Questions and Answers of Fundamentals Simulations And Advanced Topics

2.17 Show that in Algorithm 4 if the leader variable is not included in the (parent) message and the test in Line 18 is not performed, then the algorithm is incorrect.
4.9 Show why the variable Choosing[i] is needed in the bakery algorithm (Al- gorithm 10). Specifically, consider a version of Algorithm 10 in which this variable is omitted, and construct an
3.3 Is leader election possible in a synchronous ring in which all but one processor have the same identifier? Either give an algorithm or prove an impossibility result.
3.4 Consider the following algorithm for leader election in an asynchronous ring: Each processor sends its identifier to its right neighbor; every processor for- wards a message (to its right
3.5 In Section 3.3.3, we have seen a lower bound of (n log n) on the number of messages required for electing a leader in an asynchronous ring. The proof of the lower bound relies on two additional
3.6 Extend Theorem 3.5 to the case in which n is not an integral power of 2. Hint: Consider the largest n'
3.7 Modify the formal model of synchronous message passing systems to describe the non-synchronized start model of Section 3.4.1. That is, state the conditions that executions and admissible
3.8 Prove that the order-equivalent ring R' in proof of Lemma 3.17 can always be constructed.
3.9 Recall the ring Ry from the proof of Theorem 3.18. For every partition of Rinto consecutive segments, where j is a power of 2, prove that all of these segments are order equivalent.
3.10 Consider an anonymous ring where processors start with binary inputs. 1. Prove there is no uniform synchronous algorithm for computing the AND of the input bits. Hint: Assume by way of
3.11 Derive an (n log n) lower bound on the number of messages required for leader election in the asynchronous model of communication from the lower bound for the synchronous model. In the
3.2 Prove that there is no anonymous leader election algorithm for synchronous ring systems that is uniform.
4.1 Suppose an algorithm satisfies the condition that in every admissible execution, at most one processor is in the critical section in any configuration. Show that this algorithm also satisfies the
4.8 Modify the tournament tree mutual exclusion algorithm for n processors so that it can use an arbitrary two-processor mutual exclusion algorithm as "subroutines" at the nodes of the tree. Prove
4.7 Prove that Algorithm 11 provides no deadlock.
4.6 Present an algorithm that solves the 2-mutual exclusion problem (defined in Exercise 4.2) and efficiently exploits the resources, that is, a processor does not wait when only one processor is in
4.10 Formalize the discussion at the beginning of Section 4.4.4 showing that n variables are required for no deadlock mutual exclusion if they are single- writer.
4.14 Design a fast mutual exclusion algorithm using test&set operations.
4.5 Calculate the waiting time for the algorithm presented in Section 4.4.3 using the method from Exercise 4.4. That is, calculate how long a processor waits, in the worst case, since entering the
4.4 Propose a method for measuring worst-case time complexity in the asyn- chronous shared memory model analogous to that in the asynchronous message- passing model.
4.3 (a) Prove, by induction on the length of the execution, the invariant properties of Algorithm 9, as stated in Lemma 4.3. (b) Based on the invariants, prove that Algorithm 9 provides mutual
4.2 An algorithm solves the 2-mutual exclusion problem if at any time at most two processors are in the critical section. Present an algorithm for solving the 2-mutual exclusion problem by using

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