Richard J. Lipton is the 2014 winner of the Knuth Prize. His talk today was a summary of his work, which lead to received the prize. The talk proved to be a series of short anecdotes, which are difficult to capture, but I've copied down the highlights, as best as I can.
"Do problems have labels?" For example, simulate a queue as two stacks, is this a theory problem or system problem? At the time, the qualifying exams were split by problem types, so labeling it mattered for which exam contained it. Faculty at Yale were 50/50 split on whether to mix the problem types and instead students would sit for several days of CS questions rather than a theory day, then a systems day, etc.
Finding a division, separator to a planar graph, in root time. T(n) <= C*T(n/2)
In explaining the result to Knuth, while visiting Tarjan, who responded "You've ruined my lunch." As the result destroyed the best known algorithms that were being written, at the time, in Vol. 4.
"Throw away comments are wrong" Many introductions make inaccurate statements like "non-uniform cannot imply uniform". There is the work of Karp-Lipton dealing with non-uniform circuits and the uniform nature of algorithms. The proof was later handed out on tote-bags at CCC 2010.
"Think in High Dimensions" Binary search in high dimensional space, still logarithmic in the number of elements. For example, take a planar graph and split it by the intersections, each slab is linear and can be quickly searched.
"Learn New Tools" Now, one tool is "Probabilistic method" published on June 28, 1974, which shortly thereafter was a Yale seminar. "By an Elementary Calculation" means to Erdos to use Sterling's approximation, which in one case required taking the approximation to 7 places. Before learning this method, had been asked about the problem of Extendible Hashing, and had no idea and put it out of mind. Later asked about it again, and the problem solved easily (or perhaps two days of proofs).
"Guess Right" One problem in solving problems in the community is that we are guessing wrong. "It is really hard to prove false statements." Take the problem of detecting whether a sequence of a_nb_m has n = m? Possible using a multi-pass scan with a probablistic FSM. Can do with one-way (i.e., single pass)?
"Need a Trick" Solving a problem of vector addition, with fixed counters, with adding and subtracting (where cannot subtract from 0). 1 counter is decidable, 2 counters is not. But if there is no test for whether the counter is 0. Proved it takes EXPSPACE-hard. Pair counters, so add is subtract and vice versa.
"My Favorite Two Results" - Proving that a a^-1 = 1, in long sequence (abaaaba^-1...) can be done in LOGSPACE. Do so by replacing the a, b with matrices, then modulo prime. Given the distributed law and applying in any order, prove that it always stops on any expression.
"Future" Old problems, yes. But dream of finding proofs to math problems that use CS theory tricks.
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