On the Computational Complexity and Effectiveness of N-hub Shortest-Path Routing

Transcription

1 1 On the Comptational Complexity and Effectiveness of N-hb Shortest-Path Roting Reven Cohen Gabi Nakibli Dept. of Compter Sciences Technion Israel Abstract In this paper we stdy the comptational complexity and effectiveness of a concept we term N-hb Shortest- Path Roting in IP networks. N-hb Shortest-Path Roting allows the ingress node of a roting domain to determine p to N intermediate nodes ( hbs ) throgh which a packet will pass before reaching its final destination. This facilitates better tilization of the network resorces, while allowing the network roters to contine to employ the simple and well-known shortest-path roting paradigm. Althogh this concept has been proposed in the past, this paper is the first to investigate it in depth. We apply N-hb Shortest-Path Roting to the problem of minimizing the maximm load in the network. We show that the reslting roting problem is NP-complete and hard to approximate. However, we propose efficient algorithms for solving it both in the online and the offline contexts. Or reslts show that N-hb Shortest-Path Roting can increase network tilization significantly even for N = 1. Hence, this roting paradigm shold be considered as a powerfl mechanism for ftre datagram roting in the Internet. I. INTRODUCTION Intra-AS roting in the Internet is based on the hop-by-hop shortest-path paradigm. The sorce of a packet specifies the destination address, and each roter along the rote forwards the packet to a neighbor located closest to the destination. Since the roting is sally static, i.e., the cost of a path is dependent on the network topologies rather than on the dynamics of the network traffic, a single rote is sed for every sorce-destination pair. The shortest-path roting paradigm is known to be simple and efficient. It does not place a heavy processing brden on the roters and sally reqires at most one entry per destination network in every roter. However, while this scheme finds the shortest path for each pair of nodes and ths minimizes the bandwidth consmed by every packet, it does not garantee fll tilization of the network resorces nder high traffic loads. When the network load is not niformly distribted, some of the roters introdce an excessive delay while others are ndertilized. In some cases this non-optimized se of network resorces may introdce not only excessive delays bt also incr a high packet loss rate. Mch research has been condcted in a search for an alternative roting paradigm that wold address this drawback of shortest-path roting. The soght paradigm shold tilize the network resorces more efficiently and minimize the probability of congestion, thereby achieving better delaythroghpt behavior than traditional shortest-path roting. In path 1 (shortest) A B C D path (throgh G) E F G H path 3 (throgh F,B) Fig. 1. An example of N-hb roting addition, sch a scheme shold be practical in terms of the volme of control information exchanged by the roters, the memory reqirement, the processing brden imposed by every packet, and so forth. Finally, sch a scheme shold interoperate seamlessly with network roters that contine to employ the shortest-path roting paradigm. Most of the roting schemes proposed in the past are able to employ more than one path between every sorcedestination pair. Generally, these schemes base their roting decisions on the load imposed on every network link. When a particlar link, or an area, becomes congested, some of the rotes are modified. Some roting schemes find an alternate data path only when the standard path is highly congested [1]. In [] [4], alternate rotes are fond for every sorcedestination pair even if the standard rote is not heavily loaded. Several loop-free paths are fond in advance and the load is distribted between them. However, de to the complexity of these schemes, their increased processing brden, and their considerable deviation from the conventional shortest-path roting paradigm, not one of them has been adopted for the Internet. A major drawback of many proposed roting schemes is that they mst be deployed over the lion s share of the roting domain in order to be effective. This paper investigates a roting scheme that takes advantage of a concept we refer to as N-hb Shortest-Path Roting, or simply N-hb roting. This concept can be implemented sing several existing IP mechanisms, as will be discssed in Section II. N-hb roting allows the ingress roter of a roting domain to determine one or more intermediate nodes ( hbs ) that a packet will traverse before reaching its final destination. Fig. 1 illstrates this concept. The figre shows three paths for a packet whose sorce and destination

2 nodes are A and D. The first path, path-1, is the shortest path. Path- ses node G as a single hb. Packets are roted first on the shortest path from A to G and then on the shortest path from G to D. Sch a rote is likely to improve the throghpt if the links B C or C D are heavily loaded while the links B G, G H and H D are ndertilized. Path-3 ses hbs: F and B. Packets are roted first on the shortest path from A to F, then on the shortest path from F to B, and finally on the shortest path from B to D. It is evident from the example above that N-hb roting is a generalization of shortest-path roting, becase shortest-path roting is eqivalent to N-hb roting with N = 0. Using the concept of N-hb roting, the roting protocol gains better control over the roting process, while the network roters contine to employ the shortest-path paradigm for bilding their roting tables. Althogh this concept is not employed today in the Internet, we think it is a powerfl tool that shold be investigated in the context of traffic engineering and QoS. It is important to note the practical benefits of N-hb Shortest-Path Roting over virtal-circit roting. First, N- hb roting can be implemented in networks that sally do not employ virtal-circit roting technologies (sch as MPLS [5]). In particlar, it can be implemented in sensor networks and ad hoc (mobile) networks. Second, when virtalcircit roting is sed, only one or two rotes are sally established between every two roters. Therefore, it is not possible to react to changes in the traffic