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.

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$index)
##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 6961 7813 9170 1226 3839 7008 9424 2876 3767  7456
## [2,] 8990  125 8362 6248 4779 2685 9581 7234 3846  5162
## [3,] 9326 3930 1608 7055 2299 2535 9984 3870 6416  4849
## [4,] 3103 9028 5464 6662 2537 4710  553 2122 6900  9294
## [5,] 3242  718 6475 6813 9395 2844 7682 2232 1385  6772
## [6,] 2221  352 6141 7122  591 1269  480 7758 4629  1807
head(fout$distance)
##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
## [1,] 0.7948161 0.8700061 0.9444299 0.9736093 1.0045227 1.0259050 1.0373362
## [2,] 0.7987711 0.8186981 0.8391710 0.8547602 0.8618549 0.8623643 0.8785859
## [3,] 0.9887107 1.0396488 1.0614358 1.0758956 1.0850965 1.0995462 1.0998265
## [4,] 1.0073244 1.0550710 1.0565176 1.0693396 1.0985015 1.1051071 1.1145887
## [5,] 0.9491025 0.9575265 0.9926144 1.0278524 1.0385156 1.0391379 1.0672122
## [6,] 1.0073630 1.0495905 1.0893822 1.0962145 1.1590247 1.1655495 1.1744522
##           [,8]      [,9]     [,10]
## [1,] 1.0566652 1.0574998 1.0726549
## [2,] 0.8855493 0.8856993 0.8926532
## [3,] 1.1020565 1.1131608 1.1305606
## [4,] 1.1284751 1.1318379 1.1322208
## [5,] 1.0704776 1.0827966 1.0901881
## [6,] 1.1923896 1.1974756 1.2143994

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] 9326 3930 1608 7055 2299 2535 9984 3870 6416 4849

… with the following distances to those neighbors:

fout$distance[3,]
##  [1] 0.9887107 1.0396488 1.0614358 1.0758956 1.0850965 1.0995462 1.0998265
##  [8] 1.1020565 1.1131608 1.1305606

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$index)
##      [,1] [,2] [,3] [,4] [,5]
## [1,] 3896 7662 2570 1502 7470
## [2,] 4784 8439 9245 4159 3410
## [3,] 7134 2162 1791 8784 3087
## [4,] 1039 1661 3299 2363 7723
## [5,] 9271 1113 8542 9340 7815
## [6,] 8201 3139  555 6463 9062
head(qout$distance)
##           [,1]      [,2]      [,3]     [,4]     [,5]
## [1,] 0.9698087 1.0042420 1.0136370 1.030114 1.030365
## [2,] 0.8719075 0.9062713 0.9257835 1.011957 1.013228
## [3,] 0.9806023 0.9935614 1.0215643 1.062581 1.064352
## [4,] 1.0010272 1.0449616 1.1036209 1.129081 1.161795
## [5,] 1.0027726 1.0117239 1.0292581 1.029434 1.046989
## [6,] 0.8826955 1.0486813 1.0554361 1.065353 1.092932

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] 7134 2162 1791 8784 3087

… with the following distances to those neighbors:

qout$distance[3,]
## [1] 0.9806023 0.9935614 1.0215643 1.0625811 1.0643518

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,] 9326 3930 1608 7055 2299
## [2,] 3103 9028 5464 6662 2537
## [3,] 3242  718 6475 6813 9395
## 
## $distance
##           [,1]      [,2]      [,3]     [,4]     [,5]
## [1,] 0.9887107 1.0396488 1.0614358 1.075896 1.085097
## [2,] 1.0073244 1.0550710 1.0565176 1.069340 1.098501
## [3,] 0.9491025 0.9575265 0.9926144 1.027852 1.038516

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 clustering2.

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 version 4.2.1 Patched (2022-07-09 r82577)
## Platform: x86_64-apple-darwin17.0 (64-bit)
## Running under: macOS Big Sur ... 10.16
## 
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRlapack.dylib
## 
## locale:
## [1] C/en_US.UTF-8/en_US.UTF-8/C/en_GB/en_US.UTF-8
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
## [1] BiocParallel_1.32.0  BiocNeighbors_1.16.0 knitr_1.40          
## [4] BiocStyle_2.26.0    
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_1.0.9          magrittr_2.0.3      BiocGenerics_0.44.0
##  [4] lattice_0.20-45     R6_2.5.1            rlang_1.0.6        
##  [7] fastmap_1.1.0       stringr_1.4.1       tools_4.2.1        
## [10] parallel_4.2.1      grid_4.2.1          xfun_0.34          
## [13] cli_3.4.1           jquerylib_0.1.4     htmltools_0.5.3    
## [16] yaml_2.3.6          digest_0.6.30       bookdown_0.29      
## [19] Matrix_1.5-1        BiocManager_1.30.19 S4Vectors_0.36.0   
## [22] sass_0.4.2          codetools_0.2-18    cachem_1.0.6       
## [25] evaluate_0.17       rmarkdown_2.17      stringi_1.7.8      
## [28] compiler_4.2.1      bslib_0.4.0         stats4_4.2.1       
## [31] jsonlite_1.8.3

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): 2351–58.
Yianilos, P. N. 1993. “Data Structures and Algorithms for Nearest Neighbor Search in General Metric Spaces.” In SODA, 93:311–21. 194.

  1. Except in the presence of ties, see ?"BiocNeighbors-ties" for details.↩︎

  2. The algorithm type is automatically determined when BNINDEX is specified, so there is no need to also specify BNPARAM in the later functions.↩︎