Ranged/Filtered Cross Join with R data.table

Question

I want to cross-join two data tables without evaluating the full cross join, using a ranging criterion in the process. In essence, I would like CJ with filtering/ranging expression.

Can someone suggest a high performing approach avoiding the full cross join?

See test example below doing the job with the evil full cross join.

library(data.table)

# Test data.
dt1 <- data.table(id1=1:10, D=2*(1:10), key="id1")
dt2 <- data.table(id2=21:23, D1=c(5, 7, 10), D2=c(9, 12, 16), key="id2")

# Desired filtered cross-join data table by hand: D1 <= D & D <= D2.
dtfDesired <- data.table(
    id1=c(3, 4, 4, 5, 6, 5, 6, 7, 8)
  , id2=c(rep(21, 2), rep(22, 3), rep(23, 4))
  , D1=c(rep(5, 2), rep(7, 3), rep(10, 4))
  , D=c(6, 8, 8, 10, 12, 10, 12, 14, 16)
  , D2=c(rep(9, 2), rep(12, 3), rep(16, 4))
)
setkey(dtfDesired, id1, id2)

# My "inefficient" programmatic attempt with full cross join.
fullCJ <- function(dt1, dt2) {
  # Full cross-product: NOT acceptable with real data!
  dtCrossAll <- CJ(dt1$id1, dt2$id2)
  setnames(dtCrossAll, c("id1", "id2"))
  # Merge all columns.
  dtf <- merge(dtCrossAll, dt1, by="id1")
  dtf <- merge(dtf, dt2, by="id2")
  setkey(dtf, id1, id2)
  # Reorder columns for convenience.
  setcolorder(dtf, c("id1", "id2", "D1", "D", "D2"))
  # Finally, filter the cases I want.
  dtf[D1 <= D & D <= D2, ]
}

dtf <- fullCJ(dt1, dt2)

# Print results.
print(dt1)
print(dt2)
print(dtfDesired)
all.equal(dtf, dtfDesired)

Test data output

> # Print results.
> print(dt1)
    id1  D
 1:   1  2
 2:   2  4
 3:   3  6
 4:   4  8
 5:   5 10
 6:   6 12
 7:   7 14
 8:   8 16
 9:   9 18
10:  10 20
> print(dt2)
   id2 D1 D2
1:  21  5  9
2:  22  7 12
3:  23 10 16
> print(dtfDesired)
   id1 id2 D1  D D2
1:   3  21  5  6  9
2:   4  21  5  8  9
3:   4  22  7  8 12
4:   5  22  7 10 12
5:   5  23 10 10 16
6:   6  22  7 12 12
7:   6  23 10 12 16
8:   7  23 10 14 16
9:   8  23 10 16 16
> all.equal(dtf, dtfDesired)
[1] TRUE

So now the challenge is to write the filtered cross join in a way that can scale to millions of rows!

Below are a collection of alternative implementations including those suggested in answers and comments.

# My "inefficient" programmatic attempt looping manually.
manualIter <- function(dt1, dt2) {
  id1Match <- NULL; id2Match <- NULL; dtf <- NULL;
  for (i1 in seq_len(nrow(dt1))) {
    # Find matches in dt2 of this dt1 row.
    row1 <- dt1[i1, ]
    id1 <- row1$id1
    D <- row1$D
    dt2Match <- dt2[D1 <= D & D <= D2, ]
    nMatches <- nrow(dt2Match)
    if (0 < nMatches) {
      id1Match <- c(id1Match, rep(id1, nMatches))
      id2Match <- c(id2Match, dt2Match$id2)
    }
  }
  # Build the return data.table for the matching ids.
  dtf <- data.table(id1=id1Match, id2=id2Match)
  dtf <- merge(dtf, dt1, by="id1")
  dtf <- merge(dtf, dt2, by="id2")
  setkey(dtf, id1, id2)
  # Reorder columns for convenience & consistency.
  setcolorder(dtf, c("id1", "id2", "D1", "D", "D2"))
  return(dtf)
}

dtJoin1 <- function(dt1, dt2) {
  dtf <- dt1[, dt2[D1 <= D & D <= D2, list(id2=id2)], by=id1]
  dtf <- merge(dtf, dt1, by="id1")
  dtf <- merge(dtf, dt2, by="id2")
  setkey(dtf, id1, id2)
  setcolorder(dtf, c("id1", "id2", "D1", "D", "D2")) # Reorder columns for convenience & consistency.
  return(dtf)
}

dtJoin2 <- function(dt1, dt2) {
  dtf <- dt2[, dt1[D1 <= D & D <= D2, list(id1=id1, D1=D1, D=D, D2=D2)], by=id2]
  setkey(dtf, id1, id2)
  setcolorder(dtf, c("id1", "id2", "D1", "D", "D2")) # Reorder columns for convenience & consistency.
  return(dtf)
}

# Install Bioconductor IRanges (see bioTreeRange below).
source("http://bioconductor.org/biocLite.R")
biocLite("IRanges")

