Question

Show that the median of five numbers can be found by making six comparisons.

Short answer

With strategic comparisons, it is possible to find the median of five numbers with six comparisons. The process involves forming two pairs, comparing within and between pairs, comparing with the leftover number, and a final comparison to determine the median.

Step-by-step solution

Pair Comparisons

Start by splitting the list of five numbers into two pairs and one leftover. Let's say our numbers are [A,B,C,D,E]. We can form pairs (A,B) and (C,D), with E remaining alone. Compare the numbers in each pair and ensure that for each pair, the first number is smaller. This requires two comparisons - one for each pair.

Middle Comparison

Compare the larger numbers in both pairs. In this case, you compare the larger number from the pair (A,B) with the larger number from the pair (C,D). Without loss of generality, assume that the larger number from the (A,B) pair is smaller, reorder the pairs such that (A,B) is the pair with the smaller largest element. This process requires one comparison.

Comparison with Single Number

Compare the single leftover number (E) with the middle number in the sorted list so far. Let's say that number is from pair (A,B). If E is smaller, compare it with A. If it is larger, compare it with B. This step involves two comparisons.

Final Comparison

The final step is to compare the larger number from pair (C,D) with E. This requires one comparison and will give you the median of these five numbers.

Key concepts

Algorithmic Strategy

In the realm of problem-solving, the concept of an algorithmic strategy is crucial. It refers to the methodical, step-by-step process designed to solve a problem in a finite number of steps. While tackling the problem of finding the median of five numbers, one must utilize an effective algorithmic strategy to ensure efficiency and accuracy.

The approach to finding the median, as outlined in the original exercise, is a testament to strategic thinking. By breaking down the problem into smaller, manageable steps, we avoid the potential computational heaviness of comparing each number with all others. This streamlined process requires only six comparisons, and thus demonstrates the success of a well-planned algorithmic strategy in optimizing the solution to a problem.

Comparison Operation

In algorithm design, a comparison operation is a fundamental action that determines the relative order of elements within a data set. It can ascertain whether one element is less than, greater than, or equal to another. This is particularly important when finding a median, which inherently depends on the relative ranking of numbers within a set.

Diving into the problem at hand, each comparison operation is deliberately placed to progressively narrow down the location of the median in a series of logical steps. By structuring the comparisons in pairs, and then against each other, the algorithm uses these binary decisions to efficiently locate the central value with minimal computational effort. This step-by-step processing illustrates the power of simple comparison operations when intelligently orchestrated.

Selection Algorithm

A selection algorithm is a recipe for finding the k-th smallest or largest element in a list or array. For the purpose of finding a median, which is essentially the middle element in a sorted list, a selection algorithm can be optimally designed to minimize the number of operations required.

The exercise portrays a selection algorithm specifically tailored to identify the median in a fixed-size list comprising five elements. By using strategically positioned comparisons, the algorithm selects the median with a predefined number of steps. The brilliance of this approach is its independence from full sorting. While a sorting algorithm would require more comparisons for a list of this size, the selection algorithm is engineered to find the median using only six targeted comparisons, showcasing its efficiency for the task at hand.