## Lower Bounds for Fully Dynamic Connectivity Problems in Graphs (1998)

Citations: | 32 - 5 self |

### BibTeX

@MISC{Henzinger98lowerbounds,

author = {M. R. Henzinger and M. L. Fredman},

title = {Lower Bounds for Fully Dynamic Connectivity Problems in Graphs },

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

We prove lower bounds on the complexity of maintaining fully dynamic k-edge or k-vertex connectivity in plane graphs and in (k − 1)-vertex connected graphs. We show an amortized lower bound of �(log n/k(log log n + log b)) per edge insertion, deletion, or query operation in the cell probe model, where b is the word size of the machine and n is the number of vertices in G. We also show an amortized lower bound of �(log n/(log log n + log b)) per operation for fully dynamic planarity testing in embedded graphs. These are the first lower bounds for fully dynamic connectivity problems.

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Citation Context ...ach vertex the order of its incident edges. Two nodes x and y are k-edge (k-vertex) connected if there are k edge-disjoint (vertex-disjoint) paths connecting them. For further terminology we refer to =-=[6]-=-. Given a graph G, the fully dynamic k-edge (k-vertex) connectivity problem is to execute the following operations in arbitrary order: Insert(u; v) : Add the edge (u; v) to G. Delete(u; v) : Remove th... |

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Citation Context ...re the first lower bounds for fully dynamic connectivity problems. 1 Introduction This paper shows lower bounds for fully dynamic data structures by giving reductions to the parity prefix sum problem =-=[7]-=-. We call a graph G plane if G is planar and we are given a fixed embedding of G. An embedding of a graph G is uniquely determined by fixing at each vertex the order of its incident edges. Two nodes x... |

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Citation Context ...r general graphs we present both, the best deterministic and the best randomized bounds. plane planar general, rand. general, det. connectivity O(log n) [3] O(log 2 n) [5] O(log 2 n) [10, 10] O( p n) =-=[4]-=- 2-edge-connectivity O(log 2 n) [11] O(log 2 n) [5] O(log 3 n) [8, 10] O( p n) [4] 2-vertex-connectivity O(log 2 n) [14] O( p n) [5] O( p n log 3=2 n) [9] 3-edge-connectivity O( p n) [5] O(n 2=3 ) [4]... |

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Citation Context ...e graphs, planar graphs, and general graphs. For general graphs we present both, the best deterministic and the best randomized bounds. plane planar general, rand. general, det. connectivity O(log n) =-=[3]-=- O(log 2 n) [5] O(log 2 n) [10, 10] O( p n) [4] 2-edge-connectivity O(log 2 n) [11] O(log 2 n) [5] O(log 3 n) [8, 10] O( p n) [4] 2-vertex-connectivity O(log 2 n) [14] O( p n) [5] O( p n log 3=2 n) [9... |

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Citation Context ...en tight lower bounds of \Omega\Gamma ff(m; n)) per operation are known for connectivity, 2-edge connectivity, 2-vertex connectivity, 3-edge connectivity, 3-vertex connectivity, and planarity testing =-=[17, 15]-=- by reducing these problems to the union-find problem. An earlier version of this work has appeared in [14]. In the next section we give the general ideas of the lower bound proofs. In Section 3 and S... |

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Citation Context ...neral, det. connectivity O(log n) [3] O(log 2 n) [5] O(log 2 n) [10, 10] O( p n) [4] 2-edge-connectivity O(log 2 n) [11] O(log 2 n) [5] O(log 3 n) [8, 10] O( p n) [4] 2-vertex-connectivity O(log 2 n) =-=[14]-=- O( p n) [5] O( p n log 3=2 n) [9] 3-edge-connectivity O( p n) [5] O(n 2=3 ) [4] 3-vertex-connectivity O( p n) [5] O(n) [4] 4-edge-connectivity O( p n) [5] O(nff(n)) [4] planarity testing O(log 2 n) [... |

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Citation Context ...r graphs, and general graphs. For general graphs we present both, the best deterministic and the best randomized bounds. plane planar general, rand. general, det. connectivity O(log n) [3] O(log 2 n) =-=[5]-=- O(log 2 n) [10, 10] O( p n) [4] 2-edge-connectivity O(log 2 n) [11] O(log 2 n) [5] O(log 3 n) [8, 10] O( p n) [4] 2-vertex-connectivity O(log 2 n) [14] O( p n) [5] O( p n log 3=2 n) [9] 3-edge-connec... |

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Citation Context ...en tight lower bounds of \Omega\Gamma ff(m; n)) per operation are known for connectivity, 2-edge connectivity, 2-vertex connectivity, 3-edge connectivity, 3-vertex connectivity, and planarity testing =-=[17, 15]-=- by reducing these problems to the union-find problem. An earlier version of this work has appeared in [14]. In the next section we give the general ideas of the lower bound proofs. In Section 3 and S... |

