Thoughts on Minimizing Strangers

Abstract

Let GE(x) be the set of keys greater than or equal to key x and let LE(x) be the set of keys less than or equal to x. Also, let N1(x) be the number of ones that appear in x’s row in the Shmoo chart and let N0(x) be the number of zeros that appear in x’s row in the same Shmoo chart. Careful inspection of the Shmoo chart helps us realize that N1(x) = |LE(x)| and that N0(x) = |GE(x)|. If we compare two keys, x and y, to find key-max(x,y) and key-min(x,y), then we can conclude the following: N0(max(x,y)) = min(|GE(x)|, |GE(y)|); N1(max(x,y)) = |LE(x) \/LE(y)|; N0(min(x,y)) = |GE(x) \/GE(y)|; and N1(min(x,y)) = min(|LE(x)|, |LE(y)|). Thus, to minimize the number of strangers in key-max(x,y) and key-min(x,y) one should pick a comparator that maximizes the sum of N0(max(x,y)), N1(max(x,y)), N0(min(x,y)), and N1(min(x,y)). SHOW.WORTHS displays an N-column by N-row table. The entry in column-x and row-y of the table shows the worth of comparing key-x with key-y; i.e., the number of strangers that that comparator C(x,y) eliminates.