# Solution using Bioconductor IRanges.
bioTreeRange <- function(dt1, dt2) {
  require(IRanges)
  ir1 <- IRanges(dt1$D, width=1L)
  ir2 <- IRanges(dt2$D1, dt2$D2)
  olaps <- findOverlaps(ir1, ir2, type="within")
  dtf <- cbind(dt1[queryHits(olaps)], dt2[subjectHits(olaps)])
  setkey(dtf, id1, id2)
  setcolorder(dtf, c("id1", "id2", "D1", "D", "D2")) # Reorder columns for convenience.
  return(dtf)
}

And now below is a little benchmark on a bigger data set 2-3 orders of magnitude smaller than my real underlying scenario. The real scenario fails on the full cross-join huge memory allocation.

set.seed(1)
n1 <- 10000
n2 <- 1000
dtbig1 <- data.table(id1=1:n1, D=1:n1, key="id1")
dtbig2 <- data.table(id2=1:n2, D1=sort(sample(1:n1, n2)), key="id2")
dtbig2$D2 <- with(dtbig2, D1 + 100)

library("microbenchmark")
mbenchmarkRes <- microbenchmark(
  fullCJRes <- fullCJ(dtbig1, dtbig2)
  , manualIterRes <- manualIter(dtbig1, dtbig2)
  , dtJoin1Res <- dtJoin1(dtbig1, dtbig2)
  , dtJoin2Res <- dtJoin2(dtbig1, dtbig2)
  , bioTreeRangeRes <- bioTreeRange(dtbig1, dtbig2)
  , times=3, unit="s", control=list(order="inorder", warmup=1)
)
mbenchmarkRes$expr <- c("fullCJ", "manualIter", "dtJoin1", "dtJoin2", "bioTreeRangeRes") # Shorten names for better display.

# Print microbenchmark
print(mbenchmarkRes, order="median")

And now the current benchmark results I got on my machine:

> print(mbenchmarkRes, order="median")
Unit: seconds
            expr        min         lq     median         uq        max neval
 bioTreeRangeRes 0.05833279 0.05843753 0.05854227 0.06099377 0.06344527     3
         dtJoin2 1.20519664 1.21583650 1.22647637 1.23606216 1.24564796     3
          fullCJ 4.00370434 4.03572702 4.06774969 4.17001658 4.27228347     3
         dtJoin1 8.02416333 8.03504136 8.04591938 8.20015977 8.35440016     3
      manualIter 8.69061759 8.69716448 8.70371137 8.76859060 8.83346982     3

Conclusions

• The Bioconductor tree/IRanges solution from Arun (bioTreeRangeRes) is two orders of magnitude faster than the alternatives. But the install seems to have updated other CRAN libraries (my fault, I accepted it when the install asked the question); some of them can no longer be found when loading them -- e.g., gtools and gplots.
• The fastest pure data.table option from BrodieG (dtJoin2) is probably not as efficient as I need it to be but at least is reasonable in terms of memory consumption (I will let it run overnight on my real scenario ~ 1 Million rows).
• I tried changing the data table keys (using the dates instead of id's); it did not have any impact.
• As expected, explicitly writing the loop in R (manualIter) crawls.

Answer 1

Recently, overlap joins are implemented in data.table. This is a special case where dt1's `start and end points are identical. You can grab the latest version from the github project page to try this out:

require(data.table) ## 1.9.3+
dt1[, DD := D] ## duplicate column D to create intervals
setkey(dt2, D1,D2) ## key needs to be set for 2nd argument
foverlaps(dt1, dt2, by.x=c("D", "DD"), by.y=key(dt2), nomatch=0L)

#    id2 D1 D2 id1  D DD
# 1:  21  5  9   3  6  6
# 2:  21  5  9   4  8  8
# 3:  22  7 12   4  8  8
# 4:  22  7 12   5 10 10
# 5:  23 10 16   5 10 10
# 6:  22  7 12   6 12 12
# 7:  23 10 16   6 12 12
# 8:  23 10 16   7 14 14
# 9:  23 10 16   8 16 16

Here's the results benchmarking on the same data you've shown in your post:

# Unit: seconds
#             expr         min          lq      median          uq         max neval
#            olaps  0.03600603  0.03971068  0.04341533  0.04857602  0.05373671     3
#  bioTreeRangeRes  0.11356837  0.11673968  0.11991100  0.12499391  0.13007681     3
#          dtJoin2  2.61679908  2.70327940  2.78975971  2.86864832  2.94753693     3
#           fullCJ  4.45173294  4.75271285  5.05369275  5.08333291  5.11297307     3
#          dtJoin1 16.51898878 17.39207632 18.26516387 18.60092303 18.93668220     3
#       manualIter 29.36023340 30.13354967 30.90686594 33.55910653 36.21134712     3

where dt_olaps is:

dt_olaps <- function(dt1, dt2) {
    dt1[, DD := D]
    setkey(dt2, D1,D2)
    foverlaps(dt1, dt2, by.x=c("D","DD"), by.y=key(dt2), nomatch=0L)
}