Design And Analysis Of Distributed Algorithms 1st Edition Nicola Santoro - Solutions

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For any given n-dimensional hypercube where n > 3 find a path in which each (ordered) pair of nodes in the cube appears at least once. That is, this path consists of two consecutive Hamiltonian paths.Extend Dijkstra’s k-state self-stabilization algorithm on a ring from one privilege to two
Show the state transition sequence for the following system with n = 3 and k = 5 using Dijkstra’s self-stabilizing algorithm. Assume that P 0= 3, P 1= 1, and P 2= 4.
Design a modified ping-pong type of resilient algorithm where three tokens (t 1, t 2and t 3) are used on a logical ring consisting of n process. Your algorithm should tolerate a loss of up to 2 tokens. State explicitly how your algorithm detects three different situations: (1) no token is lost,(2)
Provide a revised Misra and Chandy’s ping-pong algorithm in which the ping and the pong are circulated in opposite directions. Compare the performance and other related issues of these two algorithms.
Assume that up to two processes can enter a critical section simultaneously. Provide possible extensions to Lamport’s algorithm and the simple token-ring-based algorithm.
Show that in Ricart and Agrawala’s algorithm the CS is granted according to the increasing order of timestamps associated with requests. Explain why timestamps are not used in Maekawa’s algorithm and then modify Maekawa’s algorithm such that the grant signal is sent based on timestamps of
Modify or enhance Lamport’s mutual exclusive algorithm for the following system: Each pair of PE has only one communication path (consisting of a sequence of FIFO channels) with a fixed propagation delay. Different paths may have different delays and the delay of each path is not known when the
Use a four-process example to illustrate the snapshot of global states. It is assumed that four processes (nodes) are fully connected by bidirectional links and only one process initiates the snapshot process.
Use the extended Petri net to model the system in Figure 3.7.
Extend Lamport’s mutual exclusion algorithm for the cases where messages received at each process may not be in the same order as they are sent. We assume that the message transmission delay is bounded by time d.
Express Lamport’s mutual exclusion algorithm using DCDL.
Assume that all LC’s are initialized zero and the d’s for P a, P b, and P c, are 1, 2, 3, respectively. Does condition a Õ b Þ LC(a) < LC(b) still hold? For any other set of d’s? and why?
Provide linear logical clocks for all the events in the system given in Problem
Define a total order (Þ) events in a distributed system based on the vector logical clock associated with each event. Note that for two events a and b: if a Õb, then a Þ b.
Consider a system where processes can be dynamically created or terminated. A process can generate a new process. For example, P 1generates both P 2and P 3. Modify the happened-before relation and the linear logical clock scheme for events in such a dynamic set of processes.
Using DCDL to write a program that replaces a * b by a b and a**b by a ¯b, where a and b are any characters other than *. For example, if a 1a 2 * a 3 * *a 4 ***a 5is the input string then a1 a2 a 3 ¯ a 4 *** a 5will be the output string.
G is a sequence of integers defined by the recurrence G i= G i-1+ G i-3 for i > 1, with initial values G 0= 0, G 1= 1, and G 2= 1.Provide a DCDL implementation of G iand use one process for each G i.
A common problem that occurs when entering text is that words are mistakenly repeated, for example,“This text contains a repeating repeating word”.Design a program that detects such repetitions.
Each element is colored either red, white, or blue. Write a program to permute the elements so that all the red elements are first and all the blue ones last. The color of an element may be tested with Boolean expressions red(b[i]), white(b[i]) and blue(b[i]) which return the obvious values. The
(The Dutch National Flag by E. W. Dijkstra) Given is an array b[1 : n] for fixed n ³
(The Welfare Crook by W. Feijen) Suppose we have three long magnetic tapes each containing a list of names in alphabetical order. The first list contains the names of people working at IBM Yorktown, the second the names of students at Columbia University and the third the names of all people on
A plateau of the array is a sequence of equal values. Write a DCDL program to store in variable p the length of the longest plateau of b[1 : n].
(The Plateau Problem by D. Gries) Given is a fixed, ordered array b[1 : n], where n >
Check whether the array a[1 : n] has been partitioned about x such that
Sum the first k elements in an array a[1 : n]. Terminate the summation process if the addition of the (k + 1)th element will cause the partial p = to exceed the value of sum s =.
Provide a DCDL implementation of the producer-consumer problem. Three processes(producer, consumer, buffer) are used with n the size of the buffer. Extend your algorithm to the case in which there is more than one consumer (still with one producer) and show that the fairness among consumers is
Discuss the relative advantages and disadvantages in using replication and partition in a database system.
Calculate (a) node degree, (b) diameter, (c) bisection width, and (d) the number of links for an n × n 2-d mesh, an n × n 2-d torus, and an n-dimensional hypercube.
Discuss different ways to construct a logically shared memory system on the same physical system structure discussed in Problem 5.