pattern before the time-consming and labor-intensive bilding of new rotes. In contrast, an N-hb rote can be changed immediately according to changes in the link loads, withot having to set p additional rotes in advance. Third, N-hb roting imposes additional processing and memory brden on the hbs and the sorce edge roters only, while the other nodes employ reglar shortest-path roting. In virtal-circit roting this brden is imposed on all the nodes along the path. This is especially significant when each node has to maintain several thosands of explicit rotes. The ingress roter of a roting domain shold be responsible for determining the intermediate roter(s) throgh which the packets of each flow will be roted. To this end, the roter may se information it acqires regarding the load distribtion in the network by means of a link-state flooding protocol like OF-TE [6]. For a typical case scenario for N-hb roting in an I AS, consider a DiffServ [7] domain, which spports the Expedited Forwarding (EF) Per Hop Behavior. When an edge roter receives a packet of an EF flow (e.g., a Voice over IP flow), and N-hb roting is not spported, the roter has no option bt to forward the packet along the defalt (shortest) path or to drop it. With N-hb roting spport, however, the edge roter ses information abot the load distribtion in the entire domain, as can be obtained sing OF-TE [6], in order to determine the hb(s) that define the least congested rote. This list of hb(s) is added to the packet, and is also kept in the roter s local flow table. When sbseqent packets of the same flow are received by this roter, it identifies them as belonging to the same flow, e.g., sing the flow label of IPv6, and fetches from its table the list of hb(s) associated with this flow. Once every time-ot period, the roter checks if there is a better N-hb rote that can be sed by the considered flow. To the best of or knowledge, this paper is the first to propose a thorogh theoretical and practical investigation of N-hb Shortest-Path Roting. The contribtion of this paper is forfold. First, we define the N-hb shortest-path problem as an optimization problem, and show that from a comptational complexity perspective, N-hb is closer to virtal circit ( V -hb) roting than to shortest path (0-hb) roting. This is becase N-hb is NP-complete, and it has no polynomial approximation scheme (PTAS). Second, we develop a probabilistic approximation algorithm for the N-hb problem. Third, we show that online algorithms originally designed for mlticommodity roting maintain their competitive ratio for N-hb roting. Forth, we show that in practice, one hb for every flow is sfficient to obtain reslts that are almost eqal to those obtained by optimal algorithms for the splittable mlticommodity flow problem. These reslts are pper bonds for the reslts that can be obtained by optimal algorithms for virtal circit roting. The rest of this paper is organized as follows. In Section II we discss related work and the varios mechanisms that can be employed to implement N-hb roting. In Section III we define the N-hb roting problem and review its comptational complexity. In Section IV we present several approximation algorithms for the online context. The competitive ratio of these algorithms is discssed, and one of them is shown to have the best competitive ratio that can be obtained for this problem. In Section V we present simlation reslts that show the potential effectiveness of N-hb roting in general, and the effectiveness of the varios algorithms proposed in the paper. Finally, Section VI concldes the paper. II. N-HUB SHORTEST-PATH ROUTING IN IP NETWORKS: IMPLEMENTATION AND RELATED WORK N-hb Shortest-Path Roting can be implemented sing several existing mechanisms. A straightforward way is to take advantage of the IPv4 Loose Sorce-Roting option [8]. When this option is sed, the IP header is extended by a list of the addresses of the intermediate node(s) the packet mst traverse. However, this option, mch like any other IPv4 option, is rarely sed, mainly becase of the heavy processing brden imposed on the general prpose CPU of the roter when an IPv4 header contains any optional field. Moreover, there are some notable secrity isses related to this option [9]. In [10], it is noted that only 8% of Internet roters are sorce-roting capable. As opposed to IPv4, IPv6 [11] has a more bilt-in spport for N-hb roting. The primary header of an IPv6 packet can be followed by flexible extension headers. These headers can, for example, indicate the IP addresses of the network roters the packet shold traverse en rote to its destination. Another way to implement N-hb roting in IPv4 is to se IP-in-IP encapslation [1]. In this case, an IP header indicating the final destination is encapslated in the payload of another IP header. The latter header contains, in its destination address field, the IP address of an intermediate roter. The total

3 3 nmber of headers is therefore eqal to the nmber of hbs pls 1. N-hb roting can also be implemented throgh an overlay network [13]. In an overlay network the sorce sends a packet to the first hb, while adding to its payload information that identifies the next hbs and the final destination. Each hb ses this information to rote the packet to the next hb. Another powerfl way to implement the N-hb roting paradigm is to se MPLS [5]. MPLS is a virtal circit technology that allows an MPLS ingress node to set p a tnnel over the shortest path or over an explicit path to an egress node. An explicit path contains a list of intermediate nodes. The rote between two consective nodes in the list is either strict or loose. A loose rote may contain other nodes. Therefore, N-hb shortest path roting can be viewed as a special case of the MPLS explicit rote option. With respect to MPLS, or reslts imply that an explicit strict rote need not be specified. Rather, it is sfficient for the ingress MPLS node to inclde a single loose node in the RSVP-TE Path message. If the tnnel shold be established over the rote whose maximm load is minimized, the roting algorithms we propose can be sed. We are not aware of any work that addresses the comptational complexity and the potential effectiveness of N-hb Shortest-Path Roting, which is the core of this