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 obliviously and randomly selects a path for each packet independent of (the existence of) the other packets. Despite this obliviousness, the selected paths have $(congestion + dilation)$ within a $\mathrm{poly}(\log n)$ factor of the best possible value. More precisely, for any integer hop-bound $h$, this oblivious routing scheme selects paths of length at most $h \cdot \mathrm{poly}(\log n)$ and is $\mathrm{poly}(\log n)$-competitive in terms of $congestion$ in comparison to the best possible $congestion$ achievable via paths of length at most $h$ hops. These paths can be sampled in polynomial time.
This result can be viewed as an analogue of the celebrated oblivious routing results of Räcke [FOCS 2002, STOC 2008], which are $O(\log n)$-competitive in terms of $congestion$, but are not competitive in terms of $dilation$.
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…
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 …
A weakness of next-hop routing is that following a link or router failure there may be no routes between some source-destination pairs, or packets may get stuck in a routing loop as the protocol operates to establish new routes. In this article, we address these weaknesses by describing mechanisms to choose alternate next hops. Our first contribution is to model the scenario as the following {\sc tree augmentation} problem. Consider a mixed graph where some edges are directed and some undirected. The directed edges form a spanning tree pointing towards the common destination node. Each directe…
For a positive parameter $β$, the $β$-bounded distance between a pair of vertices $u,v$ in a weighted undirected graph $G = (V,E,ω)$ is the length of the shortest $u-v$ path in $G$ with at most $β$ edges, aka {\em hops}. For $β$ as above and $ε>0$, a {\em $(β,ε)$-hopset} of $G = (V,E,ω)$ is a graph $G' =(V,H,ω_H)$ on the same vertex set, such that all distances in $G$ are $(1+ε)$-approximated by $β$-bounded distances in $G\cup G'$. Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used for distance-related problems in a variety of computational sett…
We present an algorithm for multi-hop routing and scheduling of requests in wireless networks in the \sinr\ model. The goal of our algorithm is to maximize the throughput or maximize the minimum ratio between the flow and the demand. Our algorithm partitions the links into buckets. Every bucket consists of a set of links that have nearly equivalent reception powers. We denote the number of nonempty buckets by $\sigdiv$. Our algorithm obtains an approximation ratio of $O(\sigdiv \cdot \log n)$, where $n$ denotes the number of nodes. For the case of linear powers $\sigdiv =1$, hence the approxim…
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…
Network design problems aim to compute low-cost structures such as routes, trees and subgraphs. Often, it is natural and desirable to require that these structures have small hop length or hop diameter. Unfortunately, optimization problems with hop constraints are much harder and less well understood than their hop-unconstrained counterparts. A significant algorithmic barrier in this setting is the fact that hop-constrained distances in graphs are very far from being a metric. We show that, nonetheless, hop-constrained distances can be approximated by distributions over "partial tree metrics."…
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.
