The complexity of the characterization of networks supporting shortest-path interval routing

AbstractInterval Routing is a routing method that was proposed in order to reduce the size of the routing tables by using intervals and was extensively studied and implemented. Some variants of the original method were also defined and studied. The question of characterizing networks which support optimal (i.e., shortest path) Interval Routing has been thoroughly investigated for each of the variants and under different models, with only partial answers, both positive and negative, given so far. In this paper, we study the characterization problem under the most basic model (the one unit cost), and with the most restrictive memory requirements (one interval per edge). We prove that this problem is NP-hard (even for the restricted class of graphs of diameter at most 3). Our result holds for all variants of Interval Routing. It significantly extends some related NP-hardness result, and implies that, unless P=NP, partial characterization results of some classes of networks which support shortest path Interval Routing, cannot be pushed further to lead to efficient characterizations for these classes

Interval Routing Schemes for Circular-Arc Graphs

Interval routing is a space efficient method to realize a distributed routing function. In this paper we show that every circular-arc graph allows a shortest path strict 2-interval routing scheme, i.e., by introducing a global order on the vertices and assigning at most two (strict) intervals in this order to the ends of every edge allows to depict a routing function that implies exclusively shortest paths. Since circular-arc graphs do not allow shortest path 1-interval routing schemes in general, the result implies that the class of circular-arc graphs has strict compactness 2, which was a hitherto open question. Additionally, we show that the constructed 2-interval routing scheme is a 1-interval routing scheme with at most one additional interval assigned at each vertex and we an outline algorithm to calculate the routing scheme for circular-arc graphs in O(n^2) time, where n is the number of vertices.Comment: 17 pages, to appear in "International Journal of Foundations of Computer Science

Optimum Multi-Dimensional Interval Routing Schemes on Networks with Dynamic Cost Links

One of the fundamental tasks in any distributed computing system is routing messages between pairs of nodes. An Interval Routing Scheme (IRS) is a~space efficient way of routing messages in a network. The problem of characterizing graphs that support an IRS is a well-known problem and has been studied for some variants of IRS. It is natural to assume that the costs of links may vary over time (dynamic cost links) and to try to find an IRS which routes all messages on shortest paths (optimum IRS). In this paper, we study this problem for a~variant of IRS in which the labels assigned to the vertices are $d$-ary integer tuples ($d$-dimensional IRS). The only known results in this case are for specific graphs like hypercubes, $n$-dimensional grids, or for the 1-dimensional case. We give a complete characterization for the class of networks supporting multi-dimensional strict and linear (no cyclic intervals) interval routing schemes with dynamic cost links

SPIDER: Fault Resilient SDN Pipeline with Recovery Delay Guarantees

When dealing with node or link failures in Software Defined Networking (SDN), the network capability to establish an alternative path depends on controller reachability and on the round trip times (RTTs) between controller and involved switches. Moreover, current SDN data plane abstractions for failure detection (e.g. OpenFlow "Fast-failover") do not allow programmers to tweak switches' detection mechanism, thus leaving SDN operators still relying on proprietary management interfaces (when available) to achieve guaranteed detection and recovery delays. We propose SPIDER, an OpenFlow-like pipeline design that provides i) a detection mechanism based on switches' periodic link probing and ii) fast reroute of traffic flows even in case of distant failures, regardless of controller availability. SPIDER can be implemented using stateful data plane abstractions such as OpenState or Open vSwitch, and it offers guaranteed short (i.e. ms) failure detection and recovery delays, with a configurable trade off between overhead and failover responsiveness. We present here the SPIDER pipeline design, behavioral model, and analysis on flow tables' memory impact. We also implemented and experimentally validated SPIDER using OpenState (an OpenFlow 1.3 extension for stateful packet processing), showing numerical results on its performance in terms of recovery latency and packet losses.Comment: 8 page

Privacy-Preserving Shortest Path Computation

Navigation is one of the most popular cloud computing services. But in virtually all cloud-based navigation systems, the client must reveal her location and destination to the cloud service provider in order to learn the fastest route. In this work, we present a cryptographic protocol for navigation on city streets that provides privacy for both the client's location and the service provider's routing data. Our key ingredient is a novel method for compressing the next-hop routing matrices in networks such as city street maps. Applying our compression method to the map of Los Angeles, for example, we achieve over tenfold reduction in the representation size. In conjunction with other cryptographic techniques, this compressed representation results in an efficient protocol suitable for fully-private real-time navigation on city streets. We demonstrate the practicality of our protocol by benchmarking it on real street map data for major cities such as San Francisco and Washington, D.C.Comment: Extended version of NDSS 2016 pape