4.5 Routing Algorithms

Transcription

1 4.5 ROUTING ALGORITHMS 363 to hosts enjo the securit services provided b IPsec. On the sending side, the transport laer passes a segment to IPsec. IPsec then encrpts the segment, appends additional securit fields to the segment, and encapsulates the resulting paload in an ordinar IP datagram. (It s actuall a little more complicated than this, as e ll see in Chapter 8.) The sending host then sends the datagram into the Internet, hich transports it to the destination host. There, IPsec decrpts the segment and passes the unencrpted segment to the transport laer. The services provided b an IPsec session include: Crptographic agreement. Mechanisms that allo the to communicating hosts to agree on crptographic algorithms and kes. Encrption of IP datagram paloads. When the sending host receives a segment from the transport laer, IPsec encrpts the paload. The paload can onl be decrpted b IPsec in the receiving host. Data integrit. IPsec allos the receiving host to verif that the datagram s header fields and encrpted paload ere not modified hile the datagram as en route from source to destination. Origin authentication. When a host receives an IPsec datagram from a trusted source (ith a trusted ke see Chapter 8), the host is assured that the source IP address in the datagram is the actual source of the datagram. When to hosts have an IPsec session established beteen them, all TCP and UDP segments sent beteen them ill be encrpted and authenticated. IPsec therefore provides blanket coverage, securing all communication beteen the to hosts for all netork applications. A compan can use IPsec to communicate securel in the nonsecure public Internet. For illustrative purposes, e ll just look at a simple example here. Consider a compan that has a large number of traveling salespeople, each possessing a compan laptop computer. Suppose the salespeople need to frequentl consult sensitive compan information (for example, pricing and product information) that is stored on a server in the compan s headquarters. Further suppose that the salespeople also need to send sensitive documents to each other. Ho can this be done ith IPsec? As ou might guess, e install IPsec in the server and in all of the salespeople s laptops. With IPsec installed in these hosts, henever a salesperson needs to communicate ith the server or ith another salesperson, the communication session ill be secure. 4.5 Routing Algorithms So far in this chapter, e ve mostl explored the netork laer s forarding function. We learned that hen a packet arrives to a router, the router indexes a forarding table and determines the link interface to hich the packet is to be directed. We also learned that routing algorithms, operating in netork routers, exchange and

2 364 CHAPTER 4 THE NETWORK LAYER compute the information that is used to configure these forarding tables. The interpla beteen routing algorithms and forarding tables as shon in Figure 4.2. Having explored forarding in some depth e no turn our attention to the other major topic of this chapter, namel, the netork laer s critical routing function. Whether the netork laer provides a datagram service (in hich case different packets beteen a given source-destination pair ma take different routes) or a VC service (in hich case all packets beteen a given source and destination ill take the same path), the netork laer must nonetheless determine the path that packets take from senders to receivers. We ll see that the job of routing is to determine good paths (equivalentl, routes), from senders to receivers, through the netork of routers. Tpicall a host is attached directl to one router, the default router for the host (also called the first-hop router for the host). Whenever a host sends a packet, the packet is transferred to its default router. We refer to the default router of the source host as the source router and the default router of the destination host as the destination router. The problem of routing a packet from source host to destination host clearl boils don to the problem of routing the packet from source router to destination router, hich is the focus of this section. The purpose of a routing algorithm is then simple: given a set of routers, ith links connecting the routers, a routing algorithm finds a good path from source router to destination router. Tpicall, a good path is one that has the least cost. We ll see, hoever, that in practice, real-orld concerns such as polic issues (for example, a rule such as router x, belonging to organization Y, should not forard an packets originating from the netork oned b organization Z ) also come into pla to complicate the conceptuall simple and elegant algorithms hose theor underlies the practice of routing in toda s netorks. A graph is used to formulate routing problems. Recall that a graph G = (N,E) is a set N of nodes and a collection E of edges, here each edge is a pair of nodes from N. In the context of netork-laer routing, the nodes in the graph represent routers the points at hich packet-forarding decisions are made and the edges connecting these nodes represent the phsical links beteen these routers. Such a graph abstraction of a computer netork is shon in Figure To vie some graphs representing real netork maps, see [Dodge 202, Chesick 2000]; for a discussion of ho ell different graph-based models model the Internet, see [Zegura 997, Faloutsos 999, Li 2004]. As shon in Figure 4.27, an edge also has a value representing its cost. Tpicall, an edge s cost ma reflect the phsical length of the corresponding link (for example, a transoceanic link might have a higher cost than a short-haul terrestrial link), the link speed, or the monetar cost associated ith a link. For our purposes, e ll simpl take the edge costs as a given and on t orr about ho the are determined. For an edge (x,) in E, e denote c(x,) as the cost of the edge beteen nodes x and. If the pair (x,) does not belong to E, e set c(x,) =. Also, throughout e consider onl undirected graphs (i.e., graphs hose edges do not have a direction), so that edge (x,) is the same as edge (,x) and that c(x,) = c(,x). Also, a node is said to be a neighbor of node x if (x,) belongs to E.

