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Design And Analysis Of Distributed Algorithms 1st Edition Nicola Santoro - Solutions
Implement protocol GeneralHalving of Exercise 5.6.7, throughly test it, and run extensive experiments. Compare the experimental results with the theoretical ones.
Modify protocol Halving as follows: In iteration i,(a) discard from both Dix and Diy, all elements greater than max{mi x,mi y} and all those smaller than min{mi x,mi y}, where Dix and Diy denote the set of elements of Dx and Dy still under consideration at the beginning of stage i, and mi x and mi
(Firing in a Line of CA with 5 States) () Consider a line of CA with only one initiator (located at the end of the line). Develop a solution using only five states or prove it can not be done.
Prove that the K-selection problem can be reduced to a medianfinding problem regardless of K and of the size of the two sets.
Write protocol Halving so that itworks with any two arbitrarily sized sets with the same complexity.
Prove that after discarding the elements greater than mx from Dx and discarding the elements greater than my from Dy , the overall lower median is the lower median of the elements still under considerations.
() Consider a network whose topology is a complete binary tree where each entity has just one item. Show how to perform selection using O(n log n)messages.
() Consider a mesh network where each entity has just one item.Show how to perform selection using O(n log 32 n) messages.
() Consider a ring network where each entity has just one item.Show how to perform selection using O(n log3 n) messages.
Prove that in protocol MinMax, if a candidate x survives an even stage i, its predecessor l(i, x) becomes defeated.
Modify protocol Stages with Feedback using the min-max approach discussed in Section 3.3.7. Prove its correctness. Show that its message costs are unchanged.
Show an initial configuration for n = 9 in which protocol Stages with Feedback will require the most stages. Describe how to construct the “worst configuration” for any n.
Give a more accurate estimate of the message costs of protocol Stages with Feedback.
Prove that in protocol Stages withFeedback, the minimum distance between two candidates in stage i is d(i) ≥ 3i−1.
Prove that in protocol Stages with Feedback, the number of ring segments where no feedback will be transmitted in stage i is ni+1.
Write the rules of protocol Stages with Feedback enforcing message ordering.
Derive the ideal time complexity of protocol Stages withFeedback.
Prove that the message and time costs of Stages* are no worse that those of Stages. Produce an example in which the costs of Stages* are actually smaller.
Show that protocol Stages* correctly terminates.
Assume that in Stages* candidate x in stage i receives a message M∗ with stage j > i. Prove that if x survives, then id(x) is smaller not only of id∗but also of the ids in the messages “jumped over” by M∗.
Implement the alternating step strategy under the same restrictions and with the same cost of protocol Alternate but without closing any port.
Determine initial configurations that will force protocol Alternate to use k steps when n = Fk.
Show that the worst case number of steps of protocol Alternate is achievable for every n > 4.
Modify protocol UniAlternate using the min-max approach discussed in Section 3.3.7. Prove its correctness. Show that its message costs are unchanged.
Prove that the ideal time complexity of protocol UniAlternate is O(n).
Without changing its message cost, modify protocol UniAlternate so that it does not require Message Ordering.
Show the step-by-step execution of Alternate and of UniAlternate in the ring of Figure 3.3. Indicate for each step, the values know at the candidates.
Design an exact simulation of Stages with Feedback for unidirectional rings. Analyze its costs.
Modify protocol UniStages using the min-max approach discussed in Section 3.3.7. Prove its correctness. Show that its message costs are unchanged.
Determine the ideal time complexity of protocol UniStages.
Show the step-by-step execution of Stages and of UniStages in the ring of Figure 3.3. Indicate for each step, the values know at the candidates.
Modify protocol Alternate using the min-max approach discussed in Section 3.3.7. Prove its correctness. Show that its message costs are unchanged.
Determine the ideal time complexity of protocol Alternate.
Write the rules of protocol Stages* described in Section 3.3.4.
Modify protocol Stages using the min-max approach discussed in Section 3.3.7. Prove its correctness. Show that its message costs are unchanged.
