# Detecting exact nearest neighbors

Aaron Lun1

1Cancer Research UK Cambridge Institute, Cambridge, United Kingdom

#### Revised: 2 December 2018

#### Package

BiocNeighbors 1.2.0

# 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.

Both methods involve a component of randomness during index construction, though the k-nearest neighbors result is fully deterministic11 Except in the presence of ties, see `?findKNN` 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,] 6709 9627 8683 8907 4811 8728 9935 5093 1868   121
    ## [2,] 7593 9203 6652  961 2856 8711 4412 8907 4666  2811
    ## [3,] 8361 2534 7096 1443 8355 5135  730 9233 5057  3333
    ## [4,] 6116 9935 8994 4613 7533 7671 4938  587 9627  6673
    ## [5,] 4363 7309 3126 6247 4800 2470 1907  285 7956  3990
    ## [6,] 8871 5867 7618 1557 1016 7067  473  187  271  1517

    head(fout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
    ## [1,] 0.9134994 0.9279878 0.9321199 0.9432648 0.9759880 0.9883781 0.9904984
    ## [2,] 0.8612897 0.9499456 0.9703708 0.9782614 1.0085772 1.0099281 1.0102099
    ## [3,] 1.0378627 1.0478542 1.0709121 1.0858699 1.0869769 1.0923892 1.0965950
    ## [4,] 0.9568614 1.0717803 1.0742441 1.0990840 1.1128575 1.1139495 1.1360398
    ## [5,] 0.8285719 0.8375260 0.8930268 0.9053241 0.9169200 0.9226835 0.9358931
    ## [6,] 0.7955337 0.9129179 0.9275191 0.9435631 0.9749024 0.9931279 0.9980455
    ##           [,8]      [,9]     [,10]
    ## [1,] 0.9966502 1.0032015 1.0052059
    ## [2,] 1.0129669 1.0177056 1.0362768
    ## [3,] 1.1051209 1.1088504 1.1158236
    ## [4,] 1.1429454 1.1429834 1.1538995
    ## [5,] 0.9490294 0.9546187 0.9621188
    ## [6,] 0.9994188 1.0027133 1.0101063

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,]

    ##  [1] 8361 2534 7096 1443 8355 5135  730 9233 5057 3333

… with the following distances to those neighbors:

    fout$distance[3,]

    ##  [1] 1.037863 1.047854 1.070912 1.085870 1.086977 1.092389 1.096595 1.105121
    ##  [9] 1.108850 1.115824

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,] 4271 6336 1896   66 5255
    ## [2,]  577 2631 4942 8018 9216
    ## [3,] 1282 7391 6915 9507 6113
    ## [4,] 8115 5553 2903 1729 1585
    ## [5,] 9979 5919 7712 1312 6032
    ## [6,] 7124 1550 5867 4266 1743

    head(qout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]
    ## [1,] 0.8255378 0.9534278 0.9662206 0.9895064 0.9973368
    ## [2,] 0.9082208 0.9134062 0.9218634 0.9518809 0.9672650
    ## [3,] 0.9416898 0.9466910 0.9773505 0.9956400 0.9989399
    ## [4,] 1.0114986 1.0219494 1.0297100 1.0405589 1.0559577
    ## [5,] 0.8845057 0.8972909 0.8973070 0.9025898 0.9055773
    ## [6,] 1.0793351 1.0811115 1.1228046 1.1345023 1.1360885

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,]

    ## [1] 1282 7391 6915 9507 6113

… with the following distances to those neighbors:

    qout$distance[3,]

    ## [1] 0.9416898 0.9466910 0.9773505 0.9956400 0.9989399

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,] 8361 2534 7096 1443 8355
    ## [2,] 6116 9935 8994 4613 7533
    ## [3,] 4363 7309 3126 6247 4800
    ##
    ## $distance
    ##           [,1]     [,2]      [,3]      [,4]     [,5]
    ## [1,] 1.0378627 1.047854 1.0709121 1.0858699 1.086977
    ## [2,] 0.9568614 1.071780 1.0742441 1.0990840 1.112857
    ## [3,] 0.8285719 0.837526 0.8930268 0.9053241 0.916920

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))

    ## [1] "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 clustering22 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)

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 version 3.6.0 (2019-04-26)
    ## Platform: x86_64-pc-linux-gnu (64-bit)
    ## Running under: Ubuntu 18.04.2 LTS
    ##
    ## Matrix products: default
    ## BLAS:   /home/biocbuild/bbs-3.9-bioc/R/lib/libRblas.so
    ## LAPACK: /home/biocbuild/bbs-3.9-bioc/R/lib/libRlapack.so
    ##
    ## locale:
    ##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
    ##  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=C
    ##  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8
    ##  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C
    ##  [9] LC_ADDRESS=C               LC_TELEPHONE=C
    ## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
    ##
    ## attached base packages:
    ## [1] stats     graphics  grDevices utils     datasets  methods   base
    ##
    ## other attached packages:
    ## [1] BiocParallel_1.18.0 BiocNeighbors_1.2.0 knitr_1.22
    ## [4] BiocStyle_2.12.0
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] Rcpp_1.0.1          bookdown_0.9        digest_0.6.18
    ##  [4] stats4_3.6.0        magrittr_1.5        evaluate_0.13
    ##  [7] stringi_1.4.3       S4Vectors_0.22.0    rmarkdown_1.12
    ## [10] tools_3.6.0         stringr_1.4.0       parallel_3.6.0
    ## [13] xfun_0.6            yaml_2.2.0          compiler_3.6.0
    ## [16] BiocGenerics_0.30.0 BiocManager_1.30.4  htmltools_0.3.6

# References

Wang, X. 2012. “A Fast Exact k-Nearest Neighbors Algorithm for High Dimensional Search Using k-Means Clustering and Triangle Inequality.” Proc Int Jt Conf Neural Netw 43 (6):2351–8.

Yianilos, P. N. 1993. “Data Structures and Algorithms for Nearest Neighbor Search in General Metric Spaces.” In SODA, 93:311–21. 194.