In a 1985 paper, the pc scientist Andrew Yao, who would go on to win the A.M. Turing Award, asserted that amongst hash tables with a selected set of properties, one of the simplest ways to seek out a person factor or an empty spot is to only undergo potential spots randomly—an strategy referred to as uniform probing. He additionally said that, within the worst-case state of affairs, the place you’re looking for the final remaining open spot, you possibly can by no means do higher than x. For 40 years, most pc scientists assumed that Yao’s conjecture was true.
Krapivin was not held again by the standard knowledge for the easy purpose that he was unaware of it. “I did this with out understanding about Yao’s conjecture,” he mentioned. His explorations with tiny pointers led to a brand new form of hash desk—one which didn’t depend on uniform probing. And for this new hash desk, the time required for worst-case queries and insertions is proportional to (log x)2—far sooner than x. This outcome instantly contradicted Yao’s conjecture. Farach-Colton and Kuszmaul helped Krapivin present that (log x)2 is the optimum, unbeatable certain for the favored class of hash tables Yao had written about.
“This result’s lovely in that it addresses and solves such a traditional downside,” mentioned Man Blelloch of Carnegie Mellon.
“It’s not simply that they disproved [Yao’s conjecture], additionally they discovered the very best reply to his query,” mentioned Sepehr Assadi of the College of Waterloo. “We may have gone one other 40 years earlier than we knew the fitting reply.”
Along with refuting Yao’s conjecture, the brand new paper additionally accommodates what many think about an much more astonishing outcome. It pertains to a associated, although barely totally different, scenario: In 1985, Yao appeared not solely on the worst-case instances for queries, but in addition on the common time taken throughout all doable queries. He proved that hash tables with sure properties—together with these which are labeled “grasping,” which signifies that new parts have to be positioned within the first obtainable spot—may by no means obtain a mean time higher than log x.
Farach-Colton, Krapivin, and Kuszmaul needed to see if that very same restrict additionally utilized to non-greedy hash tables. They confirmed that it didn’t by offering a counterexample, a non-greedy hash desk with a mean question time that’s a lot, significantly better than log x. In actual fact, it doesn’t depend upon x in any respect. “You get a quantity,” Farach-Colton mentioned, “one thing that’s only a fixed and doesn’t depend upon how full the hash desk is.” The truth that you possibly can obtain a relentless common question time, whatever the hash desk’s fullness, was wholly sudden—even to the authors themselves.
The staff’s outcomes could not result in any rapid functions, however that’s not all that issues, Conway mentioned. “It’s essential to grasp these sorts of information buildings higher. You don’t know when a outcome like it will unlock one thing that permits you to do higher in follow.”
Unique story reprinted with permission from Quanta Journal, an editorially impartial publication of the Simons Basis whose mission is to boost public understanding of science by overlaying analysis developments and tendencies in arithmetic and the bodily and life sciences.
