A lower bound for metric 1-median selection

Abstract

Consider the problem of finding a point in an n-point metric space with the minimum average distance to all points. We show that this problem has no deterministic o(n 2 )-query (4 − Ω(1))-approximation algorithms.

On Las Vegas approximations for metric 1-median selection

“…Chang [2] shows that metric 1-median has a deterministic, (2h)-approximation, O(hn 1+1/h )-time and nonadaptive algorithm for all constants h ∈ Z + \ {1}, generalizing the results of Chang [1] and Wu [15]. On the other hand, he disproves the existence of deterministic (2h − )-approximation O(n 1+1/(h−1) /h)-time algorithms for all constants h ∈ Z + \ {1} and > 0 [3,4].…”

Metric 1-Median Selection: Query Complexity vs. Approximation Ratio

“…That is, we determine the best approximation ratio of deterministic O(n 1+ )-query (resp., O(n 1+ )-time) algorithms for all ∈ (0, 1). As in the previous lower bounds for deterministic algorithms [2,4], we use an adversarial method. Roughly speaking, our proof proceeds as follows:…”

“…3, that are unseen in previous lower bounds [2,4,9]. Like in [4], we need a small set S of points whose distances to other points are answered as large values during A's execution, and yet we assign a small value to the distances from a certain pointα ∈ S to many other points in item (iii). This paper is organized as follows.…”

“…Chang [3], Wu [11] and Chang [1] show that metric 1-median has a deterministic nonadaptive O(n 1+1/h )-time (2h)-approximation algorithm for all constants h ∈ Z + \ {1}. Furthermore, Chang [4] shows the nonexistence of deterministic o(n 2 )-time (4−Ω(1))-approximation algorithms for metric 1-median. This paper generalizes his result to show that metric 1-median has no deterministic o(n 1+1/(h−1) )-query (2h − Ω(1))-approximation algorithms for any constant h ∈ Z + \ {1}.…”

A deterministic sublinear-time nonadaptive algorithm for metric 1-median selection

“…When n is a perfect square and h = 2, our proof is equivalent to that of Theorem 1 [2]. Chang [3] shows that metric 1-median has no deterministic o(n 2 )-query (4 − Ω(1))-approximation algorithms (where an algorithm's query complexity is the number of distances that it inspects). So the approximation ratio of 4 in Theorem 1 cannot be improved to a smaller constant.…”