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systems analysis and design
Design And Analysis Of Distributed Algorithms 1st Edition Nicola Santoro - Solutions
An entity is said to be a waiter at time t if it has requested a token and it has not yet received it at time t . Prove that at any time t during the execution of protocol Arrow, in L[t ] any terminal path leads either to the entity holding the token or to a waiter.
Prove that during the execution of protocol Arrow, every request will be delivered to its target within finite time.
Compare experimentally the performance of protocols Arrow and OnDemandTraversal under different load conditions. Investigate the impact of the structure of the spanning tree on their performace.
Prove that under Temporal Constraint, in protocol AskAll every request receives permissions from all entities within finite time.
Show how to use using virtual clocks to ensure that property Temporal Constraint holds.
() Write the set of rules corresponding to Strategy AskAll using logical clocks to impose total order among requests. Implement and throughly test the corresponding protocol AskAllClocks. Compare the experimental results with the theoretical bounds.
() Write the set of rules of protocol AskQuorum and prove its correctness.
() Implement and throughly test protocol AskQuorum of Exercise 9.6.21. Compare the experimental results with the theoretical bounds.
Calculate the coefficient α for the coterie Square when n is not a perfect square.
() Let n = p2. Construct a coterie with α = 1 where each quorum has size precisely p.
Devise a method so that the entities in the core can execute the ring-election protocol without sending any message to noncore entities.
(Pipeline in Trees: Min) Write the protocol for finding the minimum of all the values in a tree using the 2-bit communicator and pipeline. Prove its correctness. Determine its costs.
(Maximum Finding I) () Consider a ring of known size n. Each entity has a positive integer value; they all start at the same time, but their values are not necessarily distinct. The maximum-finding problem is the one of having all the entities with the largest value become maximum and all the
(Pipeline in Trees: Max) Write the protocol for finding the maximum of all the values in a tree using the 2-bit communicator and pipeline. Prove its correctness. Determine its costs.
(Size Communicator) Consider the class of communicators that use the first quantum to communicate the total number of bits that will be transmitted.Determine the minimum cost that can be achieved and design the corresponding protocol.
(2-BitPattern Communicator) () Consider the class of communicators that use two successive transmissions of 1 to denote termination. Determine the minimum cost that can be achieved and design the corresponding protocol.
(Maximum Finding II) () Determine whether the maximumfinding problem in a ring of known size can be solved in time linear in imax with O(n)bits.
(BitPattern Communicator) Consider the class of communicators that use a bit set to 1 to denote termination. Determine the minimum cost that can be achieved and design the corresponding protocol.
(OneBit Protocol) Determine under what conditions information can be communicated using only 1 bit and describe the corresponding OneBit protocol.
(Bit-Optimal Election I) () Show how to elect a leader in a ring with only O(n) bits without knowing n. Possibly the time should be polynomial in i or exponential in n. (Hint: Use a single iteration of DoubleWait as a preprocessing phase.)
() Consider a complete graph where at each entity at most f < n2 incident links may crash. Design a protocol to achieve unanimity using O(n2)messages.
Consider a complete graph where f < n2 entities might have crashed but no more failures will occur. Consider the Election problem and assume that all identities are known to all (nonfaulty) entities. Show how the election can be performed using O(kf ) messages, where k is the number of initiators.
(Bit-Optimal Election II) () Determine whether or not it is possible to elect a leader without knowing n with (n) bits in time sublinear in i, that is, to match the complexity achievable when n is known.
() Consider a complete networks whereF
() Consider a complete networks whereF
Prove that protocol FT-LinkElect correctly elects a leader provided k ≤ n−6 2 . (Hint: Use the results of Exercises 7.10.36, 7.10.37, and 7.10.38).
(Unison without knowingd) () Consider the unison problem when there is no known upperbound on the diameter d of the network. Prove or disprove that in this case the unison problem cannot be solved with explicit termination.