3 4.5 ROUTING ALGORITHMS u 3 v x z Figure 4.27 Abstract graph model of a computer netork Given that costs are assigned to the various edges in the graph abstraction, a natural goal of a routing algorithm is to identif the least costl paths beteen sources and destinations. To make this problem more precise, recall that a path in a graph G = (N,E) is a sequence of nodes (x, x 2,..., x p ) such that each of the pairs (x,x 2 ), (x 2,x 3 ),...,(x p-,x p ) are edges in E. The cost of a path (x,x 2,..., x p ) is simpl the sum of all the edge costs along the path, that is, c(x,x 2 ) + c(x 2,x 3 ) c(x p-,x p ). Given an to nodes x and, there are tpicall man paths beteen the to nodes, ith each path having a cost. One or more of these paths is a least-cost path. The least-cost problem is therefore clear: Find a path beteen the source and destination that has least cost. In Figure 4.27, for example, the least-cost path beteen source node u and destination node is (u, x,, ) ith a path cost of 3. Note that if all edges in the graph have the same cost, the least-cost path is also the shortest path (that is, the path ith the smallest number of links beteen the source and the destination). As a simple exercise, tr finding the least-cost path from node u to z in Figure 4.27 and reflect for a moment on ho ou calculated that path. If ou are like most people, ou found the path from u to z b examining Figure 4.27, tracing a fe routes from u to z, and someho convincing ourself that the path ou had chosen had the least cost among all possible paths. (Did ou check all of the 7 possible paths beteen u and z? Probabl not!) Such a calculation is an example of a centralized routing algorithm the routing algorithm as run in one location, our brain, ith complete information about the netork. Broadl, one a in hich e can classif routing algorithms is according to hether the are global or decentralized. A global routing algorithm computes the least-cost path beteen a source and destination using complete, global knoledge about the netork. That is, the algorithm takes the connectivit beteen all nodes and all link costs as inputs. This then requires that the algorithm someho obtain this information before actuall performing the calculation. The calculation itself can be run at one site

4 366 CHAPTER 4 THE NETWORK LAYER (a centralized global routing algorithm) or replicated at multiple sites. The ke distinguishing feature here, hoever, is that a global algorithm has complete information about connectivit and link costs. In practice, algorithms ith global state information are often referred to as link-state (LS) algorithms, since the algorithm must be aare of the cost of each link in the netork. We ll stud LS algorithms in Section In a decentralized routing algorithm, the calculation of the least-cost path is carried out in an iterative, distributed manner. No node has complete information about the costs of all netork links. Instead, each node begins ith onl the knoledge of the costs of its on directl attached links. Then, through an iterative process of calculation and exchange of information ith its neighboring nodes (that is, nodes that are at the other end of links to hich it itself is attached), a node graduall calculates the least-cost path to a destination or set of destinations. The decentralized routing algorithm e ll stud belo in Section is called a distance-vector (DV) algorithm, because each node maintains a vector of estimates of the costs (distances) to all other nodes in the netork. A second broad a to classif routing algorithms is according to hether the are static or dnamic. In static routing algorithms, routes change ver slol over time, often as a result of human intervention (for example, a human manuall editing a router s forarding table). Dnamic routing algorithms change the routing paths as the netork traffic loads or topolog change. A dnamic algorithm can be run either periodicall or in direct response to topolog or link cost changes. While dnamic algorithms are more responsive to netork changes, the are also more susceptible to problems such as routing loops and oscillation in routes. A third a to classif routing algorithms is according to hether the are loadsensitive or load-insensitive. In a load-sensitive algorithm, link costs var dnamicall to reflect the current level of congestion in the underling link. If a high cost is associated ith a link that is currentl congested, a routing algorithm ill tend to choose routes around such a congested link. While earl ARPAnet routing algorithms ere load-sensitive [McQuillan 980], a number of difficulties ere encountered [Huitema 998]. Toda s Internet routing algorithms (such as RIP, OSPF, and BGP) are load-insensitive, as a link s cost does not explicitl reflect its current (or recent past) level of congestion The Link-State (LS) Routing Algorithm Recall that in a link-state algorithm, the netork topolog and all link costs are knon, that is, available as input to the LS algorithm. In practice this is accomplished b having each node broadcast link-state packets to all other nodes in the netork, ith each link-state packet containing the identities and costs of its attached links. In practice (for example, ith the Internet s OSPF routing protocol, discussed in Section 4.6.) this is often accomplished b a link-state broadcast

