Compact Routing Schemes Research Articles

Improved Compact Routing Schemes for Random Interconnects

Random topology has been an increasingly favorable approach for designing interconnection networks, as it can provide a combination of low latency and incremental network growth that could not be provided by the traditional rigid topologies. However, the common shortest-path routing in a random interconnect poses a scalability problem, for it requires global network info to make routing decisions and so, the routing table size (RTS) can be very large. Therefore, this manuscript would aim to revisit the well-known research area of landmark-based compact routing and to improve the universal routing schemes for the specific case of random interconnects. It would propose new landmark-based compact routing schemes, using 2 heuristic techniques to select landmarks that are evenly spaced, which would reduce the RTS in the well-known Thorup and Zwick's scheme by up to 18% and produce a shorter average path length.

Certification of Compact Low-Stretch Routing Schemes

On the one hand, the correctness of routing protocols in networks is an issue of utmost importance for guaranteeing the delivery of messages from any source to any target. On the other hand, a large collection of routing schemes have been proposed during the last two decades, with the objective of transmitting messages along short routes, while keeping the routing tables small. Regrettably, all these schemes share the property that an adversary may modify the content of the routing tables with the objective of, e.g., blocking the delivery of messages between some pairs of nodes, without being detected by any node. In this paper, we present a simple certification mechanism which enables the nodes to locally detect any alteration of their routing tables. In particular, we show how to locally verify the stretch-3 routing scheme by Thorup and Zwick [SPAA 2001] by adding certificates of $\widetilde{O}(\sqrt{n})$ bits at each node in $n$-node networks, that is, by keeping the memory size of the same order of magnitude as the original routing tables. We also propose a new name-independent routing scheme using routing tables of size $\widetilde{O}(\sqrt{n})$ bits. This new routing scheme can be locally verified using certificates on $\widetilde{O}(\sqrt{n})$ bits. Its stretch is3 if using handshaking, and 5 otherwise.

WMGR: A Generic and Compact Routing Scheme for Data Center Networks

Data center networks (DCNs) connect hundreds and thousands of computers and, as a result of the exponential growth in their number of nodes, the design of scalable (compact) routing schemes plays a pivotal role in the optimal operation of the DCN. Traditional trends in the design of DCN architectures have led to solutions, where routing schemes and network topologies are interdependent, i.e., specialized routing schemes. Unlike these, we propose a routing scheme that is compact and generic, i.e., independent of the DCN topology, the word-metric-based greedy routing. In this scheme, each node is assigned to a coordinate (or label) in the word-metric space (WMS) of an algebraic group and then nodes forward packets to the closest neighbor to the destination in this WMS. We evaluate our scheme and compare it with other routing schemes in several topologies. We prove that the memory space requirements in nodes and the forwarding decision time grow sub-linearly (with respect to n, the number of nodes) in all of these topologies. The scheme finds the shortest paths in topologies based on Cayley graphs and trees (e.g. Fat tree), while in the rest of topologies, the length of any path is stretched by a factor that grows logarithmically (with respect to n). Moreover, the simulation results show that many of the paths remain far below this upper bound.

Dist-Coop: Distributed Cooperative Transmission in UWSNs using Optimization Congestion Control and Opportunistic Routing

One of the real issues in UWSN is congestion control. The need is to plan an optimized congestion control scheme which enhances the network life time and in addition limits the usage of energy in data transmission from source to destination. In this paper, we propose a routing protocol called Dist-Coop in UWSN. Dist-Coop is a distributed cooperation based routing scheme which uses mechanism for optimized congestion control in noisy links of underwater environment. It is compact, energy proficient and high throughput opportunistic routing scheme for UWSN. In this proposed protocol architecture, we present congestion control with cooperative transmission of data packets utilizing relay sensors. The final objective is to enhance the network life time and forward information utilizing cooperation procedure, limiting energy consumption amid transmission of information. At destination node, combining strategy utilized is based on Signal-to-Noise Ratio (SNRC). Simulation results of Dist-Coop scheme indicate better outcomes in terms of energy consumption, throughput and network lifetime in contrast with Co-UWSN and EH-UWSN routing protocols. Dist-Coop has expended substantially less energy and better throughput when contrasted with these protocols.

