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One possible approach using the divide and conquer method with a time complexity of O(N log N) to determine the greatest discrepancy among sixteen-digit binary numbers ranging from 1 to less than 10^5 is as follows:

1. Divide the range of numbers into two equal parts, recursively applying this step until each sub-range contains at most one number.
2. For each sub-range, compute the minimum and maximum values, as well as their absolute difference (discrepancy).
3. Merge the results obtained from the sub-ranges, comparing the discrepancies to identify the greatest one.
4. Return the greatest discrepancy found.

The main idea behind this algorithm is to exploit the properties of binary numbers, which have a fixed length of 16 digits, to reduce the search space and avoid redundant computations. By dividing the range of numbers into smaller sub-ranges, we can compute the discrepancies independently and combine them efficiently, using logarithmic time for each recursive step. The final result is then obtained by comparing the discrepancies found in each sub-range, selecting the highest one. This approach is effective for larger data sets and can be extended to other domains, such as text search or numerical optimization.