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Citation Context ...est randomized bounds. plane planar general, rand. general, det. connectivity O(log n) [3] O(log 2 n) [5] O(log 2 n) [10, 10] O( p n) [4] 2-edge-connectivity O(log 2 n) [11] O(log 2 n) [5] O(log 3 n) =-=[8, 10]-=- O( p n) [4] 2-vertex-connectivity O(log 2 n) [14] O( p n) [5] O( p n log 3=2 n) [9] 3-edge-connectivity O( p n) [5] O(n 2=3 ) [4] 3-vertex-connectivity O( p n) [5] O(n) [4] 4-edge-connectivity O( p n... |

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Citation Context ...eneral graphs. For general graphs we present both, the best deterministic and the best randomized bounds. plane planar general, rand. general, det. connectivity O(log n) [3] O(log 2 n) [5] O(log 2 n) =-=[10, 10]-=- O( p n) [4] 2-edge-connectivity O(log 2 n) [11] O(log 2 n) [5] O(log 3 n) [8, 10] O( p n) [4] 2-vertex-connectivity O(log 2 n) [14] O( p n) [5] O( p n log 3=2 n) [9] 3-edge-connectivity O( p n) [5] O... |

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Citation Context ...he best deterministic and the best randomized bounds. plane planar general, rand. general, det. connectivity O(log n) [3] O(log 2 n) [5] O(log 2 n) [10, 10] O( p n) [4] 2-edge-connectivity O(log 2 n) =-=[11]-=- O(log 2 n) [5] O(log 3 n) [8, 10] O( p n) [4] 2-vertex-connectivity O(log 2 n) [14] O( p n) [5] O( p n log 3=2 n) [9] 3-edge-connectivity O( p n) [5] O(n 2=3 ) [4] 3-vertex-connectivity O( p n) [5] O... |

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Citation Context ...tructure. We define the helpful parity prefix sum problem and reduce the helpful parity prefix sum problem to the dynamic problem. The parity prefix sum problem is defined as follows: Given an array A=-=[1]-=-; : : : ; A[n] of integers mod 2 with initial value zero execute an arbitary sequence of the following operations: Add(l): Increase A[l] by 1. Sum(l): Return S l mod 2, where S l = P l i=1 A[i]. The h... |

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Citation Context ...not depend on the "hardness" of the (k-1)-edge ((k-1)-vertex) connectivity problem. Related Work. Miltersen et al. independently showed the same lower bound for the fully dynamic connectivit=-=y problem [13]-=-. Eppstein [2] recently simplified the proof and applied it to grid graphs. We review next some related work. Westbrook and Tarjan [16] gave a lower bound for a (stronger) variant of the dynamic conne... |

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Citation Context ...] O( p n) [5] O( p n log 3=2 n) [9] 3-edge-connectivity O( p n) [5] O(n 2=3 ) [4] 3-vertex-connectivity O( p n) [5] O(n) [4] 4-edge-connectivity O( p n) [5] O(nff(n)) [4] planarity testing O(log 2 n) =-=[12]-=- O( p n) [5] Figure 1: The best upper bounds for fully dynamic problems. We establish lower bounds on the time per operation for the above problems in the cell probe model. We show the lower bounds in... |

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Citation Context ...he "hardness" of the (k-1)-edge ((k-1)-vertex) connectivity problem. Related Work. Miltersen et al. independently showed the same lower bound for the fully dynamic connectivity problem [13].=-= Eppstein [2]-=- recently simplified the proof and applied it to grid graphs. We review next some related work. Westbrook and Tarjan [16] gave a lower bound for a (stronger) variant of the dynamic connectivity, 2-edg... |

1 |
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Citation Context ...3] O(log 2 n) [5] O(log 2 n) [10, 10] O( p n) [4] 2-edge-connectivity O(log 2 n) [11] O(log 2 n) [5] O(log 3 n) [8, 10] O( p n) [4] 2-vertex-connectivity O(log 2 n) [14] O( p n) [5] O( p n log 3=2 n) =-=[9]-=- 3-edge-connectivity O( p n) [5] O(n 2=3 ) [4] 3-vertex-connectivity O( p n) [5] O(n) [4] 4-edge-connectivity O( p n) [5] O(nff(n)) [4] planarity testing O(log 2 n) [12] O( p n) [5] Figure 1: The best... |

1 |
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Citation Context ... the same lower bound for the fully dynamic connectivity problem [13]. Eppstein [2] recently simplified the proof and applied it to grid graphs. We review next some related work. Westbrook and Tarjan =-=[16]-=- gave a lower bound for a (stronger) variant of the dynamic connectivity, 2-edge-connectivity, and 2-vertex-connectivity problems. Their model of a dynamic algorithm requires that a query returns the ... |