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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.)

    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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Introduction to Distributed Algorithms

Introduction to Distributed Algorithms

Gerard Tel 1st Edition

Chapter 4, Problem 7

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Additionally, there should be nodes \( u \) and \( v \) such that a packet from \( u \) to \( v \) is delivered after \( N-1 \) hops using the depth-first search iterative lengthening search (ILS). The graph should be outerplanar. Show more…

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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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