Determine the ideal time complexity of protocol Stages.
Minimal Chordal Ring () Find a chordal ring with k = 2 where it is possible to elect a leader with O(n) messages.
Oriented Butterfly. Design an election protocol for an oriented butterfly. Determine its complexity. Implement and test your protocol.
Oriented Cube-Connected Cycles () Design an election protocol for an oriented CCC using O(n) messages. Implement and test your protocol.
Linear Election in Hypercubes. () Prove or disprove that it is possible to elect a leader in an hypercube in O(n) messages even when it is not oriented.
Unoriented Hypercubes. () Design a protocol that can elect a leader in a hypercube with arbitrary labelling using O(n log log n) messages. Implement and test your protocol.
Bidirectional Oriented Rings. () Prove or disprove that there is an efficient protocol for bidirectional oriented rings that cannot be used nor simulated neither in unidirectional rings nor in general bidirectional ones with the same or better costs.
MinMax+ Variations () In protocol MinMax+ we use “promotion by distance” only in the even stages and “promotion by witness” only in the odd stages. Determine what would happen if we use 1. only “promotion by distance” but in every stage;2. only “promotion by witness” but in every
Bidirectional MinMax () Design a bidirectional version of Min-Max with the same costs.
Better Stages () Construct a protocol based on electoral stages that guarantees ni ≤ ni−1 b with cn messages transmitted in each stage, where c log b
Alternating Steps () Design a conflict resolution mechanism for the alternating steps strategy to cope lack of orientation in the ring. Analyze the complexity of the resulting protocol
Unlabelled Chordal Rings () Show how to elect a leader in the chordal ring of
Improved Time () Show how to elect a leader using O(m +n log n) messages but only O(n) ideal time units.
Distances in MinMax+ () In computing the cost of protocol MinMax+ we have used dis(i) = Fi+2. Determine what will be the cost if we use dis(i) = 2i instead.
Show an initial configuration for n = 8 in which protocol Stages will require the most messages. Describe how to construct the “worst configuration”for any n.
Prove that in protocol Stages with Feedback, the minimum distance between two candidates in stage i is d(i) ≥ 2i−1.
Show that in protocol Stages, there will be at most one enqueued message per closed port.
Expand the rules of protocol Stages described in Section 3.3.4, so as to enforce message ordering.
Modify protocol AsFar so to use strategy Elect Minimum Initiator.Determine the average number of messages assuming that any subset of k∗ entities is equally likely to be the initiators.
Modify protocol All the Way so to use strategy Elect Minimum Initiator.
Show that the time costs of protocol All the Way will be at most 2n − 1. Determine also the minimum cost and the condition that will cause it.
Design an efficient single-initiator protocol to find the minimum value in a ring. Prove its correctness and analyze its costs.
Modify protocol MinF-Tree (presented in Section 2.6.2) so as to implement strategy Elect Minimum Initiator in a tree. Prove its correctness and analyze its costs. Show that, in the worst case, it uses 3n + k − 4 ≤ 4n − 4 messages.
Write the rules of protocol Stages with Feedback assuming message ordering.
Josephus Problem. Consider the following set of electoral rules.In stage i, a candidate x sends its id and receives the id from its two neighboring candidates, r(i, x) and l(i, x): x does not survive this stage if and only if its id is larger than both received ids. Analyze the corresponding
Prove that in the YO-YO protocol, during an iteration, no sink or internal node will become a source.
() Prove that it is possible to elect a leader in a complete graph using O(n) messages with any sense of direction.
Prove that to solve SPT under IR, a message must be sent on every link.
Show how to transform a spanning-tree construction algorithm C so as to elect a leader with at most O(n) additional messages.
Prove that under IR, the problem of finding the smallest of the entities’ values is computationally equivalent to electing a leader and has the same message complexity.
Prove that all the rings R(2), . . . , R(k) where messages are sent by protocol Kelect do not have links in common.
