[+] Integer: sorting in linear time for w = Ω(lg^{2+ε} n), priority queues  
This lecture is about the stateoftheart in sorting and priority
queues on a word RAM. An equivalence by Thorup shows that any sorting
algorithm can be transformed into a priority queue with operations taking
1/nth the time to sort. So these are really one and the same problem.
The best results we know for sorting in linear time (and thus for constanttime priority queues) is when w = O(lg n) and when w = Ω(lg^{2+ε} n). The first result is just radix sort. The second result is the main topic of the lecture: a fancy wordRAM algorithm called signature sorting. It uses a combination of hashing, merge sort, and parallel sorting networks. The range of w in between lg and lg^{2+ε} remains unsolved. The best algorithm so far runs in O(n √lg lg n) expected time.  
Lecture notes, page 1/7 •
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