Consider a computer structure with both shared memory and local memory and discuss different ways to construct a logically distributed memory system.
What is the advantage of using two different views (physical and logical) of a distributed system?
Why is a standard like DCE important?
Give several practical examples on the use of distributed systems.
Use your own words to explain the differences between distributed systems, multiprocessors, and network systems.
Determine the number of messages of protocol Speed if the waiting function is f (v) = cv, for an integer c > 2.
Determine the number of messages of protocol Speed if the waiting function is f (v) = vc, for an integer c > 1.
Modify protocol Speed so that even if the entities do not start simultaneously, a leader is elected with O(n) messages.
Prove that Protocol Speed requires 2in time units.
Modify protocol C2 so that it communicates any integer i, positive or negative, transmitting 2 bits and O(|i|) time units.
Construct a protocol R2 that communicates any positive integer I transmitting 2 bits and only 2 + I 4 time units.
Consider protocol TwoBits when each packet containsc > 1 bits. Use the content of the packets to convey information about the value i to be communicated.Determine the time costs that can be achieved.
Construct a protocol R3 that communicates any positive integer I transmitting 3 bits and only √I + 3 time units.
Consider a system where packets containc > 1 bits. Modify protocol R3 using the content of the packets so as to reduce the time costs. Determine the amount of savings that can be achieved.
Prove that the communicator described in Section 6.2.1 uses at most O(i 1k ) time units.
Use the content of the transmitted bits so as to reduce the time costs of the communicator described in Section 6.2.1. Show how a time cost of at most (k − 1)(I/4)1 k−1 + k clock ticks can be achieved.
Prove that communicator Orderk uses f (I, k) + k + 1 time to communicate I , where f (I, k) is the smallest integer t such that t+k k
Prove that communicator Orderk+ uses g(I, k) + k + 1 time to communicate I , where g(I, k) is the smallest integer t such that I2k+1 t+k k
Prove that ω(t, k) = b (2 + 2)
Prove that any protocol using k + 1 corruptible bits to communicate values from U requiresadditional time on the average. (F(1014) (101 - i (i + 1 + 1)) + i+k- 0
Prove that protocol Orderk is average-case optimal.
Let β(t, k) denote the size of the largest set for which the two-party communication problem can always be solved transmitting k + 1 reliable bits and at most t additional time units. Prove that B(t, k) = 2k+1 t+k k
Let r(w, k) denote the number of additional time units needed in the worst case to solve the two-party communication problem for Zw with k + 1 bits, in absence of corruptions. Let β(t, k) be as in Exercise 6.6.17, and let g(w, k) be the smallest integer t such that β(t, k) ≥ x. Prove that r(w,
Let β(t, k) be as in Exercise 6.6.17, and let g(w, k) be the smallest integer t such that β(t, k) ≥ x. Prove that r(w, k) = g(w, k). Prove that any solution protocol using k + 1 reliable bits to communicate values from U requires i+k-1 (i ) 2k+1 k-1 g(U, k)+k k time on the average.
Prove that protocol Order+k is average-case optimal.
() A communicator is monotonically increasing if, whenever I > J, the communication sequence for I is lexicographically smaller than that for J . Communicators Orderk are optimal and corruption tolerant; however, they are not monotonically increasing fork > 2; thus, they cannot be used in pipeline
Communicators Order+k are optimal but not monotonically increasing fork > 2; thus, they can not be used in pipeline for computing the minimum.Determine a class MonotoneOrder+k of optimal communicators that are monotonically increasing.
Write a protocol for finding the largest value in a chain using the 2-bit communicator and pipeline. Prove its correctness.
Minimum-Finding in Pipeline. Write a protocol for finding the smallest value in a chain using the 2-bit communicator and pipeline. Prove its correctness.Determine its costs.
Sum-Finding in Pipeline. Write a protocol for finding the sum of all the values in a chain using the 2-bit communicator and pipeline. Prove its correctness. Determine its costs.
Protocol SynchStages is the transformation of Stages using communicator TwoBits. Add pipeline to this protocol to convey information from a candidate to a neighboring one. Prove its correctness. Analyze its costs; in particular, determine the reduction in time with respect to the nonpipelined
Modify protocolWait so that it finds the minimum value only among the initiators.
Determine the smallest waiting function that allows protocol Wait to work correctly without simultaneous initiation: (a) in a unidirectional ring; (b) in a bidirectional ring.
Determine the smallest waiting function that allows protocol Wait to work correctly with simultaneous initiation: (1) in a a × b mesh; (2) in a a × b torus; (3) in a k-dimensional hypercube; (4) in a complete network.
Determine the smallest waiting function that allows protocol Wait to work correctly without simultaneous initiation: (1) in a a × b mesh; (2) in a a × b torus; (3) in a k-dimensional hypercube.
Prove that protocol Wait would work even if a quantity n ≥ n is used instead of n.
