You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
There is growing interest in representing image data and feature descriptors using compact binary codes for fast near neighbor search. Although binary codes are motivated by their use as direct indices (addresses) into a hash table, codes longer than 32 bits are not being used as such, as it was thought to be ineffective. We introduce a rigorous way to build multiple hash tables on binary code substrings that enables exact k-nearest neighbor search in Hamming space. The approach is storage efficient and straightforward to implement. Theoretical analysis shows that the algorithm exhibits sub-linear run-time behavior for uniformly distributed codes. Empirical results show dramatic speedups over a linear scan baseline for datasets of up to one billion codes of 64, 128, or 256 bits.

Cited Abstract(s):
[1]: Recent indexing techniques inspired by source coding have been shown successful to index billions of high-dimensional vectors in memory. In this paper, we propose an approach that re-ranks the neighbor hypotheses obtained by these compressed-domain indexing methods. In contrast to the usual post-verification scheme, which performs exact distance calculation on the short-list of hypotheses, the estimated distances are refined based on short quantization codes, to avoid reading the full vectors from disk. We have released a new public dataset of one billion 128-dimensional vectors and proposed an experimental setup to evaluate high dimensional indexing algorithms on a realistic scale. Experiments show that our method accurately and efficiently re-ranks the neighbor hypotheses using little memory compared to the full vectors representation.
* -0.2cm (Top) Curves show the ( @math ) number of distinct hash table indices (buckets) within a Hamming ball of radius @math , for different code lengths. With 64-bit codes there are about 1B buckets within a Hamming ball with a 7-bit radius. Hence with fewer than 1B database items, and a search radius of 7 or more, a hash table would be less efficient than linear scan. Using hash tables with 128-bit codes is prohibitive for radii larger than 6. (Bottom) This plot shows the expected search radius required for @math search as a function of @math , based on a dataset of 1B SIFT descriptors. Binary codes with 64 and 128 bits were obtained by random projections (LSH) from the SIFT descriptors [1] . Standard deviation bars help show that large search radii are often required. * -0.2cm