Optimal routing in double loop networks

AbstractIn this paper, we study the problem of finding the shortest path in circulant graphs with an arbitrary number of jumps. We provide algorithms specifically tailored for weighted undirected and directed circulant graphs with two jumps which compute the shortest path. Our method only requires O(logN) arithmetic operations and the total bit complexity is O(log2NloglogNlogloglogN), where N is the number of the graph’s vertices. This elementary and efficient shortest path algorithm has been derived from the Closest Vector Problem (CVP) of lattices in dimension two and with an ℓ1 norm

Preserving Link Privacy in Social Network Based Systems

A growing body of research leverages social network based trust relationships to improve the functionality of the system. However, these systems expose users' trust relationships, which is considered sensitive information in today's society, to an adversary. In this work, we make the following contributions. First, we propose an algorithm that perturbs the structure of a social graph in order to provide link privacy, at the cost of slight reduction in the utility of the social graph. Second we define general metrics for characterizing the utility and privacy of perturbed graphs. Third, we evaluate the utility and privacy of our proposed algorithm using real world social graphs. Finally, we demonstrate the applicability of our perturbation algorithm on a broad range of secure systems, including Sybil defenses and secure routing.Comment: 16 pages, 15 figure

Shortest Paths Avoiding Forbidden Subpaths

In this paper we study a variant of the shortest path problem in graphs: given a weighted graph G and vertices s and t, and given a set X of forbidden paths in G, find a shortest s-t path P such that no path in X is a subpath of P. Path P is allowed to repeat vertices and edges. We call each path in X an exception, and our desired path a shortest exception-avoiding path. We formulate a new version of the problem where the algorithm has no a priori knowledge of X, and finds out about an exception x in X only when a path containing x fails. This situation arises in computing shortest paths in optical networks. We give an algorithm that finds a shortest exception avoiding path in time polynomial in |G| and |X|. The main idea is to run Dijkstra's algorithm incrementally after replicating vertices when an exception is discovered.Comment: 12 pages, 2 figures. Fixed a few typos, rephrased a few sentences, and used the STACS styl

TOPOLOGY CONTROL ALGORITHMS FOR RULE-BASED ROUTING

In this dissertation, we introduce a new topology control problem for rule- based link-state routing in autonomous networks. In this context, topology control is a mechanism to reduce the broadcast storm problem associated with link-state broadcasts. We focus on a class of topology control mechanisms called local-pruning mechanisms. Topology control by local pruning is an interesting multi-agent graph optimization problem, where every agent/router/station has access to only its local neighborhood information. Every agent selects a subset of its incident link-state in- formation for broadcast. This constitutes the pruned link-state information (pruned graph) for routing. The objective for every agent is to select a minimal subset of the local link-state information while guaranteeing that the pruned graph preserves desired paths for routing. In topology control for rule-based link-state routing, the pruned link-state information must preserve desired paths that satisfy the rules of routing. The non- triviality in these problems arises from the fact that the pruning agents have access to only their local link-state information. Consequently, rules of routing must have some property, which allows specifying the global properties of the routes from the local properties of the graph. In this dissertation, we illustrate that rules described as algebraic path problem in idempotent semirings have these necessary properties. The primary contribution of this dissertation is identifying a policy for pruning, which depends only on the local neighborhood, but guarantees that required global routing paths are preserved in the pruned graph. We show that for this local policy to ensure loop-free pruning, it is sufficient to have what is called an inflatory arc composition property. To prove the sufficiency, we prove a version of Bellman's optimality principle that extends to path-sets and minimal elements of partially ordered sets. As a motivating example, we present a stable path topology control mecha- nism, which ensures that the stable paths for routing are preserved after pruning. We show, using other examples, that the generic pruning works for many other rules of routing that are suitably described using idempotent semirings

Optimizing Flow Thinning Protection in Multicommodity Networks with Variable Link Capacity

International audienceFlow thinning (FT) is a concept of a traffic routing and protection strategy applicable to communication networks withvariable capacity of links. In such networks, the links do not attain their nominal (maximum) capacity simultaneously, so in atypical network state only some links are fully available whereas on each of the remaining links only a fraction of itsmaximum capacity is usable. Every end-to-end traffic demand is assigned a set of logical tunnels whose total capacity isdedicated to carry the demand’s traffic. The nominal (i.e., maximum) capacity of the tunnels, supported by the nominal(maximum) link capacity, is subject to state-dependent thinning to account for variable capacity of the links fluctuating belowthe maximum. Accordingly, the capacity available on the tunnels is also fluctuating below their nominal levels and hence theinstantaneous traffic sent between the demand’s end nodes must accommodate to the current total capacity available onits dedicated tunnels. The related multi-commodity flow optimization problem is NP-hard and its noncompact linearprogramming formulation requires path generation. For that, we formulate an integer programming pricing problem, atthe same time showing the cases when the pricing is polynomial. We also consider an important variant of FT, affinethinning, that may lead to practical FT implementations. We present a numerical study illustrating traffic efficiency of FT andcomputational efficiency of its optimization models. Our considerations are relevant, among others, for wireless meshnetworks utilizing multiprotocol label switching tunnels

Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane

Given a set $P$ of $n$ points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. A TSP instance is rectilinear when the points lie in the plane and the distance considered between two points is the $l_1$ distance. In this paper, a fixed-parameter algorithm for the Rectilinear TSP is presented and relies on techniques for solving TSP on bounded-treewidth graphs. It proves that the problem can be solved in $O\left(nh7^h\right)$ where $h \leq n$ denotes the number of horizontal lines containing the points of $P$. The same technique can be directly applied to the problem of finding a shortest rectilinear Steiner tree that interconnects the points of $P$ providing a $O\left(nh5^h\right)$ time complexity. Both bounds improve over the best time bounds known for these problems.Comment: 24 pages, 13 figures, 6 table