To be pedantic, not all problems in P are equally hard. Sorting for example is harder than finding the median.
The provable lower bound for the worst-case performance of searching is O(n log n). Finding the median takes linear time i.e. O(n).
Sorting and finding a median reduce to each other polynomially, of course.
Interesting subsets of polynamial problems are e.g. linear problems, or highly parallelizeable problems whose runtime tends to O(log n) as the number of processors tends to infinity. Adding a list of numbers is not only linear but also highly parallelizeable. Proving a problem to be not parallelizeable like that, is also interesting.
There's also the question whether randomness helps. It does in practice, but we don't know whether access to random bits helps in theory.
The provable lower bound for the worst-case performance of searching is O(n log n). Finding the median takes linear time i.e. O(n).
Sorting and finding a median reduce to each other polynomially, of course.
Interesting subsets of polynamial problems are e.g. linear problems, or highly parallelizeable problems whose runtime tends to O(log n) as the number of processors tends to infinity. Adding a list of numbers is not only linear but also highly parallelizeable. Proving a problem to be not parallelizeable like that, is also interesting.
There's also the question whether randomness helps. It does in practice, but we don't know whether access to random bits helps in theory.