Finding number of elements in one vector that are less than an element in another vector

Question

Say we have a couple vectors

a <- c(1, 2, 2, 4, 7)
b <- c(1, 2, 3, 5, 7)

For each element b[i] in b I want find the number of elements in a that's less than b[i], or, equivalent, I want to know the rank of b_i in c(b[i], a).

there are a couple naive ways I can think of, e.g. doing either of the following length(b) times:

min_rank(c(b[i], a))
sum(a < b[i])

What's the best way to do this if length(a) = length(b) = N where N is large?

EDIT:

To clarify, I'm wondering if there's a more computationally efficient way to do this, i.e. if I can do better than quadratic time in this case.

Vectorization is always cool though ;), thanks @Henrik!

Running time

a <- rpois(100000, 20)
b <- rpois(100000, 10)

system.time(
  result1 <- sapply(b, function(x) sum(a < x))
)
# user  system elapsed 
# 71.15    0.00   71.16

sw <- proc.time()
  bu <- sort(unique(b))
  ab <- sort(c(a, bu))
  ind <- match(bu, ab)
  nbelow <- ind - 1:length(bu)
  result2 <- sapply(b, function(x) nbelow[match(x, bu)])
proc.time() - sw

# user  system elapsed 
# 0.46    0.00    0.48 

sw <- proc.time()
  a1 <- sort(a)
  result3 <- findInterval(b - sqrt(.Machine$double.eps), a1)
proc.time() - sw

# user  system elapsed 
# 0.00    0.00    0.03 

identical(result1, result2) && identical(result2, result3)
# [1] TRUE

Answer 1

Assuming that a is weakly sorted increasingly, use findInterval:

a <- sort(a)
## gives points less than or equal to b[i]
findInterval(b, a)
# [1] 1 3 3 4 5
## to do strictly less than, subtract a small bit from b
## uses .Machine$double.eps (the smallest distinguishable difference)
findInterval(b - sqrt(.Machine$double.eps), a)
# [1] 0 1 3 4 4

Answer 2

If you're really optimising this process for large N, then you may want to remove duplicate values in b at least initially, and then you can sort and match:

bu <- sort(unique(b))
ab <- sort(c(a, bu))
ind <- match(bu, ab)
nbelow <- ind - 1:length(bu)

As we've merged a and b values into ab, the match includes all a less than the specific value of b together with all b's, so that's why we remove the cummulative count of b on the final line. I suspect this may be faster for large sets - it should be if match is internally optimised for sorted lists, which one would hope to be the case. It should then be a trivial matter to map back nbelow to your original set of bs

Answer 3

I don't claim this is "the best way", but it's a way. sapply applies the (anonymous) function to each element of b.

 sapply(b, function(x) sum(a < x))
 # [1] 0 1 3 4 4