# 1 Introduction

The BiocNeighbors package implements a few algorithms for exact nearest neighbor searching:

• The k-means for k-nearest neighbors (KMKNN) algorithm (Wang 2012) uses k-means clustering to create an index. Within each cluster, the distance of each of that cluster’s points to the cluster center are computed and used to sort all points. Given a query point, the distance to each cluster center is determined and the triangle inequality is applied to determine which points in each cluster warrant a full distance calculation.
• The vantage point (VP) tree algorithm (Yianilos 1993) involves constructing a tree where each node is located at a data point and is associated with a subset of neighboring points. Each node progressively partitions points into two subsets that are either closer or further to the node than a given threshold. Given a query point, the triangle inequality is applied at each node in the tree to determine if the child nodes warrant searching.
• The exhaustive search is a simple brute-force algorithm that computes distances to between all data and query points. This has the worst computational complexity but can actually be faster than the other exact algorithms in situations where indexing provides little benefit, e.g., data sets with few points and/or a very large number of dimensions.

Both KMKNN and VP-trees involve a component of randomness during index construction, though the k-nearest neighbors result is fully deterministic1 Except in the presence of ties, see ?"BiocNeighbors-ties" for details..

# 2 Identifying k-nearest neighbors

The most obvious application is to perform a k-nearest neighbors search. We’ll mock up an example here with a hypercube of points, for which we want to identify the 10 nearest neighbors for each point.

nobs <- 10000
ndim <- 20
data <- matrix(runif(nobs*ndim), ncol=ndim)

The findKNN() method expects a numeric matrix as input with data points as the rows and variables/dimensions as the columns. We indicate that we want to use the KMKNN algorithm by setting BNPARAM=KmknnParam() (which is also the default, so this is not strictly necessary here). We could use a VP tree instead by setting BNPARAM=VptreeParam().

fout <- findKNN(data, k=10, BNPARAM=KmknnParam())
head(fout$index) ## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] ## [1,] 1935 2356 2362 57 8076 6373 5699 2483 9504 9040 ## [2,] 9789 1782 9921 1314 4105 9344 9640 9096 5722 7435 ## [3,] 2166 3428 1158 2401 9600 1977 2356 5826 3971 3140 ## [4,] 7713 915 5590 1773 2314 2003 8328 5043 5424 2663 ## [5,] 8997 3311 1852 4290 9367 5674 9882 4370 5598 4785 ## [6,] 4614 9224 4163 6303 9040 2594 2314 6288 5133 706 head(fout$distance)
##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
## [1,] 0.9095515 0.9113790 0.9183992 0.9648601 0.9725504 0.9770221 0.9804174
## [2,] 0.8808309 1.0099397 1.0236221 1.0281230 1.0290028 1.0438050 1.0465617
## [3,] 0.8330738 1.0292092 1.0337894 1.0443245 1.0453908 1.0485869 1.0514509
## [4,] 1.0081959 1.0588291 1.0667738 1.0885254 1.0887729 1.1135694 1.1175354
## [5,] 0.9006528 0.9119734 0.9333579 0.9358952 0.9677931 0.9791606 0.9905248
## [6,] 0.8677079 1.0352460 1.0541254 1.0677597 1.0713551 1.0728613 1.0746965
##           [,8]     [,9]    [,10]
## [1,] 0.9971581 1.009139 1.010270
## [2,] 1.0677903 1.073756 1.075368
## [3,] 1.0534295 1.080317 1.080381
## [4,] 1.1239404 1.141637 1.159081
## [5,] 1.0121908 1.013311 1.013700
## [6,] 1.0756643 1.077918 1.086240

Each row of the index matrix corresponds to a point in data and contains the row indices in data that are its nearest neighbors. For example, the 3rd point in data has the following nearest neighbors:

fout$index[3,] ##  2166 3428 1158 2401 9600 1977 2356 5826 3971 3140 … with the following distances to those neighbors: fout$distance[3,]
##   0.8330738 1.0292092 1.0337894 1.0443245 1.0453908 1.0485869 1.0514509
##   1.0534295 1.0803172 1.0803810

Note that the reported neighbors are sorted by distance.

