Fast Exact Nearest Neighbour Matching in High Dimensions Using 𝑑-D Sort

Algorithm 1

𝑘NN Query Algorithm for the 𝑑-D Sort Structure.

  Input: x, 𝑜, q, 𝑘, 𝑟, 𝜖

  Output: 𝑛_𝑟, 𝑖_𝑟, 𝑒_𝑟

(1.1) begin

(1.2)  Sort query, q in ascending order, yielding order array 𝑜_𝑞.

(1.3)  Select prime dimension, 𝑗←𝑜_𝑞{𝑑}.

(1.4)  Select data order vector 𝑜_𝑗=𝑜{𝑗} as the search order.

(1.5)  Compute 𝑟_𝑢 and 𝑟_𝑙.

(1.6)  Compute 𝑔 according to (5).

(1.7)  Compute initial search range, [𝑡_𝑢,𝑡_𝑙], according to (6).

(1.8)  Use binary search to find  𝑖₀←arg_𝑖min𝐱{𝑜_𝑗{𝑖}}{𝑗}−𝑞_𝑗.

(1.9)  foreach  𝑖∈[1,𝑛] in order of increasing distance from 𝑖₀   do

(1.10)   If 𝐱{𝑜_𝑗{𝑖}}{𝑗} is out of search range [𝑡_𝑢,𝑡_𝑙], terminate.

(1.11)   Compute distance 𝑒 = dist 𝐱𝑜_𝑗{𝑖},𝐪 using the ordered partial distance method.

(1.12)   if  𝑒<𝑟  then

(1.13)    Insert 𝑖, 𝑒 in results list, 𝑖_𝑟, 𝑒_𝑟.

(1.14)    Increment 𝑛_𝑟 (up to 𝑘).

(1.15)    if  𝑛_𝑟=𝑘  then

(1.16)     Reduce the search range, 𝑟←𝑒_𝑘/(1+𝜖).

(1.17)     Recompute [𝑡_𝑢,𝑡_𝑙] according to (6).

(1.18)    end

(1.19)   end

(1.20)   end

(1.21) end