Labeling Schemes for Tree Representation

Cohen, Reuven; Fraigniaud, Pierre; Ilcinkas, David; Korman, Amos; Peleg, David

doi:10.1007/11603771_2

Cited by 9 publications

(5 citation statements)

References 13 publications

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“…One can rephrase many recent results of the literature in the framework of advising schemes. For instance, a (⌈log n⌉, 0)-advising scheme, with average advice length O(log log n) bits, is described in [3] for computing a spanning tree. It is also easy to extract a (2, 1)-advising scheme for spanning tree (with average advice length 4 3 ) from the proof of the main result in [4].…”

Section: Related Workmentioning

confidence: 99%

Local MST Computation with Short Advice

2010

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We use the recently introduced advising scheme framework for measuring the difficulty of locally distributively computing a Minimum Spanning Tree (MST). An (m, t)-advising scheme for a distributed problem P is a way, for every possible input I of P, to provide an "advice" (i.e., a bit string) about I to each node so that: (1) the maximum size of the advices is at most m bits, and (2) the problem P can be solved distributively in at most t rounds using the advices as inputs. In case of MST, the output returned by each node of a weighted graph G is the edge leading to its parent in some rooted MST T of G. Clearly, there is a trivial (⌈log n⌉, 0)advising scheme for MST (each node is given the local port number of the edge leading to the root of some MST T), and it is known that any (0, t)-advising scheme satisfies t ≥Ω(√ n). Our main result is the construction of an (O(1), O(log n))-advising scheme for MST. That is, by only giving a constant number of bits of advice to each node, one can decrease exponentially the distributed computation time of MST in arbitrary graph, compared to algorithms dealing with the problem in absence of any a priori information. We also consider the average size of the advices. On the one hand, we show that any (m, 0)-advising scheme for MST gives advices of average size Ω(log n). On the other hand we design an (m, 1)-advising scheme for MST with advices of constant average size, that is one round is enough to decrease the average size of the advices from log n to constant.

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Section: Related Workmentioning

confidence: 99%

Local MST Computation with Short Advice

2010

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“…For instance, a (⌈log n⌉, 0)-advising scheme, with average advice length O(log log n) bits, is described in [3] for computing a spanning tree. It is also easy to extract a (2, 1)-advising scheme for spanning tree (with average advice length 4 3 ) from the proof of the main result in [4].…”

Section: Related Workmentioning

confidence: 99%

“…The notion of index in introduced in the description of this simple scheme for the sake of clarity, since it will be used later in the design our more efficient schemes 3. The node IDs are used to break symmetry; In the anonymous ring with all edge-weights the same, there is no way to break symmetry.…”

mentioning

confidence: 99%

Local MST computation with short advice

Fraigniaud

Korman

Lebhar

2007

Proceedings of the Nineteenth Annual ACM Symposium on Parallel Algorithms and Architectures

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We use the recently introduced advising scheme framework for measuring the difficulty of locally distributively computing a Minimum Spanning Tree (MST). An (m, t)-advising scheme for a distributed problem P is a way, for every possible input I of P, to provide an "advice" (i.e., a bit string) about I to each node so that: (1) the maximum size of the advices is at most m bits, and (2) the problem P can be solved distributively in at most t rounds using the advices as inputs. In case of MST, the output returned by each node of a weighted graph G is the edge leading to its parent in some rooted MST T of G. Clearly, there is a trivial (⌈log n⌉, 0)-advising scheme for MST (each node is given the local port number of the edge leading to the root of some MST T ), and it is known that any (0, t)-advising scheme satisfies t ≥Ω( √ n). Our main result is the construction of an (O(1), O(log n))-advising scheme for MST. That is, by only giving a constant number of bits of advice to each node, one can decrease exponentially the distributed computation time of MST in arbitrary graph, compared to algorithms dealing with the problem in absence of any a priori information. We also consider the average size of the advices. On the one hand, we show that any (m, 0)-advising scheme for MST gives advices of average size Ω(log n). On the other hand we design an (m, 1)-advising scheme for MST with advices of constant average size, that is one round is enough to decrease the average size of the advices from log n to constant.

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“…The construction is a variant of Kruskal's minimum-weight spanning tree (MST) algorithm (cf. [6]), similar to the one in [5]. It maintains a collection of trees.…”

Section: Theorem 3 There Exists An Oracle Of Size O(n) Permitting Thmentioning

confidence: 99%

Oracle size

Fraigniaud

Ilcinkas

Pełc

2006

Proceedings of the Twenty-Fifth Annual ACM Symposium on Principles of Distributed Computing

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We study the problem of the amount of knowledge about a communication network that must be given to its nodes in order to efficiently disseminate information. While previous results about communication in networks used particular partial information available to nodes, such as the knowledge of the neighborhood or the knowledge of the network topology within some radius, our approach is quantitative: we investigate the minimum total number of bits of information (minimum oracle size) that has to be available to nodes in order to perform efficient communication.It turns out that the minimum oracle size for which a distributed task can be accomplished efficiently, can serve as a measure of the difficulty of this task. We use this measure to make a quantitative distinction between the difficulty of two apparently similar fundamental communication primitives: the broadcast and the wakeup. In both of them a distinguished node, called the source, has a message, which has to be transmitted to all other nodes of the network. In the wakeup, only nodes that already got the source message (i.e., are awake) can send messages to their neighbors, thus waking them up. In the broadcast, all nodes can send control messages even before getting the source message, thus potentially facilitating its future dissemination. In both cases we are interested in accomplishing the communication task with optimal message complexity, i.e., using a number of messages linear in the number of nodes.We show that the minimum oracle size permitting the * Supported by the projects PairAPair of the ACI Masses de Données, and FRAGILE of the ACI Sécurité Informatique.wakeup with a linear number of messages in a n-node network, is Θ(n log n), while the broadcast with a linear number of messages can be achieved with an oracle of size O(n). We also show that the latter oracle size is almost optimal: no oracle of size o(n) can permit to broadcast with a linear number of messages. Thus an efficient wakeup requires strictly more information about the network than an efficient broadcast.

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Labeling Schemes for Tree Representation

Cited by 9 publications

References 13 publications

Local MST Computation with Short Advice

Local MST Computation with Short Advice

Local MST computation with short advice

Oracle size

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