Algorithms and Hardness Results for Nearest Neighbor Problems in Bicolored Point Sets
Abstract

Open image in new window : Given k sets of points \(P_1, P_2, \ldots , P_k\) in a metric space \(\mathcal X\), the goal is to choose subsets of points \(P'_i \subseteq P_i\) for \(i=1,2,\ldots ,k\) such that \(\forall \ p \in P_i\) its nearest neighbor among \(\bigcup _{j=1}^{k} P'_j \) lies in \(P'_i\) for each \(i\in [k]\) while minimizing (Note that we also enforce the condition \(P'_i\ge 1\ \forall \ i\in [k]\).) the quantity \(\sum _{i=1}^k P'_i\).

Open image in new window : Given k sets of points \(P_1, P_2, \ldots , P_k\) in a metric space \(\mathcal X\), the goal is to choose subsets of points \(P'_i \subseteq P_i\) for \(i=1,2,\ldots ,k\) such that \(\forall \ p \in P_i\) its nearest neighbor among \(\Big (\bigcup _{j=1, j\ne i}^{k} P_j\Big ) \cup P'_i \) lies in \(P'_i\) for each \(i\in [k]\) while minimizing (Note that we again enforce the condition \(P'_i\ge 1\ \forall \ i\in [k]\).) the quantity \(\sum _{i=1}^k P'_i\).
While there have been several heuristics proposed for these two problems in the computer vision and machine learning community, the only theoretical results for these problems (to the best of our knowledge) are due to Wilfong [SOCG ’91] who showed that both 3MCS(\(\mathbb {R}^2\)) and 2MSS(\(\mathbb {R}^2\)) are NPcomplete. We initiate the study of these two problems from a theoretical perspective, and obtain several algorithmic and hardness results.
On the algorithmic side, we first design an \(O(n^2)\) time exact algorithm and \(O(n\log n)\) time 2approximation for the 2MCS(\(\mathbb {R}\)) problem, i.e., the points are located on the real line. Moreover, we show that the exact algorithm also extends to the case when the points are located on the circumference of a circle. Next, we design an \(O(r^2)\) time online algorithm for the 2MCS(\(\mathbb {R}\)) problem such that \(r<n\), where n is the set of points and r is an integer. Finally, we give a PTAS for the kMSS(\(\mathbb {R}^2\)) problem. On the hardness side, we show that both the 2MCS and 2MSS problems are NPcomplete on graphs. Additionally, the problems are W[2]hard parameterized by the size k of the solution. For points on the Euclidean plane, we show that the 2MSS problem is contained in W[1]. Finally, we show a lower bound of \(\varOmega (\sqrt{n})\) bits for the storage of any (randomized) algorithm which solves both 2MCS(\(\mathbb {R}\)) and 2MSS(\(\mathbb {R}\)).
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