## Generating Low-Degree 2-Spanners (1993)

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Citations: | 11 - 0 self |

### BibTeX

@TECHREPORT{Kortsarz93generatinglow-degree,

author = {Guy Kortsarz and David Peleg},

title = {Generating Low-Degree 2-Spanners},

institution = {},

year = {1993}

}

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### Abstract

Abstract. A k-spanner of a connected (undirected unweighted) graph G = (V, E) is a subgraph G ′ consisting of all the vertices of V and a subset of the edges, with the additional property that the distance between any two vertices in G ′ is larger than that distance in G by no more than a factor of k. This paper is concerned with approximating the problem of finding a 2-spanner in a given graph, with minimum maximum degree. We first show that the problem is at least as hard to approximate as set cover. Then a randomized approximation algorithm is provided for this problem, with approximation ratio of Õ(∆1/4). We then present a probabilistic algorithm that is more efficient for sparse graphs. Our algorithms are converted into deterministic ones using derandomization.

### Citations

10922 |
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(Show Context)
Citation Context ... 2 has a t\Gammacover, i.e., does there exist a subset V 0 ` V 1 , jV 0 jst such that every vertex in V 2 has a neighbor in V 0 ? It is well known that the set cover problem is NP \Gammacomplete, cf. =-=[GJ79]-=-. We now reduce the set cover problem to LD \Gamma 2SP . Assume that V 1 = fv 1 1 ; : : : ; v 1 n 1 g and V 2 = fv 2 1 ; : : : ; v 2 n 2 g. Define the following graph G 0 = (V; E 0 ) as follows. The v... |

847 |
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(Show Context)
Citation Context ...point share exactly one line. Consider now K n = (V; V \Theta V ) where V = fv 1 ; : : : ; v n g. Let q be a prime number such that b p ncsqs2b p nc. (Such a prime exists by Bertrand's postulate, cf. =-=[HW56]-=-.) Thus, n ! q 2 + q + 1 ! 5n. Let P = (P; L) be a projective plane of order q. Define a bipartite graph G = (L; P; E) where (l i ; p j ) 2 E iff p j 2 l i . Define the following spanning subgraph H =... |

714 |
A Measure of Asymptotic Efficiency of Tests for a Hypothesis Based on a Sum of Observations
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(Show Context)
Citation Context ... the set of vertices not touching e that are in a triangle with e. In the sequel we estimate the probability of the deviation of some random variables from their expectation, using the Chernoff bound =-=[Che52]. Lemma 2.2 [-=-Che52] Let X 1 ; X 2 ; : : : ; Xm be independent Bernoulli trials with IP(X i = 1) = p i . Let X = P m i=1 X i and �� = P m i=1 p i . Then IP(X ? (1 + ffi)��) ! " exp(ffi) (1 + ffi) (1+ff... |

684 | Approximation Algorithms for Combinatorial Problems - Johnson - 1974 |

628 | A threshold of ln n for approximating set cover
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(Show Context)
Citation Context ...ers V2, i.e., such that every vertex in V2 has a neighbor in S. It is known that this problem is hard to approximate. In particular, the following theorem is proved in [LY93, Fei96]. Theorem 4.1 (see =-=[Fei96]-=-). The set cover problem cannot be approximated with ratio ln n − ǫ, for any fixed ǫ > 0, unless NP ⊂ DT IME(n log log n ). Also, the following theorem is proven in [BGLR93]. Theorem 4.2 (see [BGLR93]... |

382 | On the hardness of approximating minimization problems - Lund, Yannakakis - 1993 |

333 |
Randomized rounding: A technique for provably good algorithms and algorithmic proofs
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(Show Context)
Citation Context ... inefficient) for it. Hence a different (and more involved) approach is required. The technique used in this paper for the LD \Gamma 2SP problem is a variant of the "randomized rounding" tec=-=hnique of [RT87]. Our algo-=-rithm is composed of two different procedures. The first procedure, is designed to cover edges lying on "many" triangles. The second procedure deals with the yet uncovered edges, i.e., edges... |

279 |
Probabilistic construction of deterministic algorithms: Approximating packing integer programs
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(Show Context)
Citation Context ...below by some constant.) Therefore, with probability at least 1 − 1/n 3 , d 1 r(v) ≤ 4 · ∆ 3/4 · √ log n. By summing up the probabilities over all the vertices, the lemma follows. (We note that as in =-=[Rag88]-=- slightly better results are attainable; i.e., it is possible to show a degree bound of 2 · √ log n∆ 3/4 + o(∆ 3/4 ). However, we give the simpler bound here, since in our case we already have an appr... |

