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An Illuminating Algorithm for the Light Bulb Problem

by   Josh Alman, et al.

The Light Bulb Problem is one of the most basic problems in data analysis. One is given as input n vectors in {-1,1}^d, which are all independently and uniformly random, except for a planted pair of vectors with inner product at least ρ· d for some constant ρ > 0. The task is to find the planted pair. The most straightforward algorithm leads to a runtime of Ω(n^2). Algorithms based on techniques like Locality-Sensitive Hashing achieve runtimes of n^2 - O(ρ); as ρ gets small, these approach quadratic. Building on prior work, we give a new algorithm for this problem which runs in time O(n^1.582 + nd), regardless of how small ρ is. This matches the best known runtime due to Karppa et al. Our algorithm combines techniques from previous work on the Light Bulb Problem with the so-called `polynomial method in algorithm design,' and has a simpler analysis than previous work. Our algorithm is also easily derandomized, leading to a deterministic algorithm for the Light Bulb Problem with the same runtime of O(n^1.582 + nd), improving previous results.


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