Gordon T. Wilfong

The stable paths problem and interdomain routing

Dynamic routing protocols such as RIP and OSPF essentially implement distributed algorithms for solving the shortest paths problem. The border gateway protocol (BGP) is currently the only interdomain routing protocol deployed in the Internet. BGP does not solve a shortest paths problem since any interdomain protocol is required to allow policy-based metrics to override distance-based metrics and enable autonomous systems to independently define their routing policies with little or no global coordination. It is then natural to ask if BGP can be viewed as a distributed algorithm for solving some fundamental problem. We introduce the stable paths problem and show that BGP can be viewed as a distributed algorithm for solving this problem. Unlike a shortest path tree, such a solution does not represent a global optimum, but rather an equilibrium point in which each node is assigned its local optimum.We study the stable paths problem using a derived structure called a dispute wheel, representing conflicting routing policies at various nodes. We show that if no dispute wheel can be constructed, then there exists a unique solution for the stable paths problem. We define the simple path-vector protocol (SPVP), a distributed algorithm for solving the stable paths problem. SPVP is intended to capture the dynamic behavior of BGP at an abstract level. If SPVP converges, then the resulting state corresponds to a stable paths solution. If there is no solution, then SPVP always diverges. In fact, SPVP can even diverge when a solution exists. We show that SPVP will converge to the unique solution of an instance of the stable paths problem if no dispute wheel exists.

Autonomous robot vehicles

Contents: Guidance (Kinematics, Control, and Trajectory Generation.- Sensors.- Navigation (Position and Copurse Estimation).- Map Representation.- Sensing Strategies.- Motion Planning.- Systems.

An analysis of BGP convergence properties

The Border Gateway Protocol (BGP) is the de facto inter-domain routing protocol used to exchange reachability information between Autonomous Systems in the global Internet. BGP is a path-vector protocol that allows each Autonomous System to override distance-based metrics with policy-based metrics when choosing best routes. Varadhan et al. [18] have shown that it is possible for a group of Autonomous Systems to independently define BGP policies that together lead to BGP protocol oscillations that never converge on a stable routing. One approach to addressing this problem is based on static analysis of routing policies to determine if they are safe. We explore the worst-case complexity for convergence-oriented static analysis of BGP routing policies. We present an abstract model of BGP and use it to define several global sanity conditions on routing policies that are related to BGP convergence/divergence. For each condition we show that the complexity of statically checking it is either NP-complete or NP-hard.

On the correctness of IBGP configuration

The Border Gateway Protocol (BGP) has two distinct modes of operation. External BGP (EBGP) exchanges reachability information between autonomous systems, while Internal BGP (IBGP) exchanges external reachability information within an autonomous system. We study several routing anomalies that are unique to IBGP because, unlike EBGP, forwarding paths and signaling paths are not always symmetric. In particular, we focus on anomalies that can cause the protocol to diverge, and those that can cause a routers chosen forwarding path to an egress point to be deflected by another router on that path. Deflections can greatly complicate the debugging of routing problems, and in the worst case multiple deflections can combine to form persistent forwarding loops. We define a correct IBGP configuration to be one that is anomaly free for every possible set of routes sent by neighboring autonomous systems. We show that determination of IBGP configuration correctness is NP-hard. However, we give simple sufficient conditions on network configurations that guarantee correctness.

A safe path vector protocol

An IP routing protocol is safe if it is guaranteed to converge in the absence of network topology changes. BGP, currently the only interdomain routing protocol employed on the Internet, is not safe in this sense. It may seem that the source of BGPs potential divergence is inherent in the requirements for any interdomain routing protocol-policy-based metrics must be allowed to override distance-based metrics, and each autonomous system must be allowed to independently define its routing policies with little or no global coordination. In this paper we present a simple path vector protocol (SPVP) that captures the underlying semantics of BGP by abstracting away all nonessential details. We then add a dynamically computed attribute to SPVP routing messages, called the route history. Protocol oscillations caused by policy conflicts produce routes whose histories contain cycles. These cycles identify the policy conflicts and the autonomous systems involved. SPVP is made safe by automatically suppressing routes whose histories contain cycles. We discuss how this safe SPVP can be used in the design of a safe BGP.

