Routing is the task of finding a path from a sender to a desired destination. In the IP "Catenet model" this reduces primarily to a matter of finding gateways between networks. As long as a message remains on a single network or subnet, any routing problems are solved by technology that is specific to the network. For example, the Ethernet and the ARPANET each define a way in which any sender can talk to any specified destination within that one network. IP routing comes in primarily when messages must go from a sender on one such network to a destination on a different one. In that case, the message must pass through gateways connecting the networks. If the networks are not adjacent, the message may pass through several intervening networks, and the gateways connecting them. Once the message gets to a gateway that is on the same network as the destination, that network's own technology is used to get to the destination.
Throughout this section, the term "network" is used generically to cover a single broadcast network (e.g., an Ethernet), a point to point line, or the ARPANET. The critical point is that a network is treated as a single entity by IP. Either no routing is necessary (as with a point to point line), or that routing is done in a manner that is transparent to IP, allowing IP to treat the entire network as a single fully-connected system (as with an Ethernet or the ARPANET). Note that the term "network" is used in a somewhat different way in discussions of IP addressing. A single IP network number may be assigned to a collection of networks, with "subnet" addressing being used to describe the individual networks. In effect, we are using the term "network" here to refer to subnets in cases where subnet addressing is in use.
A number of different approaches for finding routes between networks are possible. One useful way of categorizing these approaches is on the basis of the type of information the gateways need to exchange in order to be able to find routes. Distance vector algorithms are based on the exchange of only a small amount of information. Each entity (gateway or host) that participates in the routing protocol is assumed to keep information about all of the destinations within the system. Generally, information about all entities connected to one network is summarized by a single entry, which describes the route to all destinations on that network. This summarization is possible because as far as IP is concerned, routing within a network is invisible. Each entry in this routing database includes the next gateway to which datagrams destined for the entity should be sent. In addition, it includes a "metric" measuring the total distance to the entity. Distance is a somewhat generalized concept, which may cover the time delay in getting messages to the entity, the dollar cost of sending messages to it, etc. Distance vector algorithms get their name from the fact that it is possible to compute optimal routes when the only information exchanged is the list of these distances. Furthermore, information is only exchanged among entities that are adjacent, that is, entities that share a common network.
Although routing is most commonly based on information about networks, it is sometimes necessary to keep track of the routes to individual hosts. The RIP protocol makes no formal distinction between networks and hosts. It simply describes exchange of information about destinations, which may be either networks or hosts. (Note however, that it is possible for an implementor to choose not to support host routes. See section 3.2.) In fact, the mathematical developments are most conveniently thought of in terms of routes from one host or gateway to another. When discussing the algorithm in abstract terms, it is best to think of a routing entry for a network as an abbreviation for routing entries for all of the entities connected to that network. This sort of abbreviation makes sense only because we think of networks as having no internal structure that is visible at the IP level. Thus, we will generally assign the same distance to every entity in a given network.
We said above that each entity keeps a routing database with one entry for every possible destination in the system. An actual implementation is likely to need to keep the following information about each destination:
In addition, various flags and other internal information will probably be included. This database is initialized with a description of the entities that are directly connected to the system. It is updated according to information received in messages from neighboring gateways.
The most important information exchanged by the hosts and gateways is that carried in update messages. Each entity that participates in the routing scheme sends update messages that describe the routing database as it currently exists in that entity. It is possible to maintain optimal routes for the entire system by using only information obtained from neighboring entities. The algorithm used for that will be described in the next section.
As we mentioned above, the purpose of routing is to find a way to get datagrams to their ultimate destinations. Distance vector algorithms are based on a table giving the best route to every destination in the system. Of course, in order to define which route is best, we have to have some way of measuring goodness. This is referred to as the "metric".
In simple networks, it is common to use a metric that simply counts how many gateways a message must go through. In more complex networks, a metric is chosen to represent the total amount of delay that the message suffers, the cost of sending it, or some other quantity which may be minimized. The main requirement is that it must be possible to represent the metric as a sum of "costs" for individual hops.
Formally, if it is possible to get from entity i to entity j directly (i.e., without passing through another gateway between), then a cost, d(i,j), is associated with the hop between i and j. In the normal case where all entities on a given network are considered to be the same, d(i,j) is the same for all destinations on a given network, and represents the cost of using that network. To get the metric of a complete route, one just adds up the costs of the individual hops that make up the route. For the purposes of this memo, we assume that the costs are positive integers.
Let D(i,j) represent the metric of the best route from entity i to entity j. It should be defined for every pair of entities. d(i,j) represents the costs of the individual steps. Formally, let d(i,j) represent the cost of going directly from entity i to entity j. It is infinite if i and j are not immediate neighbors. (Note that d(i,i) is infinite. That is, we don't consider there to be a direct connection from a node to itself.) Since costs are additive, it is easy to show that the best metric must be described by
D(i,i) = 0, all i D(i,j) = min [d(i,k) + D(k,j)], otherwise k
and that the best routes start by going from i to those neighbors k for which d(i,k) + D(k,j) has the minimum value. (These things can be shown by induction on the number of steps in the routes.) Note that we can limit the second equation to k's that are immediate neighbors of i. For the others, d(i,k) is infinite, so the term involving them can never be the minimum.