paper. Several roting schemes that are similar in one way or another to ors have been leveraged in other works, e.g., [13] [15], bt their focs is entirely different. In [16], [17], the athors present a mlti-path roting scheme called two-phase roting. In this scheme, traffic originating at a sorce node is roted over a set of rotes in predetermined and static proportions. Each rote is diverted from the sorce to an intermediate node before reaching the destination. This approach is shown to provide load balancing and bandwidth efficiency even with highly variable traffic. In [18] the athors explore the deployment of this roting scheme in optical networks, in order to increase roting resiliency. In [19] the athors stdy the throghpt performance of that roting scheme. Or paper 1 investigates the effectiveness and comptational complexity of the general form of two-phase roting. Frthermore, we consider nonstatic roting in which intermediate nodes are determined according to crrent traffic conditions, while addressing the online setting of the problem. In [0], the athors investigate the effectiveness of selfish roting in Internet-like environments. Selfish roting allows the host to determine the path according to a criterion that maximizes its profit. This work specifically addresses a setting where sorces choose N-hb rotes in an overlay network. Their main conclsion is that selfish hosts can achieve reslts similar to those achieved by roting with fll control. There are two notable differences between [0] and or work. First, in or model, the host chooses rotes that do not necessarily maximize its profit. Second, [0] assmes absolte knowledge of flow demands that do not change over time, while we deal with the more practical online scenario where flow demands 1 An early version of or paper, pblished in Infocom 004, predates Ref. [16] [19] are not known in advance. As already said, the main benefit gained from determining more intermediate nodes (hbs) for a rote between a sorcedestination pair is better control over network load distribtion, with little deviation from the traditional shortest-path roting paradigm. More specifically, the roters contine bilding their roting tables sing the shortest-path information they acqire throgh a conventional roting protocol. However, the network is capable of roting a packet over less congested areas. Moreover, it can be employed effectively even if a small fraction of the network roters spport it. This is becase traffic can be diverted to less congested areas withot the spport of the core roters. The trade-off between the simplicity of traditional datagram (shortest-path) roting and the efficiency of virtal-circit roting is well known. However, both schemes can be viewed as special cases of N-hb roting: with N = 0 for shortest-path roting and N = V for virtal circit roting. Hence, N-hb roting, where 0 N V, offers a compromise between these two extremes (see Fig. ). As the nmber of allowed hbs grows, the nmber of possible rotes between each sorce-destination pair increases, and the flexibility/efficiency of the roting scheme increases as well. However, we pay for the increased efficiency by sacrificing some of the inherent simplicity of shortest-path roting at each hb. In practice, as shown in Section V, the performance achieved with a single hb is very close to the optimal performance of virtalcircit roting. Hence, 1-hb roting can be viewed as a roting protocol that offers the performance of virtal-circit roting with only slight deviation from traditional shortest-path roting. III. PROBLEM DEFINITION AND COMPLEXITY A. Problem Definition In this paper we focs on applying the N-hb Shortest-Path Roting paradigm to a traffic engineering task. Or specific aim is to minimize the maximm load in the network. We deal with the roting problem of minimizing the maximm load imposed on a single link by determining p to N intermediate nodes throgh which the packets of each flow will be roted. Note that we do not assme any constraint regarding the criteria sed for classifying packets to flows. A similar objective minimizing the maximm load imposed on a single link was addressed in the past mainly in the context of the mlticommodity flow problem [1], [] and the Virtal Circit Roting problem [3] [5]. Maximizing the load on a single link does not always garantee perfect load balancing and minimm average delay. However, it was shown in the past to yield good performance becase the delay on a link grows exponentially with the load. Moreover, this objective is easier to analyze from a theoretical point of view. As a conter-example, consider the topology in Fig. 3 and sppose there are 3 flows as follows: 1) A flow from node A to node E, with a bandwidth demand of 1. ) A flow from node A to node B, with a bandwidth demand of.

4 4 "N hb Shortest Path Roting" 0 hb (shortest path roting) 1 hb hb 3 hb V hb (nsplitable mlticommodity flow) (virtal circit roting) simplicity flexibility/efficiency Fig.. N-hb roting as a compromise between efficiency and simplicity 3) A flow from node B to node E, with a bandwidth demand of 1. An algorithm that minimizes the maximm load may prodce a soltion that rotes flows 1 and 3 via node C. This soltion yields a greater delay of the packets of flow 1 and flow 3 than a soltion obtained by an algorithm that tries to minimize the average delay. The latter soltion might rote flow 1 throgh roter C and flow 3 throgh roter D. One may consider the average load over all the edges in the graph as a better objective for minimizing the average delay of the packets. However, this objective is achieved with static shortest-path roting which, as mentioned above, is known to be inefficient for non-niform traffic patterns in which some areas in the AS are more congested than others. Another possible objective is minimizing the variance of the loads on the network links. However, this objective does not take into accont the actal load on the links. It may therefore yield very long and possibly non-simple rotes in order to ensre that all the links will be eqally tilized. In or model the network is represented by a directed graph. The roters in the network are represented by the vertices of the graph and the links by the edges. The bandwidth of a link is represented