GATE CS 2014 Set 1 Q.49 (CS Q.39): Searching and Sorting Algorithms

The minimum number of comparisons required to find the minimum and the maximum of 100 numbers is _________________.

Your answer

Official answer and solution

Solution

Use the optimal pairwise comparison method to find both minimum and maximum of an array simultaneously.

For finding both the minimum and maximum of $n$ numbers, the standard divide-and-conquer / pairwise comparison algorithm proceeds as follows:

Pair up the $n = 100$ elements into $n / 2 = 50$ pairs.
Compare elements in each pair: this requires $\frac{n}{2} = 50$ $\frac{n}{2} = 50$ comparisons.
- The 50 winners are candidates for the maximum.
- The 50 losers are candidates for the minimum.
Find the maximum among the 50 maximum candidates: this requires $50 - 1 = 49$ comparisons.
Find the minimum among the 50 minimum candidates: this requires $50 - 1 = 49$ comparisons.

Total number of comparisons:

\text{Total} = 50 + 49 + 49 = 148

In general, for even $n$ , the optimal number of comparisons is:

\frac{3n}{2} - 2 = \frac{3(100)}{2} - 2 = 150 - 2 = 148

Final answer

148

Common trap

Finding minimum first in $n-1 = 99$ comparisons and maximum separately in $n-2 = 98$ comparisons, totaling $197$ comparisons, which is not minimal.

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