradigm that uses static precomputed routing tables for selecting routing paths within a network. Existing oblivious routing schemes with polylogarithmic competitive ratio for general networks are tree-based, in the sense that routing is performed according to a convex combination of trees. However, this restriction to trees leads to a construction that has time quadratic in the size of the network and does not parallelize well. In this paper we study oblivious routing schemes based on electrical routing. In particular, we show that general networks with $n$ vertices and $m$ edges admit a routing scheme that has competitive ratio $O(\log^2 n)$ and consists of a convex combination of only $O(\sqrt{m})$ electrical routings. This immediately leads to an improved construction algorithm with time $\tilde{O}(m^{3/2})$ that can also be implemented in parallel with $\tilde{O}(\sqrt{m})$ depth.
We prove the existence of an oblivious routing scheme that is $\mathrm{poly}(\log n)$-competitive in terms of $(congestion + dilation)$, thus resolving a well-known question in oblivious routing. Concretely, consider an undirected network and a set of packets each with its own source and destination. The objective is to choose a path for each packet, from its source to its destination, so as to minimize $(congestion + dilation)$, defined as follows: The dilation is the maximum path hop-length, and the congestion is the maximum number of paths that include any single edge. The routing scheme ob…
We present novel oblivious routing algorithms for both splittable and unsplittable multicommodity flow. Our algorithm for minimizing congestion for \emph{unsplittable} multicommodity flow is the first oblivious routing algorithm for this setting. As an intermediate step towards this algorithm, we present a novel generalization of Valiant's classical load balancing scheme for packet-switched networks to arbitrary graphs, which is of independent interest. Our algorithm for minimizing congestion for \emph{splittable} multicommodity flow improves upon the state-of-the-art, in terms of both running…
The problem of routing in graphs using node-disjoint paths has received a lot of attention and a polylogarithmic approximation algorithm with constant congestion is known for undirected graphs [Chuzhoy and Li 2016] and [Chekuri and Ene 2013]. However, the problem is hard to approximate within polynomial factors on directed graphs, for any constant congestion [Chuzhoy, Kim and Li 2016]. Recently, [Chekuri, Ene and Pilipczuk 2016] have obtained a polylogarithmic approximation with constant congestion on directed planar graphs, for the special case of symmetric demands. We extend their result by …
We provide universally-optimal distributed graph algorithms for $(1+\varepsilon)$-approximate shortest path problems including shortest-path-tree and transshipment. The universal optimality of our algorithms guarantees that, on any $n$-node network $G$, our algorithm completes in $T \cdot n^{o(1)}$ rounds whenever a $T$-round algorithm exists for $G$. This includes $D \cdot n^{o(1)}$-round algorithms for any planar or excluded-minor network. Our algorithms never require more than $(\sqrt{n} + D) \cdot n^{o(1)}$ rounds, resulting in the first sub-linear-round distributed algorithm for transship…
We introduce the notion of balance for directed graphs: a weighted directed graph is $α$-balanced if for every cut $S \subseteq V$, the total weight of edges going from $S$ to $V\setminus S$ is within factor $α$ of the total weight of edges going from $V\setminus S$ to $S$. Several important families of graphs are nearly balanced, in particular, Eulerian graphs (with $α= 1$) and residual graphs of $(1+ε)$-approximate undirected maximum flows (with $α=O(1/ε)$). We use the notion of balance to give a more fine-grained understanding of several well-studied routing questions that are considerably …
Consider the robust network design problem of finding a minimum cost network with enough capacity to route all traffic demand matrices in a given polytope. We investigate the impact of different routing models in this robust setting: in particular, we compare \emph{oblivious} routing, where the routing between each terminal pair must be fixed in advance, to \emph{dynamic} routing, where routings may depend arbitrarily on the current demand. Our main result is a construction that shows that the optimal cost of such a network based on oblivious routing (fractional or integral) may be a factor of…
Stability is an important issue in order to characterize the performance of a network, and it has become a major topic of study in the last decade. Roughly speaking, a communication network system is said to be stable if the number of packets waiting to be delivered (backlog) is finitely bounded at any one time. In this paper, we introduce a new family of combinatorial structures, which we call universally strong selectors, that are used to provide a set of transmission schedules. Making use of these structures, combined with some known queuing policies, we propose a packet-oblivious routing a…
