algorithm - Removing even/odd numbers to find missing number in an array of array of numbers represented in binary -


this related problem 5.7 (under bit manipulation) in 5th edition of cracking coding interview (if question not appropriate so, please let me know correct site , i'll move it):

an array a[1..n] contains integers 0 n except 1 number missing. in problem, cannot access entire integer in single opera-tion. elements of represented in binary, , operation can use access them “fetch jth bit of a[i]”, takes constant time. write code find missing integer. can in o(n) time?

the algorithm applied this:

  1. check lsbs of numbers in list.
  2. count occurrence of 1's , 0's in lsbs. if count(0)<=count(1), lsb of missing number 0. else 1.
  3. remove numbers lsb not matching result found in step 2.
  4. repeat 1 4, , progressively check next lsb in each iteration.

can explain logic behind step 3? removes odd/even numbers current list (depending on bit found missing number) , uses modified list in next iteration. why this?

step 3 meant (vastly) improve runtime of algorithm. if step 3 included, overall algorithm binary search algorithm using lsb branching decider. if step 3 omitted, still binary search, 1 implemented in way not become logarithmically faster on each pass (which exceed o(n) bound).

incidentally, written seems there's bit shift missing, or term lsb being used in rather liberal way.


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