Compact routing messages in self-healing trees

Existing compact routing schemes, e.g., Thorup and Zwick [4] and Chechik [6] often have no means to tolerate failures, once the system has been set up and started. This paper presents, to our knowledge, the first self-healing compact routing scheme. Besides, our schemes are developed for low memory nodes and are compact schemes, meaning they require only O(log2⁡n) bits memory.We introduce two algorithms of independent interest: The first is CompactFT, a novel compact version of the self-healing algorithm Forgiving Tree of Hayes et al. [7] that uses only O(log⁡n) bits local memory. The second algorithm (CompactFTZ) combines CompactFT with Thorup–Zwick's tree-based compact routing scheme [4] to produce a compact self-healing routing scheme. In the self-healing model, the adversary deletes nodes one at a time and the affected nodes self-heal locally by adding few edges. We introduce the bounded-memory self-healing model, where the memory each node need to use for the self-healing algorithm is bounded. CompactFT recovers from each attack in only O(1) time and Δ messages, with only +3 degree increase and O(log⁡Δ) graph diameter increase, over any sequence of deletions (Δ is the initial maximum degree).Additionally, CompactFTZ guarantees delivery of a packet sent from sender s as long as the receiver t has not been deleted, with only an additional O(ylog⁡Δ) latency, where y is the number of nodes that have been deleted on the path between s and t. If t has been deleted, s gets informed and the packet is removed from the network. CompactFTZ uses only O(log⁡n) bits memory for local fields (such as routing tables) and O(log2⁡n) bits for the routing labels, thus requiring O(log2⁡n) bits overall.

Scale-Free Compact Routing Schemes in Networks of Low Doubling Dimension

We consider compact routing schemes in networks of low doubling dimension, where the doubling dimension is the least value α such that any ball in the network can be covered by at most 2 α balls of half radius. There are two variants of routing-scheme design: (i) labeled (name-dependent) routing, in which the designer is allowed to rename the nodes so that the names (labels) can contain additional routing information, for example, topological information; and (ii) name-independent routing, which works on top of the arbitrary original node names in the network, that is, the node names are independent of the routing scheme. In this article, given any constant ϵ ∈ (0, 1) and an n -node edge-weighted network of doubling dimension α ∈ O (loglog n ), we present —a (1 + ϵ)-stretch labeled compact routing scheme with ⌈log n ⌉-bit routing labels, O (log 2 n /loglog n )-bit packet headers, and ((1/ϵ) O (α) log 3 n )-bit routing information at each node; —a (9 + ϵ)-stretch name-independent compact routing scheme with O (log 2 n /loglog n )-bit packet headers, and ((1/ϵ) O (α) log 3 n )-bit routing information at each node. In addition, we prove a lower bound: any name-independent routing scheme with o ( n (ϵ/60) 2 ) bits of storage at each node has stretch no less than 9 − ϵ for any ϵ ∈ (0, 8). Therefore, our name-independent routing scheme achieves asymptotically optimal stretch with polylogarithmic storage at each node and packet headers. Note that both schemes are scale-free in the sense that their space requirements do not depend on the normalized diameter Δ of the network. We also present a simpler nonscale-free (9 + ϵ)-stretch name-independent compact routing scheme with improved space requirements if Δ is polynomial in n .

K-Chordal Graphs: From Cops and Robber to Compact Routing via Treewidth

Cops and robber games, introduced by Winkler and Nowakowski (in Discrete Math. 43(2---3), 235---239, 1983) and independently defined by Quilliot (in J. Comb. Theory, Ser. B 38(1), 89---92, 1985), concern a team of cops that must capture a robber moving in a graph. We consider the class of k-chordal graphs, i.e., graphs with no induced (chordless) cycle of length greater than k, k?3. We prove that k?1 cops are always sufficient to capture a robber in k-chordal graphs. This leads us to our main result, a new structural decomposition for a graph class including k-chordal graphs. We present a polynomial-time algorithm that, given a graph G and k?3, either returns an induced cycle larger than k in G, or computes a tree-decomposition of G, each bag of which contains a dominating path with at most k?1 vertices. This allows us to prove that any k-chordal graph with maximum degree Δ has treewidth at most (k?1)(Δ?1)+2, improving the O(Δ(Δ?1)k?3) bound of Bodlaender and Thilikos (Discrete Appl. Math. 79(1---3), 45---61, 1997. Moreover, any graph admitting such a tree-decomposition has small hyperbolicity). As an application, for any n-vertex graph admitting such a tree-decomposition, we propose a compact routing scheme using routing tables, addresses and headers of size O(klogΔ+logn) bits and achieving an additive stretch of O(klogΔ). As far as we know, this is the first routing scheme with O(klogΔ+logn)-routing tables and small additive stretch for k-chordal graphs.