5 4.5 ROUTING ALGORITHMS 367 algorithm [Perlman 999]. We ll cover broadcast algorithms in Section 4.7. The result of the nodes broadcast is that all nodes have an identical and complete vie of the netork. Each node can then run the LS algorithm and compute the same set of least-cost paths as ever other node. The link-state routing algorithm e present belo is knon as Dijkstra s algorithm, named after its inventor. A closel related algorithm is Prim s algorithm; see [Cormen 200] for a general discussion of graph algorithms. Dijkstra s algorithm computes the least-cost path from one node (the source, hich e ill refer to as u) to all other nodes in the netork. Dijkstra s algorithm is iterative and has the propert that after the kth iteration of the algorithm, the least-cost paths are knon to k destination nodes, and among the least-cost paths to all destination nodes, these k paths ill have the k smallest costs. Let us define the folloing notation: D(v): cost of the least-cost path from the source node to destination v as of this iteration of the algorithm. p(v): previous node (neighbor of v) along the current least-cost path from the source to v. N : subset of nodes; v is in N if the least-cost path from the source to v is definitivel knon. The global routing algorithm consists of an initialization step folloed b a loop. The number of times the loop is executed is equal to the number of nodes in the netork. Upon termination, the algorithm ill have calculated the shortest paths from the source node u to ever other node in the netork. Link-State (LS) Algorithm for Source Node u Initialization: 2 N = {u} 3 for all nodes v 4 if v is a neighbor of u 5 then D(v) = c(u,v) 6 else D(v) = 7 8 Loop 9 find not in N such that D() is a minimum 0 add to N update D(v) for each neighbor v of and not in N : 2 D(v) = min( D(v), D() + c(,v) ) 3 /* ne cost to v is either old cost to v or knon 4 least path cost to plus cost from to v */ 5 until N = N

6 368 CHAPTER 4 THE NETWORK LAYER VideoNote Dijkstra s algorithm: discussion and example As an example, let s consider the netork in Figure 4.27 and compute the least-cost paths from u to all possible destinations. A tabular summar of the algorithm s computation is shon in Table 4.3, here each line in the table gives the values of the algorithm s variables at the end of the iteration. Let s consider the fe first steps in detail. In the initialization step, the currentl knon least-cost paths from u to its directl attached neighbors, v, x, and, are initialized to 2,, and 5, respectivel. Note in particular that the cost to is set to 5 (even though e ill soon see that a lesser-cost path does indeed exist) since this is the cost of the direct (one hop) link from u to. The costs to and z are set to infinit because the are not directl connected to u. In the first iteration, e look among those nodes not et added to the set N and find that node ith the least cost as of the end of the previous iteration. That node is x, ith a cost of, and thus x is added to the set N. Line 2 of the LS algorithm is then performed to update D(v) for all nodes v, ielding the results shon in the second line (Step ) in Table 4.3. The cost of the path to v is unchanged. The cost of the path to (hich as 5 at the end of the initialization) through node x is found to have a cost of 4. Hence this loer-cost path is selected and s predecessor along the shortest path from u is set to x. Similarl, the cost to (through x) is computed to be 2, and the table is updated accordingl. In the second iteration, nodes v and are found to have the least-cost paths (2), and e break the tie arbitraril and add to the set N so that N no contains u, x, and. The cost to the remaining nodes not et in N, that is, nodes v,, and z, are updated via line 2 of the LS algorithm, ielding the results shon in the third ro in the Table 4.3. And so on.... When the LS algorithm terminates, e have, for each node, its predecessor along the least-cost path from the source node. For each predecessor, e also step N D(v),p(v) D(),p() D(x),p(x) D(),p() D(z),p(z) 0 u 2,u 5,u,u ux 2,u 4,x 2,x 2 ux 2,u 3, 4, 3 uxv 3, 4, 4 uxv 4, 5 uxvz Table 4.3 Running the link-state algorithm on the netork in Figure 4.27