197 | A new polynomial time algorithm for linear programming - Karmarkar - 1984 |

174 | On sparse spanners of weighted graphs - Althöfer, Das, et al. - 1993 |

154 |
Graph spanners
- Peleg, Schäffer
- 1989
(Show Context)
Citation Context ...E 0 ) the largest degree in the subgraph (V; E 0 ). We sometimes write \Delta for \Delta(G). We make use of an alternative characterization of k \Gammaspanners, given in the following simple lemma of =-=[PS89]-=-. Lemma 2.1 [PS89] The subgraph G 0 = (V; E 0 ) is a k \Gamma spanner of the graph G = (V; E) iff dist(u; v; G 0 )sk for every (v; u) 2 E. Thus the LD \Gamma 2SP problem can be restated as follows: we... |

127 |
An optimal synchronizer for the hypercube
- Peleg, Ullman
- 1989
(Show Context)
Citation Context ...nnecting them in G 0 . We refer to k as the stretch factor of G 0 . In the Euclidean setting, spanners were studied in [Cai91, DFS87, DJ89, LL89]. Spanners for general graphs were first introduced in =-=[PU89]-=-, where it was shown that for every n\Gammavertex hypercube there exists a 3-spanner with no more than 7n edges. Spanners were used in [PU89] to construct a new type of synchronizer for an asynchronou... |

119 | There are planar graphs almost as good as the complete graph - Chew - 1989 |

115 | A polynomial algorithm in linear programming - Khachian - 1979 |

109 | Delaunay graphs are almost as good as complete graphs - Dobkin, Friedman, et al. - 1990 |

79 | New sparseness results on graph spanners - Chandra, Das, et al. - 1995 |

74 |
An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica, 2:385–393, 1982. 3-partition problem, 24 ℓ-cover of a graph, 23 ℓ-edge-cover, 17 Connectivity S-connectivity, 1 edge-connectivity, 1 element-connectivity, 1, 4 nod
- Wolsey
(Show Context)
Citation Context ... L ≤ |V2|, find a cover C ⊂ V1 of V2 and an assignment ϕ with maximum load bounded by L (i.e., such that no vertex in V1 is assigned to more than L vertices of V2). We recall the following theorem of =-=[Wol82]-=-.GENERATING LOW-DEGREE 2-SPANNERS 1455 Theorem 9.1 (see [W82]). The bounded load set cover problem can be approximated with ratio O(log |V2|). Given an instance of the SLD-2SP problem, where our aim ... |

57 | Which triangulations approximate the complete graph? International Sym[24 - Das, Joseph - 1991 |

44 | Generating sparse 2-spanners
- Kortsarz, Peleg
- 1994
(Show Context)
Citation Context ... Namely, a good spanner is one with low stretch and as few edges as possible. For the problem of finding a 2\Gammaspanner which is as sparse as possible, a logarithmic-ratio approximation is given in =-=[KP92]-=-. Focusing on optimizing the sparsity measure may result in a spanner with high vertex degrees. In terms of applications this might mean a large local load on a single vertex, increasing the cost of i... |

36 | There are planar graphs almost as good as the complete graphs and as short as the minimum spanning trees, Symposium on Optimal Algorithms - Levcopoulos, Lingas - 1989 |

31 | D.: Efficient broadcast and light-weight spanners - Awerbuch, Baratz, et al. - 1992 |

25 | Additive graph spanners - Liestman, Shermer |

10 | Approximating euclidean distances by small degree graphs. Discrete & Computational Geometry, 11:213–233 - Soares - 1994 |

8 |
Degree-constrained pyramid spanners
- Liestman, Richards
- 1995
(Show Context)
Citation Context ...sider the question of choosing a k \Gammaspanner G 0 with minimum \Delta(G 0 ), for some parameter k. We call this problem LD \Gamma kSP . The problem of designing low degree spanners is addressed in =-=[LR90]-=-, for the special case where the underlying graph is the pyramid. The problem of designing small degree spanners for Euclidean and geometric graphs is studied in [CDNS92, Soa92]. This paper treats LD ... |

2 | Tree 2-spanners - Cai - 1991 |

2 | On the ratio of integral and fractional covers - Lovász - 1975 |