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

Abstract

We give a deterministic O(hn 1+1/h )-time (2h)-approximation nonadaptive algorithm for 1-median selection in n-point metric spaces, where h ∈ Z + \ {1} is arbitrary. Our proof generalizes that of Chang [2].

On Las Vegas approximations for metric 1-median selection

“…By Fact 1, Indyk median satisfies condition (2) in Theorem 17. But it does not satisfy condition (1).…”

“…All randomized O(1)-approximation algorithms for metric k-median take Ω(nk) time [7,11]. 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 with fewer queries

“…, n − σ − 1}. Similar to [3], we do so by dynamic programming. For this purpose, we need the following recurrences for g(·, ·) and f (·, ·), whose very cumbersome (but naïve) proofs are in Appendices A-B, respectively: Lemma 6 (cf.…”

“…, t − 1}, for a total of nt distances. Instead, Chang's [3]d is d i, i + s 2 t 2 + s 1 t + s 0 mod n ≡ d i, i + s 2 t 2 mod n + d i + s 2 t 2 mod n, i + s 2 t 2 + s 1 t mod n + d i + s 2 t 2 + s 1 t mod n, i + s 2 t 2 + s 1 t + s 0 mod n for all i ∈ {0, 1, . .…”

“…, h − 1}. Then calculate Unlike in line 5 of Approx.-Median, Chang's [3] pseudo-distance function requires to query for d(i, i+st k mod n) for all i ∈ {0, 1, . .…”