# 3 Querying k-nearest neighbors

Another application is to identify the k-nearest neighbors in one dataset based on query points in another dataset. Again, we mock up a small data set:

nquery <- 1000
ndim <- 20
query <- matrix(runif(nquery*ndim), ncol=ndim)

We then use the queryKNN() function to identify the 5 nearest neighbors in data for each point in query.

qout <- queryKNN(data, query, k=5, BNPARAM=KmknnParam())
head(qout$index) ## [,1] [,2] [,3] [,4] [,5] ## [1,] 9532 5741 6032 3592 1218 ## [2,] 578 9667 4461 3273 5279 ## [3,] 5562 3333 1980 6898 2117 ## [4,] 2093 7215 670 3094 3414 ## [5,] 1521 6832 7785 1833 3370 ## [6,] 7410 6967 3505 6095 2710 head(qout$distance)
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,] 0.8899789 0.9336598 1.0216367 1.0930681 1.1071916
## [2,] 0.8409194 0.8982352 0.9818410 0.9975233 1.0060628
## [3,] 0.6049765 0.8877391 0.9093684 0.9267864 0.9368828
## [4,] 0.8781357 0.8907984 0.9068256 0.9382745 0.9781155
## [5,] 0.8517183 0.9315949 0.9406789 0.9647713 0.9707514
## [6,] 0.7912862 0.8507538 0.8785286 0.8936067 0.9549804

Each row of the index matrix contains the row indices in data that are the nearest neighbors of a point in query. For example, the 3rd point in query has the following nearest neighbors in data:

qout$index[3,] ##  5562 3333 1980 6898 2117 … with the following distances to those neighbors: qout$distance[3,]
##  0.6049765 0.8877391 0.9093684 0.9267864 0.9368828

Again, the reported neighbors are sorted by distance.

# 4 Further options

Users can perform the search for a subset of query points using the subset= argument. This yields the same result as but is more efficient than performing the search for all points and subsetting the output.

findKNN(data, k=5, subset=3:5)
## $index ## [,1] [,2] [,3] [,4] [,5] ## [1,] 2166 3428 1158 2401 9600 ## [2,] 7713 915 5590 1773 2314 ## [3,] 8997 3311 1852 4290 9367 ## ##$distance
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,] 0.8330738 1.0292092 1.0337894 1.0443245 1.0453908
## [2,] 1.0081959 1.0588291 1.0667738 1.0885254 1.0887729
## [3,] 0.9006528 0.9119734 0.9333579 0.9358952 0.9677931

If only the indices are of interest, users can set get.distance=FALSE to avoid returning the matrix of distances. This will save some time and memory.

names(findKNN(data, k=2, get.distance=FALSE))
##  "index"

It is also simple to speed up functions by parallelizing the calculations with the BiocParallel framework.

library(BiocParallel)
out <- findKNN(data, k=10, BPPARAM=MulticoreParam(3))

For multiple queries to a constant data, the pre-clustering can be performed in a separate step with buildIndex(). The result can then be passed to multiple calls, avoiding the overhead of repeated clustering2 The algorithm type is automatically determined when BNINDEX is specified, so there is no need to also specify BNPARAM in the later functions..

pre <- buildIndex(data, BNPARAM=KmknnParam())
out1 <- findKNN(BNINDEX=pre, k=5)
out2 <- queryKNN(BNINDEX=pre, query=query, k=2)

The default setting is to search on the Euclidean distance. Alternatively, we can use the Manhattan distance by setting distance="Manhattan" in the BiocNeighborParam object.

out.m <- findKNN(data, k=5, BNPARAM=KmknnParam(distance="Manhattan"))

Advanced users may also be interested in the raw.index= argument, which returns indices directly to the precomputed object rather than to data. This may be useful inside package functions where it may be more convenient to work on a common precomputed object.

# 5 Session information

sessionInfo()
## R Under development (unstable) (2021-10-19 r81077)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 20.04.3 LTS
##
## Matrix products: default
## BLAS:   /home/biocbuild/bbs-3.15-bioc/R/lib/libRblas.so
## LAPACK: /home/biocbuild/bbs-3.15-bioc/R/lib/libRlapack.so
##
## locale:
##   LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
##   LC_TIME=en_GB              LC_COLLATE=C
##   LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8
##   LC_PAPER=en_US.UTF-8       LC_NAME=C
##  LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
##
## attached base packages:
##  stats     graphics  grDevices utils     datasets  methods   base
##
## other attached packages:
##  BiocParallel_1.29.0  BiocNeighbors_1.13.0 knitr_1.36
##  BiocStyle_2.23.0
##
## loaded via a namespace (and not attached):
##   Rcpp_1.0.7          magrittr_2.0.1      BiocGenerics_0.41.0
##   lattice_0.20-45     R6_2.5.1            rlang_0.4.12
##   fastmap_1.1.0       stringr_1.4.0       tools_4.2.0
##  parallel_4.2.0      grid_4.2.0          xfun_0.27
##  jquerylib_0.1.4     htmltools_0.5.2     yaml_2.2.1
##  digest_0.6.28       bookdown_0.24       Matrix_1.3-4
##  BiocManager_1.30.16 S4Vectors_0.33.0    sass_0.4.0
##  evaluate_0.14       rmarkdown_2.11      stringi_1.7.5
##  compiler_4.2.0      bslib_0.3.1         stats4_4.2.0
##  jsonlite_1.7.2