Chapter 4, Routing Algorithms Video Solutions, Introduction to Distributed Algorithms

Problem 1

Assume that routing tables are updated after each topolugical change in such a way that they are cycle-frec even durng updates. Does thas guarantee that packets are always delvered even when the network is subject to a possibly infinite number of topological changes?
Prove that no routing algorithm can guarantee delivery of packets under continuing topologrcal changes.

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Problem 2

$ A$ student proposes to omit the sending of $\langle$ nys, $w\rangle$ messages from Algorithm 4.6; he argues that a node knows that a neighbor is not a son in $T_w$ if no $\langle\mathbf{y s}, w\rangle$ message is received from that neighbor.
Is it possible to modify the algorithm in this way? What happens to the complexity of the algorithm?

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Problem 3

Prove that the foilowing assertion is an invariant of the Chandy-Musra algorithm for computing paths towards $v_0$ (Algorithm 4.7).
$$
\begin{gathered}
\forall u, w:\left\langle\text { mydist, } v_0, d\right\rangle \in M_{\mathrm{tw} u} \Rightarrow d\left(w, v_0\right) \leq d \\
\wedge \forall u: d\left(u, v_0\right) \leq D_u\left[v_0\right]
\end{gathered}
$$
Give an example of an execution for which the number of messages is exponential in the number of channels of the network.

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24:35

Problem 4

Give the values of all variables in a terminal configuration of the Netchange algorithm when the algorithm is applied to a network of the following topology:
After a terminal configuration has been reached, a channel between $A$ and $F$ is added. What messages does $F$ send to $A$ when processing the ?repair, $A$ ? notificatron? What messages dces $A$ send upon receipt of these messages from $F$ ?
(FIGURE CAN'T COPY)

Chris Trentman

Numerade Educator

Problem 5

Give an example to demonstrate that Lemma 4.22 does not hold for networks with asymmetric channel cost.

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02:24

Problem 6

Does there exist an ILS that does not use all channels for routing? Does there exist a valid one? An optimal one?

James Kiss

Numerade Educator

Problem 7

Give a graph $G$ and a depth-first search tree $T$ of $G$ such that $G$ has $N=n^2$ nodes, the diameter of $G$ and the depth of $T$ are $O(n)$, and there are nodes $u$ and $v$ such that a packet from $u$ to $v$ is delivered after $N-1$ hops with the depth-first search ILS.
(The graph can be chosen in such a way that $G$ is outerplanar, which implies (by Theorem 4.37) that $G$ actually has an optimal ILS.)

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01:38

Problem 8

Give the depth-first search ILS for a ring of $N$ nodes. Find nodes $u$ and $v$ such that $d(u, v)=2$, and the scheme uses $N-2$ hops to deliver a packet from $u$ to $v$.

Victor Salazar

Numerade Educator

02:31

Problem 9

Prove that the minimality of the tree $T$ in the proof of Theorem 4.48 implies that it has at most $m$ leaves. Prove that any tree unth $m$ leaves has at most $m-2$ branch points.

Victoria Dollar

Numerade Educator