Chapter Questions
Give an example of a PIF algorithm for systems with synchronous message passing that does not allow computation of infima (com-pare Theorem 6.7 and 6.12). Your example may be suitable for a partacular topology only.
In a purtial order $(X, \leq)$ an element $b$ is called a botium if for all $c \leqq X, b \leq c$. not contarn a bottom. Where?Cin you bite an algorithm that computes infima in a partial order with bottom, and is not a wave algorithm?
Give two partial orders on the natural numbers for which the infimum function is (1) the gre:test conmon disisor, and (2) ti e least comimon ancestor.Give partial orders on the collection of subsets of a universe $U$ for which the infirmim function is (1) the intersection of sets, and (2) the union of sets.
Prove the Infimum Theorem (Theorem 6.13).
Show that in each computation of the tree algorithm (Algorithm 6.9) exactly two processes decade.
Use the echo algorithm (Algorithm 6.5) to write an algorthm that computes a prefix labeling scheme (see Subsection 4.4.3) for an arbitrary network using $2|E|$ messages and $\mathrm{O}(N)$ time units.Can you give an algorithm that computes the labeling scheme in $\mathrm{O}(D)$ time? ( $D$ is the diumeter of the network.)
Show that the relationship in Lemma 6.19 also holds if messages can get lost in the channel pq, but not if messages can be duplicated. What step in the proof fails if messages can be duplicated?
Apply the construction of Theorem 6.12 to the phase algorithm so as to obtain an algorithm that computes the maximum over the (integer) inputs of all processes.What are the message, time, and bit complexities of your algorithm?
Suppose you want to use a wave algorithm in a network where duplication of messages may occur.(1) What modifications should be made to the echo algorithm?(2) What modificatinns should be made to Ftnn's algornthm?
A complete bipartite graph is a graph $G=(V, E)$ where $V=V_1 \cup V_2$ with $V_1 \cap V_2=\varnothing$ and $E=V_1 \times V_2$.Gue a $2 x$-traversal algorithm for complete bipartite networks.
Prove or disprove: The traversal of a hypercube without sense of direction requres $\Theta(N \log N)$ messages.
Give an example of a computation of Tarry's algorithm in which the resulting tree $T$ is not a DFS tree.
Write an algorithm that computes the depth-first search interval labeling scheme (see Subsection 4.4.2) for an arbitrary connected network.Can it be done in $\mathrm{O}(N)$ time units? Can it be done using $\mathrm{O}(N)$ messages?
Assume that the depth-first search algorithm with neighbor knowledge is used in a system where each process knows not only the ident?ties of its neighbors but also the set of all process identities $(\mathbb{P})$. Show that messages of $N$ bits each are sufficient in this case.
Adapt the echo algorithm (Algorithm 6.5) to compute the sum over the inputs of the processes.
Assume the processes in the networks depicted in Figure 6.21 have unique identities and each process has an integer input. Simulate on both networks a computation of the phase algorithm, computing the set $S=$ $\left\{\left(p, j_p\right): p \in \mathbb{P}\right\}$ and the sum over the inputs.
What is the chain-time complexity of the phase algorithm for cliques (Algorithm 6.8)?