Consider a set of synchronous entities connected in a complete graph. Show how the existence of both digital signatures and secrete sharing can be used to implement a global source of random bits unbiased and visible to all entities.
Consider a set of asynchronous entities connected in a complete graph. Show how the existence of both private channels and a trusted dealer can be used to implement a global source of random bits unbiased and visible to all entities.
Consider a set of asynchronous entities connected in a complete graph. Show how the existence of both digital signatures and a trusted dealer can be used to implement a global source of random bits unbiased and visible to all entities.
Complete the description of protocol Committee and prove its correctness.
Prove that in any connected graph G we have T∗(G) =O(diam(G)cedge(G)).
Consider a computation C that circulates k tokens among the entities in a system where tokens (but not messages) can be lost while in transit. The problem we need to solve is the detection of whether one or more tokens are lost.Adapt the general protocol we designed for detecting stable properties
Using the transformation of Problem 8.6.14, determine the cost of TD when GC is the References Count algorithm.
() Show how to transform automatically a garbage collection algorithm GC into a termination detection protocol TD. Analyze the cost of TD.
() Construct a computation Ck, k ≥ 0 such that M(Ck) ≥ k and to detect global termination of C, every protocol must send at leastM(C) messages.
() Write the set of rules of protocol MultiShrink for global termination detection with multiple initiators. Implement and throughly test your protocol.Compare the experimental results with the theoretical bounds.
() Write the set of rules of protocol Shrink for global termination detection with a single initiator. Implement and throughly test your protocol. Compare the experimental results with the theoretical bounds.
() Write the set of rules corresponding to strategy RepeatQuery+when Q is TerminationQuery and there are multiple initiators. Implement and throughly test your protocol. Compare the experimental results with the theoretical bounds.
() For the problem of personal deadlock detection with multiple initiators consider the strategy to integrate into the solution an election process among the initiators. Design a protocol for the Generalized request model to implement efficiently this strategy; its total cost should be o(km)
() For the problem of personal deadlock detection with multiple initiators consider the strategy to integrate into the solution an election process among the initiators. Design a protocol for the p-OF-q request model to implement efficiently this strategy; its total cost should be o(km) messages
(Firing in a Line of CA with 6 States) () Finite cellular automata (CA) can only have a constant memory size, which means they cannot store a counter. The goal is thus to solve the firing squad problem with the least amount of time and to do so with the least amount of memory. The measure we use
() For the problem of personal deadlock detection with multiple initiators consider the strategy to integrate into the solution an election process among the initiators. Design a protocol for the OR-request model to implement efficiently this strategy; its total cost should be o(km) messages in
() Modify protocol LockGrant so that, with a single initiator, it works correctly also in a dynamic wait-for graph. Prove the correctness and analyze the cost of the modified protocol.
() For the problem of personal deadlock detection with multiple initiators consider the strategy to integrate into the solution an election process among the initiators. Design a protocol for the AND-request model to implement efficiently this strategy; its total cost should be o(km) messages in
() In protocol LockGrant employ Shout+ instead of Shout, so as to use at most 4|E(x0)| messages in the worst case. Write the corresponding set of rules.Implement and throughly test your protocol. Compare the experimental results with the theoretical bounds.
Implement protocol LockGrant, both for personal and for collective deadlock detections. Throughly test your protocol. Compare the experimental results with the theoretical bounds.
() For the problem of personal deadlock detection with multiple initiators consider the strategy to integrate into the solution an election process among the initiators. Design a protocol for the single-request model to implement efficiently this strategy; its total cost should be o(kn) messages
Write the set of rules of protocol Dead Check implementing the simple check strategy for personal and for collective deadlock detection in the single resource model. Implement and throughly test your protocol. Compare the experimental results with the theoretical bounds.
() Write the rules of protocol SingleDetectResolve for deadlock detection and resolution in single-request systems. Implement it and throughly test it. Compare its experimental performance with the theoretical bounds.
() Design an algorithm to occasionally reduce the values of the vector clocks. Your protocol should not destroy any causal relationship between the events occurring after the reduction. Prove its correctness and analyze its performance.