by the capacity of the corresponding edge. The sorce and destination of each flow are represented by their edge roters. For every flow there is a traffic demand. We now give a formal definition of the N-hb roting problem. Let G = (V,E) be a directed graph. Each edge, e E, has a capacity (e), where : E R +. Let F V V be a set of flows between pairs of sorce and destination nodes. Each flow f F has a traffic reqirement A Fig. 3. B C D An example of a network topology E T(f), where T : F R +. Let s f and d f denote the sorce and destination of flow f respectively. For each flow f F, find an ordered seqence of N hbs, denoted by h f 1,hf,...,hf N, where hf i V, sch that the packets of f are roted over s f h f 1 hf... hf N df, where a b denotes the shortest path from node a to node b on G, and the maximm relative load imposed on every edge in E is minimized. The relative load on edge e is defined as f e P f T(f) (e), where P f is the path chosen to rote flow f. In Appendix II we prove that the N-hb roting problem is NP-complete. B. On the Approximation Hardness of N-hb One common way to get arond an NP-complete problem is to develop a polynomial time algorithm that finds a nearoptimal soltion for the problem, namely an approximation algorithm. Usally, when the worst-case performance of an approximation algorithm is bonded, the average-case performance is very close to the optimm. Algorithm A is an approximation algorithm for an optimization problem Π if for any inpt I it rns in polynomial time in the length of I and otpts a feasible soltion A(I) for the problem. In the context of N-hb, a feasible soltion is a soltion where the rote between each sorce-destination pair traverses at most N hbs, while the rote between two consective hbs is the shortest path between them. An algorithm A for a minimization problem (like N-hb, where we seek to minimize the maximm load) is said to have an approximation ratio of ρ, if for any inpt I, vale(a(i))/vale(opt(i)) ρ. An algorithm A for a minimization problem is an approximation scheme for Π if it takes as an inpt not only the instance I of the problem, bt also a vale ǫ > 0 sch that for any fixed ǫ, vale(a(i))/vale(opt(i)) 1 + ǫ. An approximation scheme A is said to be a polynomial time approximation scheme (PTAS) [6] if for each fixed ǫ there is a polynomial approximation algorithm derived from A with an approximation ratio of 1 + ǫ. If the rnning time of the approximation algorithm is also polynomial in the vale of 1/ǫ, then A is said to be a flly polynomial approximation scheme (FPTAS) [6]. It can be easily shown that there is no FPTAS for N-hb nless P=N P. However, in what follows we show a stronger

5 5 inapproximability reslt. Definition 1: Let Π be a minimization problem. The decision problem Π K is the problem of deciding for a given instance I whether the optimm vale of Π K (I) K. Corollary 1: Unless P=N P, N-hb does not permit a PTAS and cannot be approximated within ǫ for ǫ > 0. Proof: Let Π be an integer minimization problem. Sppose that the decision problem Π K is NP-hard for some constant K. Then, from [6] we know that nless P=N P, there is no PTAS for Π and there is no polynomial algorithm with an approximation ratio that is strictly less than 1+1/K. Consider the Integer N-hb problem defined earlier. Obviosly, in a feasible soltion of this problem the maximm load has an integer vale eqal to 1 or more. However, by Corollary 3 (in Appendix II), the problem of deciding whether the optimm vale of Integer N-hb is eqal to 1 is also NP-hard. Hence, Integer N-hb, and sbseqently N-hb, does not permit a PTAS and cannot be approximated within ǫ for any ǫ > 0. Appendix III presents an asymptotic PTAS for the problem. The algorithm gives an approximation factor that decreases as the congestion in the network increases. IV. ONLINE APPROXIMATION ALGORITHMS We now consider the more practical online version of N- hb, where roting decisions for the flows are performed one at a time withot prior knowledge of ftre flows. We consider three online approximation algorithms, originally developed for the nsplittable mlticommodity flow problem [6]. We slightly modify these algorithms in order to apply them to the N-hb Shortest-Path roting problem. We prove that their competitive ratios for the nsplittable mlticommodity flow problem is the same as for the N-hb Shortest-Path roting problem. The competitive ratio of an online algorithm is defined as the worst case ratio, over all seqences of flows, between the vale of the soltion fond by the algorithm and the vale of the soltion fond by an optimal offline algorithm. See [7] for frther details. For the sake of completeness we give a formal definition of the nsplittable mlticommodity flow problem. Let G = (V,E) be a directed graph. Each edge, e E, has a capacity of (e), where : E R +. Let F V V be an ordered set of flows between pairs of sorce and destination nodes. Each flow f F has a traffic reqirement T(f), where T : F R +. Rote every flow f F, in the order the flows are received, on a single arbitrary rote in G, while minimizing the maximm relative load imposed on every edge. This problem, also known as Roting of Permanent Virtal Circits, is NP-complete. The splittable version of this problem, which allows the traffic of each flow to be split over mltiple rotes, is known to be in P. The only difference between the N-hb problem and the nsplittable mlticommodity flow problem is that in the former the set of possible rotes for each sorce-destination pair is restricted while in the latter it is not. Hence, the nsplittable mlticommodity flow problem can be viewed as a V -hb roting problem. Corollary : The best competitive ratio that can be achieved by an online algorithm for N-hb has a lower bond of Ω(log V ). Proof: In [3] this lower bond is proven for the nsplittable mlticommodity flow problem. In this proof a specific network and a specific seqence of flows are considered. For this specific instance, the maximm load imposed on an edge by an offline algorithm is 1, whereas the maximm load log V imposed by an online algorithm is at least. Since all the rotes in the considered network have a length of at most three edges, each of them can be represented as a 1-hb rote. Hence, this proof is also valid