Nearest Common Ancestors: A Survey and a New Algorithm for a Distributed Environment

Alstrup, Stephen; Gavoille, Cyril; Kaplan, Haim; Rauhe, Theis

doi:10.1007/s00224-004-1155-5

Cited by 109 publications

(199 citation statements)

References 60 publications

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“…2001;Abiteboul at al. 2006;Kaplan et al 2002], nearest common ancestor in trees [Alstrup et al 2004;Peleg 2000] and various other tree functions, such as center and separation level [Peleg 2000]. See [Gavoille and Peleg 2003] for a survey on static labeling schemes.…”

Section: Problem and Motivationmentioning

confidence: 99%

Labeling Schemes for Vertex Connectivity

Korman

2007

Automata, Languages and Programming

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This paper studies labeling schemes for the vertex connectivity function on general graphs. We consider the problem of assigning short labels to the nodes of any n-node graph is such a way that given the labels of any two nodes u and v, one can decide whether u and v are k-vertex connected in G, i.e., whether there exist k vertex disjoint paths connecting u and v. The paper establishes an upper bound of k 2 log n on the number of bits used in a label. The best previous upper bound for the label size of such a labeling scheme is 2 k log n.

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Section: Problem and Motivationmentioning

confidence: 99%

Labeling Schemes for Vertex Connectivity

Korman

2007

Automata, Languages and Programming

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“…For example, between any two vertices in a tree, there is a unique simple path connecting them, and the unique path can be found in a local manner by first traversing up the tree towards the root, and traversing down the tree at the lowest common ancestor. There is a simple labeling scheme such that one can use routing table of O(log n) bits at each node to support routing on a tree [3].…”

Section: Introductionmentioning

confidence: 99%

Resilient and Low Stretch Routing through Embedding into Tree Metrics

Gao

Zhou

2011

Lecture Notes in Computer Science

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Abstract. Given a network, the simplest routing scheme is probably routing on a spanning tree. This method however does not provide good stretch -the route between two nodes can be much longer than their shortest distance, nor does it give good resilience -one node failure may disconnect quadratically many pairs. In this paper we use two trees to achieve both constant stretch and good resilience. Given a metric (e.g., as the shortest path metric of a given communication network), we build two hierarchical well-separated trees using randomization such that for any two nodes u, v, the shorter path of the two paths in the two respective trees gives a constant stretch of the metric distance of u, v, and the removal of any node only disconnect the routes between O(1/n) fraction of all pairs. Both bounds are in expectation and hold true as long as the metric follows certain geometric growth rate (the number of nodes within distance r is a polynomial function of r), which holds for many realistic network settings such as wireless ad hoc networks and Internet backbone graphs. The algorithms have been implemented and tested on real data.

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“…We note that all the schemes we present use asymptotically optimal labels. Most of the 1-query labeling schemes obtained in this section use the label-NCA labeling scheme π N CA = M N CA , D N CA described in [5]. Given an n-node, the marker algorithm M N CA assigns each vertex v a distinct label M N CA (v) using O(log n) bits.…”

Section: Labeling Schemes With One Querymentioning

confidence: 99%

“…That is, recall (Section 3) that our 1-query labeling schemes utilize components that are schemes in the traditional model (with no queries). That is, some utilize π N CA , the label-NCA labeling scheme of [5] and some utilize π des , the routing scheme of [24]. Hence, one can first convert these components to the dynamic setting.…”

Section: Lemma 35mentioning

confidence: 99%

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Labeling Schemes with Queries

Korman

Kutten

Structural Information and Communication Complexity

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We study the question of "how robust are the known lower bounds of labeling schemes when one increases the number of consulted labels". Let f be a function on pairs of vertices. An f -labeling scheme for a family of graphs F labels the vertices of all graphs in F such that for every graph G ∈ F and every two vertices u, v ∈ G, the value f (u, v) can be inferred by merely inspecting the labels of u and v.This paper introduces a natural generalization: the notion of f -labeling schemes with queries, in which the value f (u, v) can be inferred by inspecting not only the labels of u and v but possibly the labels of some additional vertices. We show that inspecting the label of a single additional vertex (one query) enables us to reduce the label size of many labeling schemes significantly. In particular, we show that to support the distance function on n-node trees as well as the flow function on n-node general graphs, O(log n + log W )-bit labels are sufficient and necessary, where W is the maximum (integral) capacity of an edge. We note that it was shown that any labeling scheme (without queries) supporting either the flow function on general graphs or the distance function on trees, must have label size Ω(log 2 n + log n log W ). Using a single query, we also show a routing labeling scheme in the fixed-port model using O(log n)-bit labels, while the lower bound on the label size of any such routing labeling scheme (without queries) is Ω(log 2 n/log log n). We note that all our labeling schemes with queries have asymptotically optimal label size.We then show several extensions. In particular, we show that our labeling schemes with queries on trees can be extended to the dynamic scenario by some adaptations of known model translation methods. Second, we show that the study of the queries model can help with the traditional model too. That is, using ideas from our routing labeling scheme with one query, we show how to construct a 3-approximation routing scheme without queries in the fixed-port model with Θ(log n)-bit labels. Finally, we turn to a non-distributed environment and first translate known results on finding NCA to supporting distance queries. We then use the above to show that one can preprocess a general weighted graph using almost linear space so that flow queries can be answered in almost constant time.

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Nearest Common Ancestors: A Survey and a New Algorithm for a Distributed Environment

Cited by 109 publications

References 60 publications

Labeling Schemes for Vertex Connectivity

Labeling Schemes for Vertex Connectivity

Resilient and Low Stretch Routing through Embedding into Tree Metrics

Labeling Schemes with Queries

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