Chapter $5$

$$

TP Routing Algorithms

$$

$5\mathrm{.}1$ Subject

We model a computer network with a directed graph where we are only concerned whether there is at least one link from a data center to another data center. We use a binary matrix A $($composed of $0$ and $1)$ of size n $$ n where the indices of rows and columns correspond to different numbers of data centers encountered. A $1$ at the location $($i$,$ j$)$ of the matrix means that there is a link from data center i to the data center j$.$ If there is no link between these two data centers, then A$($i$,$ j$)=0.$ Adjacency matrix A may be read in a file as given below :

$$

$0$

$1$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$1$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

Table $5\mathrm{.}1$: Adjacency Matrix

$$

$$

$1.$ Imagine there are communication costs to transmit data from one data center to another one. Our first objective is to construct a routing table for each data center such that the transmission costs between any pair of centers is minimized. Costs are given by a cost matrix C as given below. Program and apply an algorithm $($which one?$)$ to find $($if exists$)$ shortest path between any pairs of data centers.

$2.$ Imagine that you need to install cable between data centers to transmit data from on center to another one $($do not consider the direction of the transmission for this question$).$ The installation cost $($if cable installation is possible$)$ between any data centers is given

$0$

$1$

$5$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$4$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$2$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$4$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$2$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$3$

$0$

$3$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$1$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$6$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$7$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

<

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