Determine under what conditions protocol Wait would work if a quantity n > n is used instead of n in the waiting function.
Assuming distinct initial values, characterize what would happen to protocolWait in a ring network if each entity x uses 2id(x)2 as its waiting function.In particular, determine under what conditions the protocol would certainly work.
Under the conditions of Exercise 6.6.33, show how all the entities can efficiently detect whether the protocol does not work.
Determine the cost of computing the AND of all input values in a synchronous ring of known size n using protocol Waiting.
Describe how to efficiently use protocol Wait to compute the OR of the input values in a synchronous ring of known size n. Determine its cost.
Modify protocol Symmetry so that it works efficiently in a bidirectional square torus of known dimension. Determine its exact costs.
Modify protocol Symmetry so that it works efficiently in a unidirectional square torus of known dimension. Determine its costs.
Prove that with simultaneous initiation, protocol Symmetry can be modified so as to work correctly in every network of known girth. (Hint: Use the girth instead of n in the waiting function.)
Determine the complexity of protocol Symmetry if we use in random selection criteria b = n and choose each value with the same probability 1 n .
Modify protocol Decide so as to compute the OR of the input values in a synchronous ring of known size n. Prove its correctness and determine its cost.
Write protocol Guess and implement it; throughly test your implementation.
Show how to find imin with k overestimates using q = k M1/k questions.
Show how we can always win the guessing game in an interval of size 2q − 1 with q question if they are all allowed to be overestimates.
Show how to obtain a unique solution to the recurrence relation of expression 6.33.
Determine a function g to bound imin so that the total time for finding with k overestimates is at most 2 h( imin, k) − 1.
Modify subprotocol Decide(p) so that it willwork in every network, regardless of its topology. Assume that an upperbound on the diameter of the network is known a priori. Prove its correctness.
Modify subprotocol Decide(p) so that protocol Guess works correctly even if the entities do not start simultaneously.
Prove that, in DoubleWait, if x is being “fooled,” then both the“Wait1” and the “Wait2” message it receives are sent by the same entity.
Let the entities start the j th iteration of DoubleWait within n − 1 time units from each other. Prove that the entity with the smallest value becomes leader and all other will become defeated in that iteration.
Let the entities start the j th iteration of DoubleWait within n − 1 time units from each other. Prove that if an entity x becomes leader in this iteration, then g(j ) ≥ n > g(j − 1).
Let the entities start the j th iteration of DoubleWait within n − 1 time units from each other. Prove that if g(j ) < n, then all entities start the (j + 1)th iteration within n − 1 time units from each other.
Prove that the time used by protocol DoubleWait, with the choices of f and h specified by Expressions 6.40 and 6.41, is at most 2(n − 1) + (4 imin + 2)j j=1 g(j ).
Consider protocol DoubleWait, where f and h are as in Expressions 6.40 and 6.41, and g is superincreasing. Prove that the time is at most 2(n − 1) +(8 imin + 2) g(g−1(n)).
Consider protocol DoubleWait, where f and h are as in Expressions 6.40 and 6.41. Determine the number of bits if the time is O(n log n i).
() Determine whether or not there is a choice of g that makes DoubleWait more efficient than SynchStages in both time and bits.
Let L = (a1, b1), . . . , (ak, bk) be the k pairs of distinct labels ai, bi ∈ {1, . . . , n}. Consider now a complete network of n nodes; in this network, select 2k + 1 nodes x0, x1, , . . . , x2k. Show that it is always possible 1. to label the links between these nodes only with pairs from L
Consider exactly the same question as in Exercise 6.6.57, where, however, n is even and exactly one pair in L, say (a1, b1) is composed of identical labels, i.e., a1 = b1.
Prove that in protocol MaxWave, the largest of the local clock values (when the execution starts) will reach (properly increased) every entity, and each entity will set its local clock to such a (properly increased) time value.
Consider protocol MaxWave when the entities do not start necessarily at the same time, and let d be known. Let t be the (global) time the first entities start the execution of the protocol and let t (x) ≥ t be the global time when x starts.Modify the protocol so that (eventhough x does not know t
Determine the message cost of protocol MaxWavea. in a unidirectional ring,b. in a bidirectional ring.You may assume that n is known.
Determine the message cost of protocol MaxWave in a kdimensional hypercube.
Determine the worst-case and average-case message costs of protocol MaxWave in a tree network.
Let, in protocolMaxWave, each entity reset its local clock to 0 when it starts the protocol. Prove that in this way, the maximum value transmitted is at most 2d.
Consider the unison protocol MinWave where instead of setting the clocks to and propagating the largest value, we set the clock to and propagate the smallest value. Prove correctness, termination, and costs of protocolMinWave.
Determine the bit and time costs of protocol MaxWave if the content of a message is communicated using the 2-bit communicator.

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Design And Analysis Of Distributed Algorithms 1st Edition Nicola Santoro - Solutions

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