7 4.5 ROUTING ALGORITHMS 369 have its predecessor, and so in this manner e can construct the entire path from the source to all destinations. The forarding table in a node, sa node u, can then be constructed from this information b storing, for each destination, the next-hop node on the least-cost path from u to the destination. Figure 4.28 shos the resulting least-cost paths and forarding table in u for the netork in Figure What is the computational complexit of this algorithm? That is, given n nodes (not counting the source), ho much computation must be done in the orst case to find the least-cost paths from the source to all destinations? In the first iteration, e need to search through all n nodes to determine the node,, not in N that has the minimum cost. In the second iteration, e need to check n nodes to determine the minimum cost; in the third iteration n 2 nodes, and so on. Overall, the total number of nodes e need to search through over all the iterations is n(n + )/2, and thus e sa that the preceding implementation of the LS algorithm has orst-case complexit of order n squared: O(n 2 ). (A more sophisticated implementation of this algorithm, using a data structure knon as a heap, can find the minimum in line 9 in logarithmic rather than linear time, thus reducing the complexit.) Before completing our discussion of the LS algorithm, let us consider a patholog that can arise. Figure 4.29 shos a simple netork topolog here link costs are equal to the load carried on the link, for example, reflecting the dela that ould be experienced. In this example, link costs are not smmetric; that is, c(u,v) equals c(v,u) onl if the load carried on both directions on the link (u,v) is the same. In this example, node z originates a unit of traffic destined for, node x also originates a unit of traffic destined for, and node injects an amount of traffic equal to e, also destined for. The initial routing is shon in Figure 4.29(a) ith the link costs corresponding to the amount of traffic carried. When the LS algorithm is next run, node determines (based on the link costs shon in Figure 4.29(a)) that the clockise path to has a cost of, hile the counterclockise path to (hich it had been using) has a cost of + e. Hence s Destination Link U V X W Y Z v x z (u, v) (u, x) (u, x) (u, x) (u, x) Figure 4.28 Least cost path and forarding table for nodule u

8 370 CHAPTER 4 THE NETWORK LAYER + e 2 + e 0 z e x z x + e 0 0 e e a. Initial routing b. x, detect better path to, clockise 0 z x e c. x,, z detect better path to, counterclockise e 2+ e 2 + e z x + e 0 0 d. x,, z, detect better path to, clockise e 0 Figure 4.29 Oscillations ith congestion-sensitive routing least-cost path to is no clockise. Similarl, x determines that its ne least-cost path to is also clockise, resulting in costs shon in Figure 4.29(b). When the LS algorithm is run next, nodes x,, and z all detect a zero-cost path to in the counterclockise direction, and all route their traffic to the counterclockise routes. The next time the LS algorithm is run, x,, and z all then route their traffic to the clockise routes. What can be done to prevent such oscillations (hich can occur in an algorithm, not just an LS algorithm, that uses a congestion or dela-based link metric)? One solution ould be to mandate that link costs not depend on the amount of traffic carried an unacceptable solution since one goal of routing is to avoid

9 4.5 ROUTING ALGORITHMS 37 highl congested (for example, high-dela) links. Another solution is to ensure that not all routers run the LS algorithm at the same time. This seems a more reasonable solution, since e ould hope that even if routers ran the LS algorithm ith the same periodicit, the execution instance of the algorithm ould not be the same at each node. Interestingl, researchers have found that routers in the Internet can self-snchronize among themselves [Flod Snchronization 994]. That is, even though the initiall execute the algorithm ith the same period but at different instants of time, the algorithm execution instance can eventuall become, and remain, snchronized at the routers. One a to avoid such selfsnchronization is for each router to randomize the time it sends out a link advertisement. Having studied the LS algorithm, let s consider the other major routing algorithm that is used in practice toda the distance-vector routing algorithm The Distance-Vector (DV) Routing Algorithm Whereas the LS algorithm is an algorithm using global information, the distancevector (DV) algorithm is iterative, asnchronous, and distributed. It is distributed in that each node receives some information from one or more of its directl attached neighbors, performs a calculation, and then distributes the results of its calculation back to its neighbors. It is iterative in that this process continues on until no more information is exchanged beteen neighbors. (Interestingl, the algorithm is also self-terminating there is no signal that the computation should stop; it just stops.) The algorithm is asnchronous in that it does not require all of the nodes to operate in lockstep ith each other. We ll see that an asnchronous, iterative, self-terminating, distributed algorithm is much more interesting and fun than a centralized algorithm! Before e present the DV algorithm, it ill prove beneficial to discuss an important relationship that exists among the costs of the least-cost paths. Let d x () be the cost of the least-cost path from node x to node. Then the least costs are related b the celebrated Bellman-Ford equation, namel, d x () = min v {c(x,v) + d v ()}, (4.) here the min v in the equation is taken over all of x s neighbors. The Bellman-Ford equation is rather intuitive. Indeed, after traveling from x to v, if e then take the least-cost path from v to, the path cost ill be c(x,v) + d v (). Since e must begin b traveling to some neighbor v, the least cost from x to is the minimum of c(x,v) + d v () taken over all neighbors v. But for those ho might be skeptical about the validit of the equation, let s check it for source node u and destination node z in Figure The source node u