Distributed and compact routing using spatial distributions in wireless sensor networks

In traditional routing, the routing tables store shortest paths to all other destinations and have size linear in the size of the network, which is not scalable for resource-constrained networks such as wireless sensor networks. In this article we show that by storing selectively a much smaller set of routing paths in the routing tables one can get low-stretch, compact routing schemes. Our routing scheme includes an approximate distance oracle with which one can obtain approximate shortest path length estimates to destinations. This distance oracle can be obtained, for example, by a landmark-based scheme, or in case of sensor networks, from the geographic distance between node locations. With an approximate distance oracle one can attempt greedy routing by forwarding to the neighbor whose estimate is closer to the destination. But there is no guarantee of delivery nor of the routing path length. We augment the distance oracle by storing, for each node u , routing paths to O (log 2 n ) strategically selected nodes that serve as intermediate destinations. These nodes are selected with probability proportional to 1/ r ρ , where r is the distance to u and ρ is a suitable constant for the network. Then we derive a set of sufficient conditions to select the next step at each stage of routing, such that these conditions can be verified locally and guarantee 1+ε stretch routing on any metric. These conditions serve as the “greedy routing” or local decision rule. On graphs of bounded growth, our scheme guarantees 1+ε stretch routing with high probability, with an average routing table size of O (√n log 2 n ). This scheme is favorable for its simplicity, generality, and blindness to any global state. It demonstrates that global routing properties could emerge from purely distributed and uncoordinated routing table design.

A compact routing scheme and approximate distance oracle for power-law graphs

Compact routing addresses the tradeoff between table sizes and stretch, which is the worst-case ratio between the length of the path a packet is routed through by the scheme and the length of an actual shortest path from source to destination. We adapt the compact routing scheme by Thorup and Zwick [2001] to optimize it for power-law graphs. We analyze our adapted routing scheme based on the theory of unweighted random power-law graphs with fixed expected degree sequence by Aiello et al. [2000]. Our result is the first analytical bound coupled to the parameter of the power-law graph model for a compact routing scheme. Let n denote the number of nodes in the network. We provide a labeled routing scheme that, after a stretch--5 handshaking step (similar to DNS lookup in TCP/IP), routes messages along stretch--3 paths. We prove that, instead of routing tables with Õ ( n 1/2 ) bits ( Õ suppresses factors logarithmic in n ) as in the general scheme by Thorup and Zwick, expected sizes of O ( n γ log n ) bits are sufficient, and that all the routing tables can be constructed at once in expected time O ( n 1+γ log n ), with γ = τ-22/τ-3 + ε, where τ∈(2,3) is the power-law exponent and ε 0 (which implies ε < γ < 1/3 + ε). Both bounds also hold with probability at least 1-1/ n (independent of ε). The routing scheme is a labeled scheme, requiring a stretch--5 handshaking step. The scheme uses addresses and message headers with O (log n log log n ) bits, with probability at least 1- o (1). We further demonstrate the effectiveness of our scheme by simulations on real-world graphs as well as synthetic power-law graphs. With the same techniques as for the compact routing scheme, we also adapt the approximate distance oracle by Thorup and Zwick [2001, 2005] for stretch-3 and we obtain a new upper bound of expected Õ ( n 1+γ ) for space and preprocessing for random power-law graphs. Our distance oracle is the first one optimized for power-law graphs. Furthermore, we provide a linear-space data structure that can answer 5--approximate distance queries in time at most Õ ( n 1/4+ε ) (similar to γ, the exponent actually depends on τ and lies between ε and 1/4 + ε).