for the 1-hb problem, and for the general N-hb problem, as well. However, if we consider a more practical variant of the online version, where termination of flows is permitted, i.e., the lifetime of each flow is finite, we can show that no roting algorithm can do better or worse than a competitive ratio of Θ( E ). Theorem 1: For the online version of N-hb, when flow termination is allowed, the competitive ratio that can be achieved by any algorithm is Θ( E ). Proof: Let s consider a directed graph that has a single sorce s, connected to a single target t via n directed edges, each with capacity C. We constrct a seqence of n flow reqests, each with a traffic demand T. After all the n flows are roted, the maximm load in the network is m T C where m n. Let e be the edge with the maximm load. We now terminate all the flows that do not pass throgh e and some (m n) flows that do pass throgh e. The maximm load in the network now is n T C. The optimal offline algorithm in this sitation can maintain a maximm load of T C by roting each of the n remaining flows on a separate edge. Hence, the best competitive ratio a roting algorithm can achieve is at least Ω( E ). We now show that the worst competitive ratio any roting algorithm can achieve is O( E ). Consider a graph with E edges, each with capacity C, and a seqence of n flow reqests with traffic demand T. The maximm load that can be prodced by an online algorithm in the worst case is nt C. The maximm load that can be prodced by the optimal offline nt E C algorithm is at least. Hence, the worst competitive ratio that can be obtained is O( E ). Using Theorem, this reslt can be extended to networks with non-niform edge capacities. From Theorem 1 it follows that not mch can be done if we want to garantee some competitive ratio when flow termination is considered. However, from Corollary it follows that when flow termination is not considered, it is possible to meet the challenge of designing an algorithm that has a competitive ratio of O(log V ). In what follows we present some online algorithms for the problem. These algorithms are similar in strctre, as follows. Let f be a new flow to be roted. Let T f be the bandwidth demand of f. Let L e and U e be the crrent load and capacity of link e E, respectively. From all feasible N-hb rotes, the algorithm chooses the one that satisfies a given criterion as follows:

6 6 Algorithm-1: minimize e P a Le+T f /Ue Λ a Le Λ, where a (1,) and Λ is as explained below Algorithm-: minimize Algorithm-3: minimize { Le MAX e E L e + T f U e MAX e P L e + T f U e. e / P e P In all cases, P denotes a possible path for the considered flow. These algorithms were presented in [3] (Algorithm-1) and in [4] (Algorithm- and Algorithm-3) for the nsplittable mlticommodity flow problem, and are applied in this paper for the N-hb roting problem. We next show that the competitive ratios of the above algorithms for the N-hb problem are the same as for the nsplittable mlticommodity flow problem, and that Algorithm-1 is the best online algorithm for the N-hb problem. In Algorithm-1, Λ is an estimate for the vale of the optimal soltion. A simple dobling techniqe is sed to estimate its vale. The algorithm starts with some initial estimate. If, dring exection, the maximm load exceeds Λ by log( V ), the estimate is dobled and the algorithm is reinvoked. The algorithm assigns to each edge a weight that increases exponentially in the load that will be imposed if this edge is part of the rote selected for the considered flow. The algorithm chooses from all possible rotes for the considered flow the one with the minimm weight. A rote s weight is the sm of the weights of all its edges. The intition behind the exponential fnction weight is that as the load on an edge increases, the weight of the edge increases exponentially. Conseqently, the algorithm prefers a long non-congested rote over an exponentially shorter, bt congested, rote. The algorithm achieves a competitive ratio of O(log V ) for the nsplittable mlticommodity flow problem. To prove this, [3] ses the following axiliary potential fnction: Φ(j) = e E a Le(j) Λ (γ L e (j)/λ), (1) where L e (j) and L e(j) are the load imposed on edge e by Algorithm-1 and by an optimal offline algorithm, respectively, after the first j flows are roted, and a = 1 + 1/γ. Fnction Φ(j) is non-increasing in j since the weight of the rote chosen by the algorithm for every flow is not greater than the weight of the rote chosen by an optimal offline algorithm. Since Φ(0) γ E and L e(j)/λ 1, γ E e E (γ 1)a Le(j) Λ holds, and the competitive ratio follows. For the N- hb problem, the weight of the rote chosen by Algorithm-1 is still not greater than the weight of the rote chosen by an optimal offline algorithm. This implies that the potential fnction in Eq. 1 is also non-increasing in j. Hence, the competitive ratio of O(log( V )) holds for N-hb as well. Algorithm- ses a simple greedy approach. It chooses a rote sch that the maximm load imposed on any edge is minimized after the flow is roted. When all edge capacities are eqal, this algorithm has a competitive ratio of O( D E ), where D is the maximm ratio, over all flows, between the length of the longest and shortest rotes that can be assigned to the flow. We now show that this competitive ratio is also valid for N-hb (when all the edges have eqal capacities). In [4], where this competitive ratio is proven for the nsplittable mlticommodity flow problem, the vales of the loads are divided into levels. The load L e on edge e is said to be in the i th level if i T max /w L e (i + 1) T max /w, where T max is the maximm bandwidth reqirement and w is the capacity of the edges. The level of rote P is the maximm level over all the edges in P. The crx of the proof is that when the maximm load in the network moves p to level i, then all the edges in the network, inclding the edges of the rote chosen by the optimal offline algorithm, are at least in level i 1. Since this claim is also valid for N-hb, the competitive ratio is valid for N-hb as well. Theorem, which will be presented later in this section, shows how to adapt this competitive ratio to the general case where the edge capacities are not necessarily eqal. Algorithm-3 always chooses the rote