TY - GEN

T1 - On notions of distortion and an almost minimum spanning tree with constant average distortion

AU - Bartal, Yair

AU - Filtsert, Arnold

AU - Neiman, Ofer

PY - 2016

Y1 - 2016

N2 - Minimum Spanning Trees of weighted graphs are fundamental objects in numerous applications. In particular in distributed networks, the minimum spanning tree of the network is often used to route messages between network nodes. Unfortunately, while being most efficient in the total cost of connecting all nodes, minimum spanning trees fail miserably in the desired property of approximately preserving distances between pairs. While known lower bounds exclude the possibility of the worst case distortion of a tree being small, it was shown in [4] that there exists a spanning tree with constant average distortion. Yet, the weight of such a tree may be significantly larger than that of the MST. In this paper, we show that any weighted undirected graph admits a spanning tree whose weight is at most (1 + ρ) times that of the MST, providing constant average distortion o(i/ρ2). The constant average distortion bound is implied by a stronger property of scaling distortion, i.e., improved distortion for smaller fractions of the pairs. The result is achieved by first showing the existence of a low weight spanner with small prioritized distortion, a property allowing to prioritize the nodes whose associated distortions will be improved. We show that prioritized distortion is essentially equivalent to coarse scaling distortion via a general transformation, which has further implications and may be of independent interest. In particular, we obtain an embedding for arbitrary metrics into Euclidean space with optimal prioritized distortion.

AB - Minimum Spanning Trees of weighted graphs are fundamental objects in numerous applications. In particular in distributed networks, the minimum spanning tree of the network is often used to route messages between network nodes. Unfortunately, while being most efficient in the total cost of connecting all nodes, minimum spanning trees fail miserably in the desired property of approximately preserving distances between pairs. While known lower bounds exclude the possibility of the worst case distortion of a tree being small, it was shown in [4] that there exists a spanning tree with constant average distortion. Yet, the weight of such a tree may be significantly larger than that of the MST. In this paper, we show that any weighted undirected graph admits a spanning tree whose weight is at most (1 + ρ) times that of the MST, providing constant average distortion o(i/ρ2). The constant average distortion bound is implied by a stronger property of scaling distortion, i.e., improved distortion for smaller fractions of the pairs. The result is achieved by first showing the existence of a low weight spanner with small prioritized distortion, a property allowing to prioritize the nodes whose associated distortions will be improved. We show that prioritized distortion is essentially equivalent to coarse scaling distortion via a general transformation, which has further implications and may be of independent interest. In particular, we obtain an embedding for arbitrary metrics into Euclidean space with optimal prioritized distortion.

UR - http://www.scopus.com/inward/record.url?scp=84963652896&partnerID=8YFLogxK

M3 - Conference contribution

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 873

EP - 882

BT - 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016

A2 - Krauthgamer, Robert

T2 - 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016

Y2 - 10 January 2016 through 12 January 2016

ER -