Fault-tolerant compact routing schemes for general graphs

This paper considers compact fault-tolerant routing schemes for weighted general graphs, namely, routing schemes that avoid a set of failed (or forbidden) edges. We present a compact routing scheme capable of handling multiple edge failures. Assume a source node s contains a message M designated to a destination target t and assume a set F of edges crashes (unknown to s). Our scheme routes the message to t (provided that s and t are still connected in G∖F) over a path whose length is proportional to the distance between s and t in G∖F, to |F|3 and to some poly-log factor. The routing table required at a node v is of size proportional to the degree of v in G and some poly-log factor. This improves on the previously known fault-tolerant compact routing scheme for general graphs, which was capable of overcoming at most 2 edge failures.

Space-stretch tradeoff optimization for routing in Internet-like graphs

Compact routing intends to achieve good tradeoff between the routing path length and the memory overhead, and is recently considered as a main alternative to overcome the fundamental scaling problems of the Internet routing system. Plenty of studies have been conducted on compact routing, and quite a few universal compact routing schemes have been designed for arbitrary network topologies. However, it is generally believed that specialized compact routing schemes for peculiar network topologies can have better performance than universal ones. Studies on complex networks have uncovered that most real-world networks exhibit power-law degree distributions, i.e., a few nodes have very high degrees while many other nodes have low degrees. High-degree nodes play the crucial role of hubs in communication and inter-networking. Based on this fact, we propose two highest-degree landmark based compact routing schemes, namely HDLR and HDLR+. Theoretical analysis on random power-law graphs shows that the two schemes can achieve better spacestretch trade-offs than previous compact routing schemes. Simulations conducted on random power-law graphs and real-world AS-level Internet graph validate the effectiveness of our schemes.

HDLBR: A name-independent compact routing scheme for power-law networks

Compact routing intends to achieve a good tradeoff between routing path length and storage overhead, and is recently considered as a main alternative to overcome the fundamental scaling limitations of the Internet routing system. It is generally believed that specialized compact routing schemes for peculiar network topologies have better average performance than universal ones, and name-independent schemes are more flexible due to their natural support of the Locator/ID split principle. In this paper, we propose the highest degree landmark based routing scheme (HDLBR), which is the first specialized name-independent compact routing scheme for Internet-like power-law networks. HDLBR optimizes routing performance by selecting a few nodes with high degrees as the landmarks and making these nodes to be the entrances for mapping topologically agnostic node names to topology-aware node addresses. Simulation results show that HDLBR has very small average routing table size, in the order of O∼(n1/2) bits per node on both synthesized power-law graphs and the real AS graphs. Meanwhile, the average stretch of the HDLBR scheme is comparable with the base Abraham scheme using random coloring, and is only slightly outperformed by the customized Abraham scheme which also takes into account the degree heterogeneity of power-law networks.

Compact policy routing

The main concern in this paper is to generalize compact routing to arbitrary routing policies that favor a broader set of path attributes beyond path length. Using the formalism of routing algebras we identify the algebraic requirements for a routing policy to be realizable with sublinear size routing tables, and we show that a wealth of practical policies can be classified by our results. By generalizing the notion of stretch, we also discover the algebraic validity of compact routing schemes considered so far and we show that there are routing policies for which one cannot expect sublinear scaling even if permitting arbitrary constant stretch. Finally, we apply our methodology to the routing policies used in Internet inter-domain routing, and we show that our algebraic approach readily generalizes to this setting as well.

Compact and low delay routing labeling scheme for Unit Disk Graphs

In this paper, we propose a new compact and low delay routing labeling scheme for Unit Disk Graphs (UDGs) which often model wireless ad hoc networks. We show that one can assign each vertex of an n-vertex UDG G a compact O(log2n)-bit label such that, given the label of a source vertex and the label of a destination, it is possible to compute efficiently, based solely on these two labels, a neighbor of the source vertex that heads in the direction of the destination. We prove that this routing labeling scheme has a constant hop route-stretch (= hop delay), i.e., for each two vertices x and y of G, it produces a routing path with h(x,y) hops (edges) such that h(x,y)⩽3⋅dG(x,y)+12, where dG(x,y) is the hop distance between x and y in G. To the best of our knowledge, this is the first compact routing scheme for UDGs which not only guaranties delivery but has a low hop delay. Furthermore, our routing labeling scheme has a constant length route-stretch and a constant power route-stretch.To obtain this result, we establish a novel balanced separator theorem for UDGs, which mimics the well-known Lipton and Tarjanʼs planar balanced shortest paths separator theorem. We prove that, in any n-vertex UDG G, one can find two hop-shortest paths P(s,x) and P(s,y) such that the removal of the 3-hop-neighborhood of these paths (i.e., NG3[P(s,x)∪P(s,y)]) from G leaves no connected component with more than 2/3n vertices. This new balanced shortest-paths–3-hop-neighborhood separator theorem allows us to build, for any n-vertex UDG G, a system T(G) of at most 2log32n+2 spanning trees of G such that, for any two vertices x and y of G, there exists a tree T in T(G) with dT(x,y)⩽3⋅dG(x,y)+12. That is, the distances in any UDG can be approximately represented by the distances in at most 2log32n+2 of its spanning trees.

