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

Algorithm 3

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) solve the optimization problem of list 𝐿 in (20) to get 𝑑

(2)  if  𝑑=𝑛 then /* total sort is the optimal option */

(3)  totally sort list 𝐿

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

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

(6)  return 𝑒 /* the algorithm is completed */

(7)  else /* top-𝑑 sorting is the optimal option */

(8)   QuickSortTopK(𝐿, 1, length(𝐿), 𝑑)

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

(10)    if 𝑒 is found then

(11)    return e

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

(13)    update 𝑃 according to (5) /* the posterior probability */

(14)    update 𝐿 by exluding the top-𝑑 item from list 𝐿

(15)     FindKnee(𝐿, 𝑃)