Dynamic Representations of Sparse Graphs

Brodal, Gerth Stølting; Fagerberg, Rolf

doi:10.1007/3-540-48447-7_34

Cited by 46 publications

(65 citation statements)

References 13 publications

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“…The proof is by the induction on i. For i = 1 it is actually the result of G. Brodal and R. Fagerberg [1999]. Now we move to the induction step.…”

Section: Inserting An Edgementioning

confidence: 95%

“…Lazy Reorienting Scheme We use the result of G. Brodal and R. Fagerberg [1999]. They show how to keep bounded outdegree in an orientation of a graph with bounded arboricity.…”

Section: Inserting An Edgementioning

confidence: 99%

“…We will refer to this method as lazy reorienting scheme, as reorientation of edges is done as late as possible. They show in [Brodal and Fagerberg 1999] that for graphs of bounded arboricity this simple approach guarantees that amortized number of edge reorientations is O(1) for one insertion and O(log n) for one deletion in a sequence of operations.…”

Section: Inserting An Edgementioning

confidence: 99%

See 2 more Smart Citations

Oracles for bounded-length shortest paths in planar graphs

Kowalik

Kurowski

2006

ACM Trans. Algorithms

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We present a new approach for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E) one can build in O(|V |) time a data structure, which allows to check in O(1) time whether two given vertices are at distance at most k in G and if so a shortest path between them is returned. Graph G can be undirected as well as directed.Our data structure works in fully dynamic environment. It can be updated in O(1) time after removing an edge or a vertex while updating after an edge insertion takes polylogarithmic amortized time. Besides deleting elements one can also disable ones for some time. It is motivated by a practical situation where nodes or links of a network may be temporarily out of service.Our results can be easily generalized to other wide classes of graphs -for instance we can take any minor-closed family of graphs.

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“…The proof is by the induction on i. For i = 1 it is actually the result of G. Brodal and R. Fagerberg [1999]. Now we move to the induction step.…”

Section: Inserting An Edgementioning

confidence: 95%

“…Lazy Reorienting Scheme We use the result of G. Brodal and R. Fagerberg [1999]. They show how to keep bounded outdegree in an orientation of a graph with bounded arboricity.…”

Section: Inserting An Edgementioning

confidence: 99%

Section: Inserting An Edgementioning

confidence: 99%

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Oracles for bounded-length shortest paths in planar graphs

Kowalik

Kurowski

2006

ACM Trans. Algorithms

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show abstract

“…Also, since c(v, t) = 0 and c(t, v) = 0 we get In order to analyze our approximation algorithm we will use the following lemma. The lemma and its proof is analogous to Lemma 2 in [4] (however, our Lemma 2 implies Lemma 2 in [4] and not vice-versa, hence we include the proof for completeness). Proof.…”

Section: Reduction To a Flow Problemmentioning

confidence: 99%

“…Complexity of many graph algorithms depends heavily on the arboricity of the input graph -see e.g. [6,4,8].…”

Section: Introductionmentioning

confidence: 99%

Approximation Scheme for Lowest Outdegree Orientation and Graph Density Measures

Kowalik

2006

Algorithms and Computation

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Abstract. We deal with the problem of finding such an orientation of a given graph that the largest number of edges leaving a vertex (called the outdegree of the orientation) is small. For any ε ∈ (0, 1) we show anÕ(|E(G)|/ε) time algorithm 3 which finds an orientation of an input graph G with outdegree at most (1 + ε)d * , where d * is the maximum density of a subgraph of G. It is known that the optimal value of orientation outdegree is d * . Our algorithm has applications in constructing labeling schemes, introduced by Kannan et al. in [18] and in approximating such graph density measures as arboricity, pseudoarboricity and maximum density. Our results improve over the previous, 2-approximation algorithms by Aichholzer et al.

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A Dynamic Implicit Adjacency Labelling Scheme for Line Graphs

Morgan

2005

Lecture Notes in Computer Science

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Dynamic Representations of Sparse Graphs

Cited by 46 publications

References 13 publications

Oracles for bounded-length shortest paths in planar graphs

Oracles for bounded-length shortest paths in planar graphs

Approximation Scheme for Lowest Outdegree Orientation and Graph Density Measures

A Dynamic Implicit Adjacency Labelling Scheme for Line Graphs

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