# 9.4: Observations on Minimizing Route Cost

- Page ID
- 11195

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Does distance-vector routing actually achieve minimum costs? For that matter, does each packet incur the cost its sender expects? Suppose node A has a forwarding entry ⟨D,B,c⟩, meaning that A forwards packets to destination D via next_hop B, and expects the total cost to be c. If A sends a packet to D, and we follow it on the actual path it takes, must the total link cost be c? If so, we will say that the network has **accurate costs**.

The answer to the accurate-costs question, as it turns out, is *yes* for the distance-vector algorithm, if we follow the rules carefully, and the network is stable (meaning that no routing reports are changing, or, more concretely, that every update report now circulating is based on the current network state); a proof is below. However, if there is a routing loop, the answer is of course no: the actual cost is now infinite. The answer would also be no if A’s neighbor B has just switched to using a longer route to D than it last reported to A.

It turns out, however, that we seek the shortest route not because we are particularly trying to save money on transit costs; a route 50% longer would generally work just fine. (AT&T, back when they were the Phone Company, once ran a series of print advertisements claiming longer routes as a *feature*: if the direct path was congested, they could still complete your call by routing you the long way ‘round.) However, we *are* guaranteed that if all routers seek the shortest route – and if the network is stable – then all paths are loop-free, because in this case the network will have accurate costs.

Here is a simple example illustrating the importance of global cost-minimization in preventing loops. Suppose we have a network like this one:

Now suppose that A and B use distance-vector but are allowed to choose the shortest route *to within 10%*. A would get a report from C that D could be reached with cost 1, for a total cost of 21. The forwarding entry via C would be ⟨D,C,21⟩. Similarly, A would get a report from B that D could be reached with cost 21, for a total cost of 22: ⟨D,B,22⟩. Similarly, B has choices ⟨D,C,21⟩ and ⟨D,A,22⟩.

If A and B both choose the minimal route, no loop forms. But if A and B both use the 10%-overage rule, they would be allowed to choose the other route: A could choose ⟨D,B,22⟩ and B could choose ⟨D,A,22⟩. If this happened, we would have a routing loop: A would forward packets for D to B, and B would forward them right back to A.

As we apply distance-vector routing, each router independently builds its tables. A router might have some notion of the path its packets would take to their destination; for example, in the case above A might believe that with forwarding entry ⟨D,B,22⟩ its packets would take the path A–B–C–D (though in distance-vector routing, routers do not particularly worry about the big picture). Consider again the accurate-cost question above. This *fails* in the 10%-overage example, because the actual path is now infinite.

We now prove that, in distance-vector routing, the network will have accurate costs, provided

- each router selects what it believes to be the shortest path to the final destination, and
- the network is stable, meaning that further dissemination of any reports would not result in changes

To see this, suppose the actual route taken by some packet from source to destination, as determined by application of the distributed distance-vector algorithm, is longer than the cost calculated by the source. Choose an example of such a path with the **fewest number of links**, among all such paths in the network. Let S be the source, D the destination, and k the number of links in the actual path P. Let S’s forwarding entry for D be ⟨D,N,c⟩, where N is S’s next_hop neighbor.

To have obtained this route through the distance-vector algorithm, S must have received report ⟨D,c_{D}⟩ from N, where we also have the cost of the S–N link as c_{N} and c = c_{D} + c_{N}. If we follow a packet from N to D, it must take the same path P with the first link deleted; this sub-path has length k-1 and so, by our hypothesis that k was the length of the shortest path with non-accurate costs, the cost from N to D is c_{D}. But this means that the cost along path P, from S to D via N, must be c_{D} + c_{N} = c, contradicting our selection of P as a path longer than its advertised cost.

There is one final observation to make about route costs: any cost-minimization can occur only within a single routing domain, where full information about all links is available. If a path traverses multiple routing domains, each separate routing domain may calculate the optimum path traversing that domain. But these “local minimums” do not necessarily add up to a globally minimal path length, particularly when one domain calculates the minimum cost from one of its routers only to the other *domain* rather than to a router within that other domain. Here is a simple example. Routers BR1 and BR2 are the **border routers** connecting the domain LD to the left of the vertical dotted line with domain RD to the right. From A to B, LD will choose the shortest path to RD (not to B, because LD is not likely to have information about links within RD). This is the path of length 3 through BR2. But this leads to a total path length of 3+8=11 from A to B; the global minimum path length, however, is 4+1=5, through BR1.

In this example, domains LD and RD join at two points. For a route across two domains joined at only a single point, the domain-local shortest paths do add up to the globally shortest path.