Policy disputes in path-vector protocols

The border gateway protocol, BGP, is currently the only interdomain routing protocol employed on the Internet. As required of any interdomain protocol, BGP allows policy-based metrics to override distance-based metrics and enables each autonomous system to independently define its routing policies with little or no global coordination. Varadhan et al. (1996) have shown that there are collections of routing policies that together are not safe in the sense that they can cause BGP to diverge. That is, an unsafe collection of routing policies can result in some autonomous systems exchanging BGP routing messages indefinitely, without ever converging to a set of stable routes. In this paper we present sufficient conditions on routing policies that guarantee BGP safety. We use a new formalism, called the simple path vector protocol (SPVP), that is designed to capture the underlying semantics of any path vector protocol such as BGP. We identify a certain circular set of relationships between routing policies at various autonomous systems that we call a dispute cycle. We show that systems with no dispute cycles are guaranteed to be safe. While these include systems whose policies are consistent with shortest paths under some link metric, the class of systems with no dispute cycles is strictly larger.

Motion planning in the presence of movable obstacles

Motion planning algorithms have generally dealt with motion in a static environment, or more recently, with motion in an environment that changes in a known manner. We consider the problem of finding collision-free motions in a changeable environment. That is, we wish to find a motion for an object where the object is permitted to move some of the obstacles. In such an environment the final positions of the movable obstacles may or may not be part of the goal. In the case where the final positions of the obstacles are unspecified, the motion planning problem is shown to be NP-hard. An algorithm that runs in <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>2</supscrpt>log<italic>n</italic>) time after <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>3</supscrpt>log<supscrpt>2</supscrpt><italic>n</italic>) preprocessing time is presented when the object to be moved is polygonal and there is only one movable polygonal obstacle in a polygonal environment of complexity <italic>&Ogr;</italic>(<italic>n</italic>). In the case where the final positions of the obstacles are specified the general problem is shown to be PSPACE-hard and an algorithm is given when there is one movable obstacle with the same preprocessing time as the previous algorithm but with <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>2</supscrpt>) query time.

Designing Multihop Wireless Backhaul Networks with Delay Guarantees

As wireless access technologies improve in data rates, the problem focus is shifting towards providing adequate backhaul from the wireless access points to the Internet. Existing wired backhaul technologies such as copper wires running at DSL, T1, or T3 speeds can be expensive to install or lease, and are becoming a performance bottleneck as wireless access speeds increase. Longhaul, non-line-of-sight wireless technologies such as WiMAX (802.16) hold the promise of enabling a high speed wireless backhaul as a cost-effective alternative. However, the biggest challenge in building a wireless backhaul is achieving guaranteed performance (throughput and delay) that is typically provided by a wired backhaul. This paper explores the problem of efficiently designing a multihop wireless backhaul to connect multiple wireless access points to a wired gateway. In particular, we provide a generalized link activation framework for scheduling packets over this wireless backhaul, such that any existing wireline scheduling policy can be implemented locally at each node of the wireless backhaul. We also present techniques for determining good interference-free routes within our scheduling framework, given the link rates and cross-link interference information. When a multihop wireline scheduler with worst case delay bounds (such as WFQ or Coordinated EDF) is implemented over the wireless backhaul, we show that our scheduling and routing framework guarantees approximately twice the delay of the corresponding wireline topology. Finally, we present simulation results to demonstrate the low delays achieved using our framework.

Route oscillations in I-BGP with route reflection

We study the route oscillation problem [16, 19] in the Internal Border Gateway Protocol (I-BGP)[18] when route reflection is used. We propose a formal model of I-BGP and use it to show that even deciding whether an I-BGP configuration with route reflection can converge is an NP-Complete problem. We then propose a modification to I-BGP and show that route reflection cannot cause the modified protocol to diverge. Moreover, we show that the modified protocol converges to the same stable routing configuration regardless of the order in which messages are sent or received.

Evaluating the performance of table processing algorithms

Abstract. While techniques for evaluating the performance of lower-level document analysis tasks such as optical character recognition have gained acceptance in the literature, attempts to formalize the problem for higher-level algorithms, while receiving a fair amount of attention in terms of theory, have generally been less successful in practice, perhaps owing to their complexity. In this paper, we introduce intuitive, easy-to-implement evaluation schemes for the related problems of table detection and table structure recognition. We also present the results of several small experiments, demonstrating how well the methodologies work and the useful sorts of feedback they provide. We first consider the table detection problem. Here algorithms can yield various classes of errors, including non-table regions improperly labeled as tables (insertion errors), tables missed completely (deletion errors), larger tables broken into a number of smaller ones (splitting errors), and groups of smaller tables combined to form larger ones (merging errors). This leads naturally to the use of an edit distance approach for assessing the results of table detection. Next we address the problem of evaluating table structure recognition. Our model is based on a directed acyclic attribute graph, or table DAG. We describe a new paradigm, “graph probing,” for comparing the results returned by the recognition system and the representation created during ground-truthing. Probing is in fact a general concept that could be applied to other document recognition tasks as well.