# Detecting approximate nearest neighbors

Aaron Lun1

1Cancer Research UK Cambridge Institute, Cambridge, United Kingdom

#### Revised: 2 December 2018

#### Package

BiocNeighbors 1.22.0

# 1 Introduction

The BiocNeighbors package provides several algorithms for approximate neighbor searches:

- The Annoy (Approximate Nearest Neighbors Oh Yeah) method uses C++ code from the RcppAnnoy package. It works by building a tree where a random hyperplane partitions a group of points into two child groups at each internal node. This is repeated to construct a forest of trees where the number of trees determines the accuracy of the search. Given a query data point, we identify all points in the same leaf node for each tree. We then take the union of leaf node sets across trees and search them exactly for the nearest neighbors.
- The HNSW (Hierarchical Navigable Small Worlds) method uses C++ code from the RcppHNSW package. It works by building a series of nagivable small world graphs containing links between points across the entire data set. The algorithm walks through the graphs where each step is chosen to move closer to a given query point. Different graphs contain links of different lengths, yielding a hierarchy where earlier steps are large and later steps are small. The accuracy of the search is determined by the connectivity of the graphs and the size of the intermediate list of potential neighbors.

These methods complement the exact algorithms described previously. Again, it is straightforward to switch from one algorithm to another by simply changing the `BNPARAM` argument in `findKNN` and `queryKNN`.

# 2 Identifying nearest neighbors

We perform the k-nearest neighbors search with the Annoy algorithm by specifying `BNPARAM=AnnoyParam()`.

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

    fout <- findKNN(data, k=10, BNPARAM=AnnoyParam())
    head(fout$index)

    ##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    ## [1,] 3506 7656 9060  294 7982 5763 5474 2185 2560  5397
    ## [2,] 5151 7920 2937 7891 1893 9072 2090 7528 3900  8499
    ## [3,] 2397 7417 7970 2647 9880 8618 2825 3404  424  6403
    ## [4,] 6203 7936  530 7071 4525  899 4804 8264 5204  4180
    ## [5,] 6131 1074  952 6845 9337 2972 5813 6267 5682   477
    ## [6,] 2170   22 9687 3589 6577  582  237 1970 7244  9155

    head(fout$distance)

    ##           [,1]      [,2]      [,3]      [,4]     [,5]     [,6]     [,7]
    ## [1,] 0.8567343 0.9148225 0.9407513 0.9881785 1.039331 1.044142 1.045700
    ## [2,] 0.6668538 0.8987589 0.9708484 1.0178564 1.025155 1.025636 1.026060
    ## [3,] 0.9775957 0.9990369 1.0128471 1.0149100 1.021895 1.027442 1.027629
    ## [4,] 0.8519940 0.9045876 0.9454841 0.9798104 1.019344 1.113069 1.130685
    ## [5,] 0.9539289 0.9631653 1.0059111 1.0206730 1.069508 1.106110 1.116073
    ## [6,] 0.9079698 0.9231571 0.9897676 0.9988787 1.027678 1.049186 1.051881
    ##          [,8]     [,9]    [,10]
    ## [1,] 1.051286 1.056190 1.064790
    ## [2,] 1.044135 1.048658 1.049732
    ## [3,] 1.058460 1.064249 1.094251
    ## [4,] 1.135794 1.140484 1.170250
    ## [5,] 1.128396 1.144618 1.145611
    ## [6,] 1.080945 1.083736 1.100816

We can also identify the k-nearest neighbors in one dataset based on query points in another dataset.