It turns out that one can compute the metric by a simple algorithm based on this. Entity i gets its neighbors k to send it their estimates of their distances to the destination j. When i gets the estimates from k, it adds d(i,k) to each of the numbers. This is simply the cost of traversing the network between i and k. Now and then i compares the values from all of its neighbors and picks the smallest.
A proof is given in  that this algorithm will converge to the correct estimates of D(i,j) in finite time in the absence of topology changes. The authors make very few assumptions about the order in which the entities send each other their information, or when the min is recomputed. Basically, entities just can't stop sending updates or recomputing metrics, and the networks can't delay messages forever. (Crash of a routing entity is a topology change.) Also, their proof does not make any assumptions about the initial estimates of D(i,j), except that they must be non-negative. The fact that these fairly weak assumptions are good enough is important. Because we don't have to make assumptions about when updates are sent, it is safe to run the algorithm asynchronously. That is, each entity can send updates according to its own clock. Updates can be dropped by the network, as long as they don't all get dropped. Because we don't have to make assumptions about the starting condition, the algorithm can handle changes. When the system changes, the routing algorithm starts moving to a new equilibrium, using the old one as its starting point. It is important that the algorithm will converge in finite time no matter what the starting point. Otherwise certain kinds of changes might lead to non-convergent behavior.
The statement of the algorithm given above (and the proof) assumes that each entity keeps copies of the estimates that come from each of its neighbors, and now and then does a min over all of the neighbors. In fact real implementations don't necessarily do that. They simply remember the best metric seen so far, and the identity of the neighbor that sent it. They replace this information whenever they see a better (smaller) metric. This allows them to compute the minimum incrementally, without having to store data from all of the neighbors.
There is one other difference between the algorithm as described in texts and those used in real protocols such as RIP: the description above would have each entity include an entry for itself, showing a distance of zero. In fact this is not generally done. Recall that all entities on a network are normally summarized by a single entry for the network. Consider the situation of a host or gateway G that is connected to network A. C represents the cost of using network A (usually a metric of one). (Recall that we are assuming that the internal structure of a network is not visible to IP, and thus the cost of going between any two entities on it is the same.) In principle, G should get a message from every other entity H on network A, showing a cost of 0 to get from that entity to itself. G would then compute C + 0 as the distance to H. Rather than having G look at all of these identical messages, it simply starts out by making an entry for network A in its table, and assigning it a metric of C. This entry for network A should be thought of as summarizing the entries for all other entities on network A. The only entity on A that can't be summarized by that common entry is G itself, since the cost of going from G to G is 0, not C. But since we never need those 0 entries, we can safely get along with just the single entry for network A. Note one other implication of this strategy: because we don't need to use the 0 entries for anything, hosts that do not function as gateways don't need to send any update messages. Clearly hosts that don't function as gateways (i.e., hosts that are connected to only one network) can have no useful information to contribute other than their own entry D(i,i) = 0. As they have only the one interface, it is easy to see that a route to any other network through them will simply go in that interface and then come right back out it. Thus the cost of such a route will be greater than the best cost by at least C. Since we don't need the 0 entries, non- gateways need not participate in the routing protocol at all.
Let us summarize what a host or gateway G does. For each destination in the system, G will keep a current estimate of the metric for that destination (i.e., the total cost of getting to it) and the identity of the neighboring gateway on whose data that metric is based. If the destination is on a network that is directly connected to G, then G simply uses an entry that shows the cost of using the network, and the fact that no gateway is needed to get to the destination. It is easy to show that once the computation has converged to the correct metrics, the neighbor that is recorded by this technique is in fact the first gateway on the path to the destination. (If there are several equally good paths, it is the first gateway on one of them.) This combination of destination, metric, and gateway is typically referred to as a route to the destination with that metric, using that gateway.
The method so far only has a way to lower the metric, as the existing metric is kept until a smaller one shows up. It is possible that the initial estimate might be too low. Thus, there must be a way to increase the metric. It turns out to be sufficient to use the following rule: suppose the current route to a destination has metric D and uses gateway G. If a new set of information arrived from some source other than G, only update the route if the new metric is better than D. But if a new set of information arrives from G itself, always update D to the new value. It is easy to show that with this rule, the incremental update process produces the same routes as a calculation that remembers the latest information from all the neighbors and does an explicit minimum. (Note that the discussion so far assumes that the network configuration is static. It does not allow for the possibility that a system might fail.)
To summarize, here is the basic distance vector algorithm as it has been developed so far. (Note that this is not a statement of the RIP protocol. There are several refinements still to be added.) The following procedure is carried out by every entity that participates in the routing protocol. This must include all of the gateways in the system. Hosts that are not gateways may participate as well.