An impact of addressing schemes on routing scalability

The inter-domain routing scalability issue is a major challenge facing the Internet. Recent wide deployments of multi-homing and traffic engineering urge for solutions to this issue. So far, tunnel-based proposals and compact routing schemes have been suggested. An implicit assumption in the routing community is that structured address labels are crucial for routing scalability. This paper first systematically examines the properties of identifiers and address labels and their functional differences. It develops a simple Internet routing model and shows that a binary relation T defined on the address label set A determines the cardinality of the compact label set L. Furthermore, it is shown that routing schemes based on flat address labels are not scalable. This implies that routing scalability and routing stability are inherently related and must be considered together when a routing scheme is evaluated. Furthermore, a metric is defined to measure the efficiency of the address label coding. Simulations show that given a 3000-autonomous system (AS) topology, the required length of ad- dress labels in compact routing schemes is only 9.12 bits while the required length is 10.64 bits for the Internet protocol (IP) upper bound case. Simulations also show that the Q values of the compact routing and IP routing schemes are 0.80 and 0.95, respectively, for a 3000-AS topology. This indicates that a compact routing scheme with necessary routing stability is desirable. It is also seen that using provider allocated IP addresses in multihomed stub ASs does not significantly reduce the global routing size of an IP routing system.

F-Sensitivity Distance Oracles and Routing Schemes

An f-sensitivity distance oracle for a weighted undirected graph G(V,E) is a data structure capable of answering restricted distance queries between vertex pairs, i.e., calculating distances on a subgraph avoiding some forbidden edges. This paper presents an efficiently constructible f-sensitivity distance oracle that given a triplet (s,t,F), where s and t are vertices and F is a set of forbidden edges such that |F|≤f, returns an estimate of the distance between s and t in G(V,E∖F). For an integer parameter k≥1, the size of the data structure is O(fkn 1+1/k log (nW)), where W is the heaviest edge in G, the stretch (approximation ratio) of the returned distance is (8k−2)(f+1), and the query time is O(|F|⋅log 2 n⋅log log n⋅log log d), where d is the distance between s and t in G(V,E∖F). Our result differs from previous ones in two major respects: (1) it is the first to consider approximate oracles for general graphs (and thus obtain a succinct data structure); (2) our result holds for an arbitrary number of forbidden edges. In contrast, previous papers concern f-sensitive exact distance oracles, which consequently have size Ω(n 2). Moreover, those oracles support forbidden sets F of size |F|≤2. The paper also considers f-sensitive compact routing schemes, namely, routing schemes that avoid a given set of forbidden (or failed) edges. It presents a scheme capable of withstanding up to two edge failures. Given a message M destined to t at a source vertex s, in the presence of a forbidden edge set F of size |F|≤2 (unknown to s), our scheme routes M from s to t in a distributed manner, over a path of length at most O(k) times the length of the optimal path (avoiding F). The total amount of information stored in vertices of G is O(kn 1+1/k log (nW)log n). To the best of our knowledge, this is the first result obtaining an f-sensitive compact routing scheme for general graphs.

