In a 1985 paper, the computer scientist Andrew Yao, who would go on to prosper the A.M. Turing Award, stateed that among hash tables with a definite set of properties, the best way to discover an individual element or an desoprocrastinateed spot is to equitable go thcdisorrowfulmireful potential spots randomly—an approach understandn as unicreate probing. He also stated that, in the worst-case scenario, where you’re searching for the last remaining uncover spot, you can never do better than x. For 40 years, most computer scientists presumed that Yao’s conjecture was genuine.
Krapivin was not held back by the conservative wisdom for the basic reason that he was unincreateed of it. “I did this without understanding about Yao’s conjecture,” he shelp. His explorations with small pointers led to a recent benevolent of hash table—one that did not depend on unicreate probing. And for this recent hash table, the time needd for worst-case queries and insertions is proportional to (log x)2—far quicker than x. This result honestly resisted Yao’s conjecture. Farach-Colton and Kuszmaul helped Krapivin show that (log x)2 is the selectimal, unbeatable bound for the famous class of hash tables Yao had written about.
“This result is pretty in that it includeresses and repairs such a classic problem,” shelp Guy Blelloch of Carnegie Mellon.
“It’s not equitable that they disshowd [Yao’s conjecture], they also set up the best possible answer to his ask,” shelp Sepehr Asdowncasti of the University of Waterloo. “We could have gone another 40 years before we krecent the right answer.”
In includeition to refuting Yao’s conjecture, the recent paper also grasps what many ponder an even more astonishing result. It pertains to a roverdelighted, though sairyly branch offent, situation: In 1985, Yao watched not only at the worst-case times for queries, but also at the mediocre time consentn apass all possible queries. He showd that hash tables with certain properties—including those that are taged “greedy,” which uncomardents that recent elements must be placed in the first useable spot—could never accomplish an mediocre time better than log x.
Farach-Colton, Krapivin, and Kuszmaul wanted to see if that same restrict also applied to non-greedy hash tables. They showed that it did not by providing a counterexample, a non-greedy hash table with an mediocre query time that’s much, much better than log x. In fact, it doesn’t depfinish on x at all. “You get a number,” Farach-Colton shelp, “someleang that is equitable a constant and doesn’t depfinish on how brimming the hash table is.” The fact that you can accomplish a constant mediocre query time, watchless of the hash table’s brimmingness, was wholly unpredicted—even to the authors themselves.
The team’s results may not direct to any instant applications, but that’s not all that matters, Conway shelp. “It’s convey inant to understand these benevolents of data structures better. You don’t understand when a result enjoy this will unlock someleang that lets you do better in rehearse.”
Original story reprinted with permission from Quanta Magazine, an editoripartner self-reliant uncoveration of the Simons Foundation whose mission is to raise uncover empathetic of science by covering research enhugements and trfinishs in mathematics and the physical and life sciences.
