C. Busch and M. Mavronicolas. Impossibility results for weak threshold networks. Information Processing Letters, 63(2):85-90, July 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....by Aspnes et al. 4]asanovel class of distributed data structures that provide highly concurrent, low contention solutions to a variety of synchronization problems. Balancing networks attract a lot of interest and attention due to their nice performance and scalability properties (see, e.g. [1, 2, 3, 5, 6, 7, 8, 14, 15, 16, 18, 19, 20, 21, 22, 23]) A balancing network is constructed from elementary switches with # input wires and # output wires, called (## #) balancers. As illustrated in Figure, a (## #) balancer accepts a stream of tokens on its # input wires. The # th token to enter the balancer leaves on output wire # mod # (where # ....

....[5] considered the threshold property [4] and they showed that this property is also preserved by the introduction of antitokens. Furthermore, they showed that for regular balancing networks, where the number of input and output wires is the same for each balancer, the weak threshold property [7] is preserved too. ### ############ A fundamental question that was left open by the results in [2, 5, 21] is to formally characterize all properties of balancing networks that are preserved under the introduction of antitokens. Suchcharacterization would be used for identifying properties that ....

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C. Busch and M. Mavronicolas, \Impossibility Results for Weak Threshold Networks," Information Processing letters, Vol. 63, No. 2, pp. 85-90, July 1997.

....w designated output wires. Balancing networks in general, and counting and smoothing networks in particular, have recently attracted a lot of interest and attention. Their study has focused on efficient constructions [1, 2, 4, 8, 13, 19, 20, 25] combinatorial properties and impossibility results [1, 9, 10, 11, 12, 24, 29], linearizability properties [22, 27, 28] analysis of their performance by both theoretical and experimental means [2, 4, 8, 13, 15, 16, 18, 20, 21, 22, 23, 30, 31, 32] and applications to solving decision problems [5] The principal motivation for introducing counting and smoothing networks ....

C. Busch and M. Mavronicolas, "Impossibility Results for Weak Threshold Networks," Information Processing Letters, Vol. 63, No. 2, pp. 85--90, July 1997.

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C. Busch and M. Mavronicolas. Impossibility results for weak threshold networks. Information Processing Letters, 63(2):85-90, July 1997.

....seems to appear that the randomized balancers need to somehow remember the entire history of the random permutations in order for antitokens (resp. tokens) to trace back the paths of tokens (resp. antitokens) Other interesting properties of balancing networks include the threshold property [4, 7] and the weak threshold property [7] outlined below. Let x be the input vector and the corresponding output vector of a balancing network that has any of these properties. For the threshold property, we require that y 0 = dkx k 1 =w out e, while for the weak threshold property, we require ....

....need to somehow remember the entire history of the random permutations in order for antitokens (resp. tokens) to trace back the paths of tokens (resp. antitokens) Other interesting properties of balancing networks include the threshold property [4, 7] and the weak threshold property [7], outlined below. Let x be the input vector and the corresponding output vector of a balancing network that has any of these properties. For the threshold property, we require that y 0 = dkx k 1 =w out e, while for the weak threshold property, we require that there is some output index j, ....

C. Busch and M. Mavronicolas. Impossibility results for weak threshold networks. Information Processing Letters, 63(2):85-90, July 1997.

....path. Balancing networks can be used to construct counting networks [3] which are useful for constructing shared exact counters, and smoothing networks [3] which are useful for load balancing. Balancing networks can also be used to construct threshold networks [3] and weak threshold networks [4], which provide highlyconcurrent, low contention implementations of threshold counters. Each of these classes of networks supports some form of Increment operation, implemented by passing a token through the network. Threshold networks are interesting because there are constructions of them with ....

....#w out # ##w out #1### 1 #w out #. A straightforward calculation reveals that y w out #1 # ##y #w out # # 1 #w out #. Hence, y #w out # is a threshold vector. It follows that the step property is a subset of the threshold property. Say that a vector y #w out # is a weak threshold vector [4] if there is some output index j, possibly j ## w out # 1, such that y j # ##y #w out # # 1 #w out #.Theweak threshold property is the set of all weak threshold vectors y #w out # . As for the case of threshold vectors, it is straightforward to see that adding a constant vector to a weak ....

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C. Busch and M. Mavronicolas, "Impossibility Results for Weak Threshold Networks," Information Processing Letters, Vol. 63, No. 2, pp. 85--90, July 1997.

....the step property: the exiting tokens are distributed uniformly among the output wires and any excess tokens appear on the upper wires. On smoothing networks [1, 4] the output tokens satisfy the # smoothing property: the sum of tokens on anytwo output wires di er by at most #. Onthreshold networks [4, 7] the output sequence satis es the threshold property: the numberoftokens on the bottom wire is increased by one for every bunchof# tokens, where # is the number of output wires of the network. Based on balancing networks, simple and elegant algorithms have been developed to solveavariety of ....

....####### # ####### ## ### ##### Fix any boundedness property #. Consider any balancing network # : # ## ## # # # ##### # such that # ##### # has the boundedness property # whenever # ## ## # is a non negative input vector. Then, # has the boundedness property #. The threshold property [4, 7] is the set of all vectors # ### , such that for the entry # ### of # ### , it holds # ### = ### ### # # ###. It has been observed in [5] that the threshold property is not a boundedness property in all non trivial cases (where # # 2) Thus, Theorem 1 does not apply a fortiori to this ....

C. Busch and M. Mavronicolas, \Impossibility Results for Weak Threshold Networks, " ########### ########## ######## Vol. 63, No. 2, pp. 85-90, July 1997.

....seems to appear that the randomized balancers need to somehow remember the entire history of the random permutations in order for antitokens (resp. tokens) to trace back the paths of tokens (resp. antitokens) Other interesting properties of balancing networks include the threshold property [4, 7] and the weak threshold property [7] outlined below. Let x (w in ) and y (wout ) be the input vector and the corresponding output vector of a balancing network that has any of these properties. For the threshold property, we require that y 0 = dkx (w in ) k 1 =w out e, while for the weak ....

....need to somehow remember the entire history of the random permutations in order for antitokens (resp. tokens) to trace back the paths of tokens (resp. antitokens) Other interesting properties of balancing networks include the threshold property [4, 7] and the weak threshold property [7], outlined below. Let x (w in ) and y (wout ) be the input vector and the corresponding output vector of a balancing network that has any of these properties. For the threshold property, we require that y 0 = dkx (w in ) k 1 =w out e, while for the weak threshold property, we require that ....

C. Busch and M. Mavronicolas. Impossibility results for weak threshold networks. Information Processing Letters, 63(2):85-90, July 1997.

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