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

    qout <- queryKNN(data, query, k=5, BNPARAM=AnnoyParam())
    head(qout$index)

    ##      [,1] [,2] [,3] [,4] [,5]
    ## [1,] 2753 8516 7796 6679 6884
    ## [2,] 3289 6218 3669  309 8970
    ## [3,] 3778 9850 9023 4961 1192
    ## [4,]  432  160 6157 8795 5380
    ## [5,] 4486 7019 1370  332 1204
    ## [6,] 2369 4021 1589 8072 9372

    head(qout$distance)

    ##           [,1]      [,2]      [,3]      [,4]      [,5]
    ## [1,] 0.7266792 1.1280293 1.1945150 1.2018101 1.2285351
    ## [2,] 0.8395190 0.8967423 0.9254928 0.9273860 0.9366759
    ## [3,] 0.9343264 1.0064628 1.0398891 1.0494432 1.0512451
    ## [4,] 0.7891868 0.9628958 1.0200688 1.0563028 1.0563600
    ## [5,] 0.8804094 0.8907453 0.9851564 0.9880109 1.0105910
    ## [6,] 0.7642388 0.8639182 0.8784883 0.8930487 0.8942956

It is similarly easy to use the HNSW algorithm by setting `BNPARAM=HnswParam()`.

# 3 Further options

Most of the options described for the exact methods are also applicable here. For example:

- `subset` to identify neighbors for a subset of points.
- `get.distance` to avoid retrieving distances when unnecessary.
- `BPPARAM` to parallelize the calculations across multiple workers.
- `BNINDEX` to build the forest once for a given data set and re-use it across calls.

The use of a pre-built `BNINDEX` is illustrated below:

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

Both Annoy and HNSW perform searches based on the Euclidean distance by default. Searching by Manhattan distance is done by simply setting `distance="Manhattan"` in `AnnoyParam()` or `HnswParam()`.

Users are referred to the documentation of each function for specific details on the available arguments.

# 4 Saving the index files

Both Annoy and HNSW generate indexing structures - a forest of trees and series of graphs, respectively - that are saved to file when calling `buildIndex()`. By default, this file is located in `tempdir()`1 1 1 On HPC file systems, you can change `TEMPDIR` to a location that is more amenable to concurrent access. and will be removed when the session finishes.

    AnnoyIndex_path(pre)

    ## [1] "/tmp/RtmpMDZF9O/file24d8b43ba45760.idx"

If the index is to persist across sessions, the path of the index file can be directly specified in `buildIndex`. This can be used to construct an index object directly using the relevant constructors, e.g., `AnnoyIndex()`, `HnswIndex()`. However, it becomes the responsibility of the user to clean up any temporary indexing files after calculations are complete.

# 5 Session information

    sessionInfo()

    ## R version 4.4.0 beta (2024-04-15 r86425)
    ## Platform: x86_64-pc-linux-gnu
    ## Running under: Ubuntu 22.04.4 LTS
    ##
    ## Matrix products: default
    ## BLAS:   /home/biocbuild/bbs-3.19-bioc/R/lib/libRblas.so
    ## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.10.0
    ##
    ## locale:
    ##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
    ##  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8
    ##  [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
    ##
    ## time zone: America/New_York
    ## tzcode source: system (glibc)
    ##
    ## attached base packages:
    ## [1] stats     graphics  grDevices utils     datasets  methods   base
    ##
    ## other attached packages:
    ## [1] BiocNeighbors_1.22.0 knitr_1.46           BiocStyle_2.32.0
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] cli_3.6.2           rlang_1.1.3         xfun_0.43
    ##  [4] jsonlite_1.8.8      S4Vectors_0.42.0    htmltools_0.5.8.1
    ##  [7] stats4_4.4.0        sass_0.4.9          rmarkdown_2.26
    ## [10] grid_4.4.0          evaluate_0.23       jquerylib_0.1.4
    ## [13] fastmap_1.1.1       yaml_2.3.8          lifecycle_1.0.4
    ## [16] bookdown_0.39       BiocManager_1.30.22 compiler_4.4.0
    ## [19] codetools_0.2-20    Rcpp_1.0.12         BiocParallel_1.38.0
    ## [22] lattice_0.22-6      digest_0.6.35       R6_2.5.1
    ## [25] parallel_4.4.0      bslib_0.7.0         Matrix_1.7-0
    ## [28] tools_4.4.0         BiocGenerics_0.50.0 cachem_1.0.8