Oblivious routing has a long history in both the theory and practice of networking. In this work we initiate the formal study of oblivious routing in the context of reconfigurable networks, a new architecture that has recently come to the fore in datacenter networking. These networks allow a rapidly changing bounded-degree pattern of interconnections between nodes, but the network topology and the selection of routing paths must both be oblivious to the traffic demand matrix. Our focus is on the trade-off between maximizing throughput and minimizing latency in these networks. For every constan…
Network-wide traffic analytics are often needed for various network monitoring tasks. These measurements are often performed by collecting samples at network switches, which are then sent to the controller for aggregation. However, performing such analytics without ``overcounting'' flows or packets that traverse multiple measurement switches is challenging. Therefore, existing solutions often simplify the problem by making assumptions on the routing or measurement switch placement. We introduce AROMA, a measurement infrastructure that generates a uniform sample of packets and flows regardless …
Dynamic Flows were introduced by Ford and Fulkerson in 1958 to model flows over time. They define edge capacities to be the total amount of flow that can enter an edge {\em in one time unit}. Each edge also has a length, representing the time needed to traverse it. Dynamic Flows have been used to model many problems including traffic congestion, hop-routing of packets and evacuation protocols in buildings. While the basic problem of moving the maximal amount of supplies from sources to sinks is polynomial time solvable, natural minor modifications can make it NP-hard. One such modification is …
Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices $u$ and $v$ can be determined efficiently by merely inspecting the labels of $u$ and $v$, without using any other information. Similarly, routing labeling schemes label the vertices of a graph in a such a way that given the labels of a source node and a destination node, it is possible to compute efficiently the port number of the edge from the source that heads in the direction of the destination. One of important problems is finding natural class…
In order to investigate the routing aspects of small-world networks, Kleinberg proposes a network model based on a $d$-dimensional lattice with long-range links chosen at random according to the $d$-harmonic distribution. Kleinberg shows that the greedy routing algorithm by using only local information performs in $O(\log^2 n)$ expected number of hops, where $n$ denotes the number of nodes in the network. Martel and Nguyen have found that the expected diameter of Kleinberg's small-world networks is $Θ(\log n)$. Thus a question arises naturally: Can we improve the routing algorithms to match th…
We propose a dynamical process for network evolution, aiming at explaining the emergence of the small world phenomenon, i.e., the statistical observation that any pair of individuals are linked by a short chain of acquaintances computable by a simple decentralized routing algorithm, known as greedy routing. Previously proposed dynamical processes enabled to demonstrate experimentally (by simulations) that the small world phenomenon can emerge from local dynamics. However, the analysis of greedy routing using the probability distributions arising from these dynamics is quite complex because of …
We consider the single-source (or single-sink) buy-at-bulk problem with an unknown concave cost function. We want to route a set of demands along a graph to or from a designated root node, and the cost of routing x units of flow along an edge is proportional to some concave, non-decreasing function f such that f(0) = 0. We present a polynomial time algorithm that finds a distribution over trees such that the expected cost of a tree for any f is within an O(1)-factor of the optimum cost for that f. The previous best simultaneous approximation for this problem, even ignoring computation time, wa…
We show that in any graph, the average length of a flow path in an electrical flow between the endpoints of a random edge is $O(\log^2 n)$. This is a consequence of a more general result which shows that the spectral norm of the entrywise absolute value of the transfer impedance matrix of a graph is $O(\log^2 n)$. This result implies a simple oblivious routing scheme based on electrical flows in the case of transitive graphs.
The online (uniform) buy-at-bulk network design problem asks us to design a network, where the edge-costs exhibit economy-of-scale. Previous approaches to this problem used tree- embeddings, giving us randomized algorithms. Moreover, the optimal results with a logarithmic competitive ratio requires the metric on which the network is being built to be known up-front; the competitive ratios then depend on the size of this metric (which could be much larger than the number of terminals that arrive). We consider the buy-at-bulk problem in the least restrictive model where the metric is not known i…
