TY - GEN

T1 - Balancing degree, diameter and weight in Euclidean spanners

AU - Solomon, Shay

AU - Elkin, Michael

PY - 2010/11/19

Y1 - 2010/11/19

N2 - In a seminal STOC'95 paper, Arya et al. [4] devised a construction that for any set S of n points in ℝd and any ε > 0, provides a (1 + ε)-spanner with diameter O(logn), weight O(log2 n)w(MST(S)), and constant maximum degree. Another construction of [4] provides a (1 + ε)-spanner with O(n) edges and diameter α(n), where α stands for the inverse-Ackermann function. Das and Narasimhan [12] devised a construction with constant maximum degree and weight O(w(MST(S))), but whose diameter may be arbitrarily large. In another construction by Arya et al. [4] there is diameter O(logn) and weight O(logn) w(MST(S)), but it may have arbitrarily large maximum degree. These constructions fail to address situations in which we are prepared to compromise on one of the parameters, but cannot afford it to be arbitrarily large. In this paper we devise a novel unified construction that trades between maximum degree, diameter and weight gracefully. For a positive integer k, our construction provides a (1 + ε)-spanner with maximum degree O(k), diameter O(logk n + α(k)), weight O(k logk n logn) w(MST(S)), and O(n) edges. For k = O(1) this gives rise to maximum degree O(1), diameter O(logn) and weight O(log2 n) w(MST(S)), which is one of the aforementioned results of [4]. For k = n 1/α(n) this gives rise to diameter O(α(n)), weight O(n1/α(n) (logn) α(n)) w(MST(S)) and maximum degree O(n 1/α(n)). In the corresponding result from [4] the spanner has the same number of edges and diameter, but its weight and degree may be arbitrarily large. Our construction also provides a similar tradeoff in the complementary range of parameters, i.e., when the weight should be smaller than log2 n, but the diameter is allowed to grow beyond logn.

AB - In a seminal STOC'95 paper, Arya et al. [4] devised a construction that for any set S of n points in ℝd and any ε > 0, provides a (1 + ε)-spanner with diameter O(logn), weight O(log2 n)w(MST(S)), and constant maximum degree. Another construction of [4] provides a (1 + ε)-spanner with O(n) edges and diameter α(n), where α stands for the inverse-Ackermann function. Das and Narasimhan [12] devised a construction with constant maximum degree and weight O(w(MST(S))), but whose diameter may be arbitrarily large. In another construction by Arya et al. [4] there is diameter O(logn) and weight O(logn) w(MST(S)), but it may have arbitrarily large maximum degree. These constructions fail to address situations in which we are prepared to compromise on one of the parameters, but cannot afford it to be arbitrarily large. In this paper we devise a novel unified construction that trades between maximum degree, diameter and weight gracefully. For a positive integer k, our construction provides a (1 + ε)-spanner with maximum degree O(k), diameter O(logk n + α(k)), weight O(k logk n logn) w(MST(S)), and O(n) edges. For k = O(1) this gives rise to maximum degree O(1), diameter O(logn) and weight O(log2 n) w(MST(S)), which is one of the aforementioned results of [4]. For k = n 1/α(n) this gives rise to diameter O(α(n)), weight O(n1/α(n) (logn) α(n)) w(MST(S)) and maximum degree O(n 1/α(n)). In the corresponding result from [4] the spanner has the same number of edges and diameter, but its weight and degree may be arbitrarily large. Our construction also provides a similar tradeoff in the complementary range of parameters, i.e., when the weight should be smaller than log2 n, but the diameter is allowed to grow beyond logn.

UR - http://www.scopus.com/inward/record.url?scp=78249253932&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-15775-2_5

DO - 10.1007/978-3-642-15775-2_5

M3 - Conference contribution

AN - SCOPUS:78249253932

SN - 3642157742

SN - 9783642157745

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 48

EP - 59

BT - Algorithms, ESA 2010 - 18th Annual European Symposium, Proceedings

T2 - 18th Annual European Symposium on Algorithms, ESA 2010

Y2 - 6 September 2010 through 8 September 2010

ER -