with the minimm load. The load of a rote is defined as the maximm load over all the rote s edges. The basic idea is to make the rote selection criterion stricter than in Algorithm-. To nderstand the difference between the two criteria, consider a network with two nodes connected by three edges with eqal capacities. Sppose that the loads imposed on these edges by existing flows are 1, 4 and 6. Sppose also that the next flow to be roted has a bandwidth demand of. Algorithm- may rote this flow either on the first edge or on the second edge, becase in both cases the maximm load remains 6. In contrast, Algorithm-3 chooses the first edge becase it is the least loaded. This implies that every rote chosen by Algorithm- 3 is also a valid choice for Algorithm-, bt not vice versa. In order to increase the attractiveness of Algorithm- over that of Algorithm-3, we have modified it in the following way. When Algorithm- finds several rotes that do not increase the maximm load imposed on any edge, it does not choose one arbitrarily, as proposed in [4], bt chooses the shortest one. When Algorithm-3 is employed in networks with eqal capacities, it has a competitive ratio of O(d log V ), where d is the longest rote that can be assigned to a flow. For reasons similar to those stated earlier for Algorithm-, and others omitted here for lack of space, the same competitive ratio is garanteed when Algorithm-3 is sed for N-hb. Once again, we can se Theorem to extend this competitive ratio to the case where edge capacities are not necessarily eqal. Theorem : Let A(I) be an online algorithm for N-hb that achieves a competitive ratio of C in networks whose edges have the same capacity. Then, A(I) achieves a competitive ratio of max min C in networks whose minimm edge capacity and maximm edge capacity are min and max respectively. Proof: Let G = (V, E) represent a network, and let : E R + be the edge capacity fnction. Let OPT be the vale of an optimal offline soltion. Let G represent another network with the same strctre bt with a different

7 7 edge capacity fnction, sch that for every edge e (e) = (e)/α. Let OPT be the vale of an optimal offline soltion for G. We first prove that OPT = αopt. () Assme that OPT /α < OPT. Let S be the soltion corresponding to OPT. Since the capacity of each edge in G is α times larger than the corresponding edge in G, applying the soltion S to the original graph G wold yield a maximm load of OPT /α. This maximm load is strictly lower than OPT, in contradiction to or assmption. A similar contradiction applies when OPT /α > OPT. Let G min be a graph similar to G whose edge capacities are eqal to min. Let A min and OPT min be the vales of the soltions fond by the online algorithm and the optimal offline algorithm, respectively, for G min. Since the edge capacities do not increase, A A min, where A is the vale of a soltion fond by the online algorithm for G. Since the capacity of each edge in G min is divided by a factor that is not greater than max min, by Eq. we get that OPT min max min OPT. Since A min C OPT min holds, we conclde that A C max min OPT. We now discss the time complexity of the three algorithms. Each algorithm has to review the entire set of N-hb rotes before choosing one. There are O( V N ) sch rotes. In a naive implementation, the algorithm metric is calclated independently for each rote. Assming that the maximm length of the shortest path between two nodes in the graph is D, the longest N-hb rote is D(N+1). Hence, each algorithm has to make O( V N D(N + 1)) metric calclations. When N = 1, and D = O( V ), the time complexity is O( V ). A faster approach for algorithms 1 and 3 is to precalclate the total metric for every possible shortest path between the graph nodes, sing Dijkstra s algorithm, for example. In this case, the time complexity is O( V N (N + 1) + V 3 ), which is smaller than the former time complexity for N >. The dominant operations in the metric calclation of Algorithm 1 are division and exponential comptations for real vales. In contrast, algorithms and 3 reqire only division operations. Hence, Algorithm 1 has a higher time complexity and a longer expected rnning time. V. SIMULATION STUDY In this section we present simlation reslts for the roting algorithms discssed in the previos section. We generated roter-level networks with random capacity edges, sing Waxman s model [8] and the BRITE simlator [9]. We randomly chose a seqence of sorce-destination nodes. Each pair represents a flow to be roted in the network. The seqence of flows was generated sing Zipf. A random network topology and a random seqence of flows form one instance of the N- hb roting problem. Using an event-driven simlator, we find for each instance the maximm load in the network nder the following schemes: 1) The standard shortest-path roting scheme () sed today in IP. This is also known as minimm hop roting. ) The hypothetical optimal roting (OPT) scheme. In this scheme we find a soltion for the splittable mlticommodity flow problem presented in Section IV. Recall that this version of the problem is in P. An algorithm for OPT that is based on linear programming is presented in Appendix I. This scheme allows the traffic of a flow to be split over mltiple rotes. OPT s performance is a theoretical lower bond for N-hb, and therein lies its importance. 