() Modify algorithm VectorClocks, so as to construct and maintain virtual clocks satisfying the Complete Causal Order property without any a priori knowledge, without incurring in any initial cost, without any additional messages, and with no more information in each message that that required by
() Prove that any timestamp-based virtual clock that satisfies property Complete Causal Order must use vectors of size at least n.
() Prove Property 5.4.6.
() Prove Property 5.4.5: Any expression of E(x) can be reexpressed as the union of sub-expressions in E−(xi ).
Prove that using strategy Bitmask, entity xi can directly evaluate any expression in E−(xi ).
Show that expressions 5.38 and 5.38 are equal.
Prove that the query (D1 − D2) ∩ (D3 − (D4 ∩ D5)) can be answered immediately at both x1 and x3 if each of the sets is stored by its entity using the DSP method.
Write the set of rules of Protocol DynamicSelectSorting. Implement and test the protocol. Compare the experimental costs with the theoretical bounds.
Show how in strategy DynamicSelectSort the coordinator x can determine π from the received information in O(n3) local processing activities.
Establish for each of the storage requirements the worst-case cost of protocol SelectSort to sort a distributed set in a oriented torus of dimension p × q.Determine under what conditions the protocol is optimal for this network (Hint: Use result of Exercise 5.6.38).
Establish for each of the storage requirements the worst-case cost of protocol SelectSort to sort an equidistributed set in a oriented torus of dimension p × q. Determine under what conditions the protocol is optimal for this network.(Hint: Use result of Exercise 5.6.38).
Establish for each of the storage requirements the worst-case cost of protocol SelectSort to sort a distributed set in a labeled hypercube of dimension d.Determine under what conditions the protocol is optimal for this network (Hint: Use result of Exercise 5.6.37).
Establish for each of the storage requirements theworst-case cost of protocol SelectSort to sort an equidistributed set in a labeled hypercube of dimensiond. Determine under what conditions the protocol is optimal for this network (Hint:Use result of Exercise 5.6.37).
Establish for each of the storage requirements theworst-case cost of protocol SelectSort to sort a distributed set in a ring. Determine under what conditions the protocol is optimal for this network (Hint: Use result of Exercise 5.6.36).
Establish for each of the storage requirements the worst-case cost of protocol SelectSort to sort an equidistributed set in a ring. Determine under what conditions the protocol is optimal for this network (Hint: Use result of Exercise 5.6.36).
Establish for each of the storage requirements the worst-case cost of protocol SelectSort to sort a distributed set in a ordered line. Determine under what conditions the protocol is optimal for this network. Compare this cost with the one of protocol OddEven-LineSort.
Establish for each of the storage requirements the worst-case cost of protocol SelectSort to sort an equidistributed set in a ordered line. Determine under what conditions the protocol is optimal for this network. Compare this cost with the one of protocol OddEven-LineSort.
Write the set of rules of Protocol SelectSort. Implement and test the protocol. Compare the experimental costs with the theoretical bounds.
Show how xπ(i) can find out ki at the beginning of the ith iteration of strategy SelectSort. Initially, each entity knows only its index in the permutation(i.e., xπ(i) knows i) as well as the storage requirements.
() Determine for each of the three storage requirements (invariant, equidistributed, compacted) a lower bound, in terms of n and N on the amount of necessary messages for sorting in an oriented torus. What would be the bound for initially equidistributed sets?
() Determine for each of the three storage requirements (invariant, equidistributed, compacted) a lower bound, in terms of n and N on the amount of necessary messages for sorting in a labeled hypercube. What would be the bound for initially equidistributed sets?
Determine for each of the three storage requirements (invariant, equidistributed, compacted) a lower bound, in terms of n and N on the amount of necessary messages for sorting in a ring. What would be the bound for initially equidistributed sets?
For each of the three storage requirements (invariant, equidistributed, compacted) show a situation where (N) messages need to be sent to sort in a complete network, even when the data are initially equidistributed.
