Knee Point Search Using Cascading Top-k Sorting with Minimized Time Complexity

Algorithm 2

FindKnee(𝐿, 𝑃).

Input: Unsorted list 𝐿 of length 𝑛, probability distribution of the knee point in the sorted

list of length 𝑛: 𝑃

Output: The knee point e or the non-existence of it (𝑒 = null).

(1) /* initialize the parameters for the optimization problem */

(2) 𝑃₁←𝑃, 𝐿₁←𝐿

(3) 𝑖←1

(4) while (true)

(5) solve the optimization problem of list 𝐿_𝑖 in (18) to get 𝑑_𝑖

(6)  if  𝑑_𝑖=𝑛_𝑖  then /* total sorting is the optimal option */

(7)  totally sort list 𝐿_𝑖

(8)  /* successful if the knee point exists */

(9)  Search the sorted list for the knee point e

(10)  break /* the algorithm is completed */

(11)  else /* top-𝑑_𝑖−𝑑_𝑖−1 sort is the optimal option */

(12)  QuickSortTopK(𝐿, 𝑑_𝑖−1 + 1, 𝑛, 𝑑_𝑖--𝑑_𝑖−1)

(13)  Search the sorted list for the knee point e

(14)  if  e is found then

(15)   break

(16)  else /* Update the parameters for the next round of optimization problem */

(17)   update 𝑃_𝑖+1 according to (5) /* the posterior probability */

(18)   update 𝐿_𝑖+1 by exluding the top-𝑑_𝑖−𝑑_𝑖−1 item from list 𝐿_𝑖

(19)   𝑖←𝑖+1

(20)  return  𝑒