Generalize the answer to Exercise 3.10.68. Design an election protocol for complete graphs that, for any log n ≤ k ≤ n, uses O(nk) messages and O(n/k) time in the worst case.
Design an election protocol for complete graphs that, like CompleteElect, uses O(n log n) messages but uses only O(n/ log n) time in the worst case.
Analyze the ideal time cost of protocol CompleteElect described in Section 3.6.1.
Determine the cost of the strategy CompleteElect described in Section 3.6.1 in the worst case (Hint: Consider how many candidates there can be at level i).
Prove that the strategy CompleteElect outlined in Section 3.6.1 solves the election problem.
with O(n) messages even if the edges are arbitrarily labeled.
Write the code for, implement, and test protocol Kelect-Stages.
Modify the YO-YO protocol so that upon termination, a spanning tree rooted in the leader has been constructed. Achieve this goal without any additional messages.
District b of B has just received a Let-us-Merge message from a along merge link (a, b). From the message, b finds out that level(A) > level(B); thus, it postpones the request. In the meanwhile, the downtown D(B) chooses (a,b) as its merge link. Explain why this situation will never occur.
Time Costs. Show that protocol Mega-Merger uses at most O(n log n) ideal time units.
Find a way to avoid notification of termination by the downtown of the megacity in protocol Mega-Merger (Hint: Show that by the time the downtown understands that the mega-merger is completed, all other districts already know that their execution of the protocol is terminated).
Consider a merger message from city A arriving at neighbouring city B along merge link (a,b) in protocol Mega-Merger. Prove that if we reverse the logical direction of the links on the path from D(A) to the exit point a and direct toward B the merge link, the union of A and B will be rooted in the
Show how to elect a leader in the double cube Cn 1, 2, 4, 8..., 2 log n with O(n) messages.
Prove that in chordal ring Ctn electing a leader requires at leastn + n t log n t messages in the worst case (Hint: Reduce the problem to that of electing a leader on a ring of size n/t ).
Show how to elect a leader in the chordal ring Cn 1, 2, 3, 4..., twith On + n t log n t messages.
() Show how to elect a leader in a complete network with O(n log n) messages in the worst case but only O(n) on the average.
() Determine the average message costs of protocol Kelect-Stages.
() Consider using the ring protocol Alternate instead of Stages in Kelect. Determine what will be the cost in this case.
Prove that in the strategy CompleteElect outlined in Section 3.6.1, the territories of any two candidates in the same stage have no nodes in common.
() Prove that it is possible to elect a leader in a hypercube using O(n) messages with any sense of direction (Hint: Use long messages).
Show that the time complexity of Protocol HyperFlood is O(log3 N).
Show how to broadcast from a corner of a mesh dimensions a × b with less than 2n messages.
Prove that even if the entities known, aveA(I |n known) ≥ 1 2n log n for any election protocol A for unidirectional rings.
Write the rules of Protocol MinMax+ without assuming message ordering.
Prove that in bidirectional rings, aveA(I ) ≥ 1 2 nHn for any election protocol A.
Prove that even if the entities known, aveA(I |nknown) ≥ ( 1 4− )n log n for any election protocol A for unidirectional rings.
Prove that in protocol MinMax+, if an envelope with value v reaches an even stage i + 1, it saves at least Fi messages in stage i with respect to MinMax (Hint: Use Property 3.3.1.).
Prove Property 3.3.1.
Write the rules of Protocol MinMax+assuming message ordering.
For protocol MinMax, consider the configuration depicted in Figure 3.32. Prove that once envelope (11, 3) reaches the defeated node z, z can determine that 11 will survive this stage.
Show that the worst case number of steps of protocol MinMax is achievable.
Modify protocol MinMax so that it does not require Message Ordering. Implement your modification and throughly test your implementation.
Determine the exact complexity ofWake-Up in a mesh of dimensions a × b.
In Protocol ElectMesh, in the first stage of the election process, if an interior node receives an election message, it will reply to the sender “I am in the interior,” so that no subsequent election messages are sent to it. Explain why it is possible to achieve the same goal without sending
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