Consider an initial distribution where x1 and xn have the same number K = (N − n + 2)/2 of data items, while all other entities have just a single data item. Augment protocol OddEven-MergeSort so as to perform an invariant sort when π = 1, 2, . . . , n. Show the corresponding sorting diagram.
Prove that protocol OddEven-MergeSort correctly sorts, regardless of the storage requirement, if the initial set is equidistributed.
Prove that protocol OddEven-MergeSort is a sequence of 1 + log n iterations and that in each iteration (except the last) every data item is sent once or twice to another entity.
Write the set of rules of protocol OddEven-MergeSort. Implement the protocol and throughly test it.
Prove that whenn > 3, if the line is not sorted according to π, then protocol OddEven-LineSort terminates but does not sort the data according to π.
Consider an initial equidistribution sorted according to permutationπ = π(n), π(n − 1), . . . , π(1). Prove that, executing protocol OddEven-LineSort in this case, every data item will change location in each iteration.
Prove that there are some initial conditions under which protocol OddEven-LineSort uses N − 1 iterations to perform invariant-size sorting of N items distributed on a sorted line, regardless of the number n of entities.
Prove that OddEven-LineSort sorts an equidistributed distribution in n − 1 iterations regardless of whether the required sorting is invariant-sized, equidistributed, or compacted with all entities having the same capacity.
() Prove that OddEven-LineSort performs an equidistributed sort of any distribution on an ordered line.
() Prove that OddEven-LineSort performs a compacted sort of any distribution on an ordered line.
() Prove that OddEven-LineSort performs an invariant-sized sort of any distribution on an ordered line.
Prove that OddEven-LineSort performs an invariant-sized sort of an equidistribution on an ordered line.
Consider the system shown in Figure 5.9. How many items will x5 have(a) after a compacted sorting with w = 5?(b) after an equidistributed sorting?Justify your answer.
Prove that after the execution of Protocol CUT there will be at most min{n, } log items left under consideration.
Prove that after the execution of Cutting Tool on C(l = 2i ), only the l − 1 columns C(1),C(2), . . . , C(l − 1) might remain unchanged; all others, including C(l) will have at least n − K/l of the entries +∞.
Prove that in the execution of ProtocolREDUCE, Local Contraction is executed at the most three times.
Prove that the number of iterations performed by Protocol Filter until there are no more than n elements left under consideration is at most 2.41 log(N/n).
() Prove that the expected number of iterations performed by Protocol RandomRandomSelect until there are less than n items left under consideration is at most 4 3log log + 1 .
Write Protocol RandomRandomSelect ensuring that each iteration uses at most 4(n − 1) + r(s) messages and 5r(s) ideal time units. Implement the protocol and throughly test your implementation.
() Determine the number of iterations if we terminate protocol RandomFlipSelect, as soon as the search space contains at most cn items, where c is a fixed constant. Determine the total cost of this truncated execution followed by an execution of protocol Rank.
() Prove that the expected number of iterations performed by Protocol RandomFlip until termination is less than ln() + ln(n) + O(1).
Prove that in the worst case, the number of iterations performed by Protocol RandomFlipSelect until termination is N.
() Determine the number of iterations if we terminate protocol RandomSelect, as soon as the search space contains at most cn items, where c is a fixed constant. Determine the total cost of this truncated execution followed by an execution of protocol Rank.
() Prove that the expected number of iterations performed by Protocol RandomSelect until termination is at most 1.387 logN + O(1).
Random Item Selection () Modify the protocol of Exercise 2.9.52 so that it can be used to select uniformly at random an element still under consideration in each iteration of Strategy RankSelect. Your protocol should use at most 2(n − 1) + dT (s, x) messages and 2r(s) + dT (s, x) ideal time
() Extend the technique of protocol Halving to work with three sets, Dx , Dy , and Dz. Write the corresponding protocol, prove its correctness, and analyze its complexity.
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