3) Algorithm-1, Algorithm- and Algorithm-3, as presented in Section IV. To solve the linear programs for OPT, we sed the Lp Solve software [30]. Throghot the simlation stdy, we assigned a random demand with a fixed average to each flow. Hence, there is a strong correlation between the nmber of flows the roting protocol has to handle and the load imposed on the network. We therefore se the nmber of flows as or offered load metric. Figre 4 depicts simlation reslts of the roting schemes OPT,, and the first online algorithm (Algorithm-1) presented in Section IV. These simlations were carried ot in a medim size backbone network (50 roters). Algorithm-1 is implemented with N = 1. The most important finding in these graphs, and probably in the research so far, is that the performance of 1-hb is very close to that of OPT, and the improvement over is significant. Algorithm-1 redces the maximm load in the network by p to 73%. We also simlated Algorithm- and Algorithm-3 with N = 1. However, the performance of these algorithms is slightly lower than that of Algorithm-1. The inferior performances of Algorithm- and Algorithm-3 can be attribted to the fact that they do not take into accont the length of the chosen rotes. Longer rotes impose, of corse, greater load on the network. We now compare the performance of the varios algorithms in networks with different topologies. Fig. 4(a) shows simlation reslts for backbone networks with low link density ( E / V = ), whereas Fig. 4(b) shows the reslts for backbone networks with higher link density ( E / V = 5). Note that as the link density increases, the nmber of rotes between two nodes also increases. As expected, the maximm load prodced by all the roting schemes decreases as the link density increases. However, while the maximm load prodced by the shortest-path roting decreases on the average by only 5%, the maximm loads prodced by the optimal roting scheme and by Algorithm-1 for 1-hb decrease by 65%. Since the shortest-path roting scheme ses only one path for a sorce-destination pair, the increase in the nmber of rotes between two nodes is insignificant. In contrast, the optimal roting scheme and the 1-hb based roting algorithms can rote different flows of a sorce-destination pair over different rotes, in response to traffic conditions. Note that the ability to se varios rotes for a single sorce-destination pair is especially important for networks with hot-spots. Figre 5 depicts simlation reslts for a small roting domain having V = 10 and E = 0. This roting domain has the same link density as the roting domain corresponding to Figre 4(a). This allows s to compare the performance of

8 8 9 8 Algorithm-1 OPT 9 8 Algorithm-1 OPT Maximm Load 5 4 Maximm Load Offered Load Offered Load (a) V = 50, E = 100 (b) V = 50, E = 50 Fig. 4. Performance of Algorithm-1 for 1-hb for different link densities 9 8 Algorithm-1 OPT 9 8 OPT 1-Hb, -Hb & 3-Hb Maximm Load 5 4 Maximm Load Offered Load Offered Load Fig. 5. Performance of the Algorithm-1 for 1-hb for small network ( V = 10, E = 0) Fig. 6. Performance of Algorithm-1 for 1-hb, -hb and 3-hb ( V = 5 and E = 75) N-hb roting in roting domains with different nmbers of roters. Note first that the increase in the nmber of roters has only a negligible effect on the difference in performance between the 1-hb and the optimal roting schemes. This is despite the fact that the nmber of nrestricted rotes between two end nodes increases exponentially with the nmber of roters, whereas the nmber of 1-hb rotes increases only linearly. One might expect the maximm loads prodced by the varios roting schemes to be higher in small roting domains than in larger ones, and the relative difference in the performances of the N-hb roting scheme and shortestpath roting to decrease. Interestingly, however, the maximm loads prodced are actally smaller than in Figre 4(a) and the relative difference between 1-hb and shortest-path is similar to that of Figre 4(a). This is attribted to the fact that the average nmber of links a flow has to traverse decreases for smaller roting domains. Hence, each flow consmes fewer network resorces, thereby redcing the maximm loads prodced by the varios roting schemes. We now examine the performance of the N-hb roting scheme with different vales for N, i.e., with different nm- bers of possible hbs. Figre 6 depicts simlation reslts of Algorithm-1 for the 1-hb, -hb and 3-hb schemes for a roting domain having V = 50 and E = 50. The most important finding is that the differences in performance for different vales of N are negligible (less than 1%). We therefore se a single crve for N = 1, N = and N = 3. This reslt is attribted to the flexibility of 1-hb roting. Adding flexibility by allowing more hbs to the roting process does not contribte to its effectiveness. However, for mch larger roting domains, we expect a visible performance difference becase the flexibility of -hb and 3-hb roting schemes increases polynomially (by powers of and 3 respectively) with the nmber of roters, whereas the flexibility of a 1-hb scheme increases only linearly. We wanted to investigate not only the specific algorithms proposed in the paper, bt also the pre concept of N-hb roting. To this end, we tested the performance of 1-hb roting with a random algorithm (RAND). This algorithm does not take into accont the aggregated load imposed on every network link or the load imposed by every connection. Rather, it selects a random hb for every new connection. We

9 9 6 5 RAND Algorithm RAND Algorithm Maximm Load 3 Maximm Load Offered Load Offered Load (a) V = 10, E = 30 (b) V = 00, E = 4000 Fig. 7. Performance of a random algorithm for 1-hb for different network sizes compared the performance of RAND with the performance of Shortest-Path () and the performance of Algorithm-1 for different roting domain sizes. Figre 7(a) depicts simlation reslts for a small roting domain with 10 roters and 30 links. It is evident that RAND performs poorly in sch domains, becase the maximm load it imposes is even higher than the maximm load imposed by. This is attribted to the fact that like, RAND selects the rotes withot knowing the distribtion of link loads. Since the rotes selected by RAND are longer than those selected by, the bandwidth consmed by RAND is higher and the maximm load imposed in the roting domain increases. However, this is not the case for a large roting domain. Figre 7(b) depicts the performance of the varios roting schemes for a domain with 00 nodes and 4000 links. The maximm load imposed by RAND is abot 40% lower than the maximm load imposed by. This redction is attribted to the fact that RAND is a symmetrybreaking procedre which better balances an offered load created with a Zipf distribtion. This load redction is not possible in a small roting domain in which the load is approximately niform and there is no advantage in roting throgh distant hbs. To validate or findings regarding the effectiveness of N- hb roting, we have also sed for or simlations an actal I topology as mapped by the RocketFel project [31]. The bandwidth for each link is determined according to [3]. Figre 8 depicts simlation reslts for the Exods I from [31]. It is evident that the