The problem asks for the absolute difference between two specific extrema: the maximum value among elements at even indices and the minimum value among elements at odd indices. A naive solution would involve two nested loops or separate scans that collect all even-indexed and odd-indexed values into auxiliary arrays, then compute the extrema, leading to O(n) time but also O(n) additional space. However, the optimal approach requires only a single pass through the array, maintaining two running variables—one for the current maximum at even indices and one for the current minimum at odd indices—thereby achieving O(n) time with O(1) extra space.

The key algorithmic insight is that the extrema can be updated incrementally: when encountering an element at an even index, compare it with the current even-max and update if larger; similarly, when encountering an odd index, compare with the current odd-min and update if smaller. This eliminates the need to store all values, which is especially important for large inputs where memory usage becomes a bottleneck. The final absolute difference is then computed in constant time after the single traversal.

This pattern exemplifies the classic “single-pass scan with running aggregates” paradigm, which is ubiquitous in interview questions involving extrema, prefix sums, or sliding windows. It demonstrates how careful state management can reduce both time and space complexity, a skill that is highly valued in production systems where performance and resource constraints are critical.