Find a missing 32bit integer among a unsorted array containing at most 4 billion ints

Question

This is the problem described in Programming pearls. I can not understand binary search method descrbied by the author. Can any one helps to elaborate? Thanks.

EDIT: I can understand binary search in general. I just can not understand how to apply binary search in this special case. How to decide the missing number is in or not in some range so that we can choose another. English is not my native language, that is one reason I can not understand the author well. So, use plain english please:)

EDIT: Thank you all for your great answer and comments ! The most important lesson I leant from solving this question is Binary search applies not only on sorted array!

Answer 1

There is some more information on this web site. Quoted from there:

"It is helpful to view this binary search in terms of the twenty bits that represent each integer. In the first pass of the algorithm we read the (at most) one million input integers and write those with a leading zero bit to one tape and those with a leading one bit to another tape. One of those two tapes contains at most 500,000 integers, so we next use that tape as the current input and repeat the probe process, but this time on the second bit. If the original input tape contains N elements, the first pass will read N integers, the second pass at most N/2, the third pass at most N/4, and so on, so the total running time is proportional to N. The missing integer could be found by sorting on tape and then scanning, but that would require time proportional to N log N."

As you can see, this is a variation on the binary search algorithm: divide the problem into two pieces and solve the problem for one of the smaller pieces.