reslts are similar to those achieved sing the Waxman model. Again, Alg-1 sing 1-hb roting performs very close to the theoretical optimm and achieves 80% improvement over the reslts achieved by shortest-path roting. We fond similar reslts when implementing other I topologies from [31]. The graphs are omitted de to lack of space. We conclde this section by looking at the problem from a different angle. Figre 9 depicts the maximm nmber of flows the network can accommodate nder each roting algorithm as a fnction of the maximm load that can be imposed on a single link. Instead of roting all the flows and finding the maximm load, we now determine the maximm nmber of Maximm Load Rand Alg-1 OPT Offered Load Fig. 8. Performance on an I (Exods). V = 80, E = 94 flows that can be roted, sbject to a maximm load constraint. A flow is rejected if roting it over the chosen rote cases the maximm load in the network to exceed the maximm tolerated load. The simlation stops when the network is satrated. The network is assmed to be satrated when 100 consective flows are rejected. Fig. 9 depicts simlation reslts for networks with V = 50 and E = 50. We can see that the 1-hb version of Algorithm-1 achieves the best reslts: it can accommodate on the average 51% more flows than. Algorithm-3 achieves 48% improvement over, and Algorithm- achieves only 34% improvement. These reslts sggest that althogh the three roting algorithms prodce similar maximm loads, as shown in the previos simlation, the difference in the qality of their roting is distinct. The higher nmber of flows accepted by Algorithm-1 and Algorithm-3 indicates their ability to better balance the load in the network, thereby achieving a higher throghpt. VI. CONCLUSIONS In this paper we stdied the effectiveness of the N-hb Shortest-Path Roting concept in IP networks. We have demonstrated that this concept offers an excellent compromise

10 10 Nmber of Roted Flows Algorithm-1 Algorithm- Algorithm Maximm Tolerated Load Fig. 9. No. of connections vs. max. tolerated load between the simplicity of shortest-path roting and the efficiency of virtal circit roting. We applied this concept to the problem of minimizing the maximm load in the network. We defined the corresponding optimization problem, and proved that it is NP-Complete even for N = 1. We also showed that it does not permit a PTAS and cannot be approximated within ǫ for ǫ > 0. However, we present in Appendix III a probabilistic asymptotic PTAS for the offline version of N-hb. We have addressed the online version of N-hb, where the set of the inpt flows is not known in advance. We showed that the best competitive ratio an online N-hb algorithm may achieve is Ω(log V ). We then presented an online algorithm that achieves this lower bond, and two additional online algorithms that have less attractive competitive ratios, bt are also less comptationally intensive. We then sed simlations to stdy the practical effectiveness of N-hb roting in general, and of the specific algorithms presented in the paper. Or main findings are as follows: The performance of N-hb Shortest-Path Roting is very close to the performance of a hypothetical optimal algorithm that splits the traffic of the same flow among mltiple rotes. The N-hb Shortest-Path Roting scheme can prodce mch better qality roting than shortest-path roting, withot the need to incorporate complicated logic into the roting process or even make the effort to learn the link load distribtion throghot the roting domain. The effect of N on the performance of N-hb is very small. Hence, even the performance of 1-hb is very close to optimal. Althogh the competitive ratio of an online algorithm is Ω(log V ), all three online algorithms proposed in this paper perform very well in practice. We therefore conclde that N-hb Shortest-Path Roting, and in particlar the N = 1 version, shold be considered as a powerfl mechanism for ftre datagram roting in the Internet. APPENDIX I A LINEAR PROGRAM FOR THE GENERAL ROUTING PROBLEM We describe a general roting problem, expressed in the form of a linear program, for the optimal roting scheme discssed in Section V. Let G = (V,E) be a directed graph representing the network. Let F be a set of flows in the network. Let T f denote the bandwidth demand of flow f, and let s f and d f denote the sorce and destination of flow f respectively. For every flow f and link e, let l f e represent the traffic load imposed on link e de to flow f. The linear program is as follows: Minimize sbject to the following constraints: (a) e E l f v in e T f if v = f d e E l f v ot e = T f if v = f s 0 otherwise v V and f F where Ev in and Ev ot are the sets of incoming and otgoing links of vertex v respectively, and (b) le f L e E. f F The first constraint ensres that the traffic flow is conserved in each vertex and it is roted from its sorce to its destination. The second constraint ensres that the load on each link does not exceed L. APPENDIX II AN NP-COMPLETENESS PROOF The 1-hb roting problem is a special case of N-hb roting. In what follows we formlate the 1-hb problem with niform capacities as a decision problem and prove that this problem is NP-complete. It is easy to see that if 1-hb with niform capacities is NP-complete, then the more general N-hb problem with arbitrary capacities is NP-complete as well. An instance for the 1-hb problem is a directed graph G = (V,E), a set F V V of flows, a fnction T of bandwidth demand for each flow, and a positive real K. The qestion is whether there exists a hb h f V for each flow f F sch that if the reqired traffic volme for f, namely T(f), is roted over s f h f d f, the total traffic roted throgh every link e E does not exceed K. Theorem 3: 1-hb is NP-complete. Proof: It is easy to see that 1-hb N P. To prove that 1-hb is NP-complete we will show a redction from SAT to 1-hb. Consider the following instance for SAT. Let U = { 1,,..., N } be a set of variables and C = {c 1,c,...,c L } a set of clases. A valid hb assignment for the 1-hb problem is an assignment that does not impose a traffic volme greater than K on any edge. We shall now transform the instance for SAT into an instance for the 1- hb problem sch that a valid hb assignment for the 1-hb problem exists if and only if C is satisfiable. L