Chapter Questions
Prove Lemma 15.4.
Prove that Dijkstra's token ring reaches a legitimate configuration in $\mathrm{O}\left(N^2\right)$ steps. Shorten the analysis by giving a single norm function, quadratically bounded in $N$, that decreases with every step of the algorithm.
Modify the ring-orientation algorithm (Algorithm 15.1) so that it will terminate after establishing its postcondition.
Design a stabilizing algorithm to orient a torus, preferably, so that the algorithm terminates.
Show that the maximal-matching algorithm can be implemented uniformly in the link-register read-all model.
Show that the maximal-matching algorithm stabilizes in $N^2+$ $\mathrm{O}(N)$ steps and show that there is an $\Omega\left(N^2\right)$ lower bound on the (worst-case) number of steps).
Design a stabilizing algorithm to construct a maximal independent set and compute the maximal number of steps before stabilization.
Give a norm function that proves the termination of Algorithm 15.4.
Prove that outerplanar graphs are three-colorable. Design a stabilizing algorithm that computes a three-coloring of an outerplanar graph.
Design a stabilizing algorithm that computes a five-coloring of a planar graph. (See [McH90] or another text on graph algorithms.)
Show that the Update algorithm can be used for election and computation of a breadth-first search spanning tree by giving an appropriate path-cost function.
Give a stabilizing algorithm for computing the network size.
Show how to compute the depth of a tree with the Update algorithm.