A Distributed Algorithm for Finding All Best Swap Edges of a Minimum-Diameter Spanning Tree

Communication in networks suffers if a link fails. When the links are edges of a tree that has been chosen from an underlying graph of all possible links, a broken link even disconnects the network. Most often, the link is restored rapidly. A good policy to deal with this sort of transient link failures is swap rerouting, where the temporarily broken link is replaced by a single swap link from the underlying graph. A rapid replacement of a broken link by a swap link is only possible if all swap links have been precomputed. The selection of high-quality swap links is essential; it must follow the same objective as the originally chosen communication subnetwork. We are interested in a minimum-diameter tree in a graph with edge weights (so as to minimize the maximum travel time of messages). Hence, each swap link must minimize (among all possible swaps) the diameter of the tree that results from swapping. We propose a distributed algorithm that efficiently computes all of these swap links, and we explain how to route messages across swap edges with a compact routing scheme. Finally, we consider the computation of swap edges in an arbitrary spanning tree, where swap edges are chosen to minimize the time required to adapt routing in case of a failure, and give efficient distributed algorithms for two variants of this problem.

Compact name-independent routing with minimum stretch

Given a weighted undirected network with arbitrary node names, we present a compact routing scheme, using a Õ (√n,) space routing table at each node, and routing along paths of stretch 3, that is, at most thrice as long as the minimum cost paths. This is optimal in a very strong sense. It is known that no compact routing using o ( n ) space per node can route with stretch below 3. Also, it is known that any stretch below 5 requires Ω(√ n ,)space per node.

On compact routing for the internet

The Internet's routing system is facing stresses due to its poor fundamental scaling properties. Compact routing is a research field that studies fundamental limits of routing scalability and designs algorithms that try to meet these limits. In particular, compact routing research shows that shortest-path routing, forming a core of traditional routing algorithms, cannot guarantee routing table (RT) sizes that on all network topologies grow slower than linearly as functions of the network size. However, there are plenty of compact routing schemes that relax the shortest-path requirement and allow for improved, sublinear RT size scaling that is mathematically provable for all static network topologies. In particular, there exist compact routing schemes designed for grids, trees, and Internet-like topologies that offer RT sizes that scale logarithmically with the network size. In this paper, we demonstrate that in view of recent results in compact routing research, such logarithmic scaling on Internet-like topologies is fundamentally impossible in the presence of topology dynamics or topology-independent (flat) addressing. We use analytic arguments to show that the number of routing control messages per topology change cannot scale better than linearly on Internet-like topologies. We also employ simulations to confirm that logarithmic RT size scaling gets broken by topology-independent addressing, a cornerstone of popular locator-identifier split proposals aiming at improving routing scaling in the presence of network topology dynamics or host mobility. These pessimistic findings lead us to the conclusion that a fundamental re-examination of assumptions behind routing models and abstractions is needed in order to find a routing architecture that would be able to scale "indefinitely.

Average stretch analysis of compact routing schemes

This paper presents some analytic results concerning the pivot interval routing ( PIR) strategy of [T. Eilam, C. Gavoille, D. Peleg, Compact routing schemes with low stretch factor, J. Algorithms 46(2) (2003) 97–114, Preliminary version appeared. in: Proceedings of the 17th ACM Symposium on Principles of Distributed Computing, June 1998, pp. 11–20.] That strategy allows message routing on every weighted n -node network along paths whose stretch (namely, the ratio between their length and the distance between their endpoints) is at most five, and whose average stretch is at most three, with routing tables of size O ( n log 3 / 2 n ) bits per node. In addition, the route lengths are at most 2 D ( ⌈ 1.5 D ⌉ for uniform weights) where D is the weighted diameter of the network. The PIR strategy can be constructed in polynomial time and can be implemented so that the generated scheme is in the form of an interval routing scheme ( IRS), using at most O ( n log n ) intervals per link. Here, it is shown that there exists an unweighted n -node graph G and an identity assignment ID for its nodes such that for every R ∈ PIR on G with a set of pivots computed by a greedy cover algorithm (respectively, a randomized algorithm), AvStr G ( R ) > 3 - o ( 1 ) (respectively, with high probability). Also, it is shown that for almost every unweighted n -node graph G , and for every R ∈ PIR on G , AvStr G ( R ) = 1.875 ± o ( 1 ) . A comparison between PIR and HCP k , the hierarchical routing strategy presented in [B. Awerbuch, A. Bar-Noy, N. Linial, D. Peleg, Improved routing strategies with succinct tables, J. Algorithms 11 (1990) 307–341.] is also given.