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Cs-621 Theory Gems
So far, we were focusing on “static” analysis of games. That is, we considered scenarios in which the game and utilities of all the players are fixed and known and our task is only to predict possible outcomes of that game when some (or all) the players are acting rationally. Today, we turn the tables: there is no predefined game, only players that have some utilities. However, the key point is...
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As we mentioned in the last lecture, one of the core components of our algorithm for distinct elements problem is a space-efficient construction of 2-wise independent functions. Formally, a k-wise independent hash function f : [m]→ [T ] is a randomized function that provides the guarantee that, for any k distinct elements j1, . . . , jk ∈ [m] and any k possible values t1, . . . , tk ∈ [T ], the...
متن کاملCs-621 Theory Gems
So far, we have seen streaming algorithms for two important variants of Lp-norm estimation problem: L0-norm estimation (the distinct elements problem) and L2-norm estimation. We also noted that the L1norm estimation problem (at least, when we do not allow element deletions) corresponds to just computing the length of the stream and thus can be trivially solved in O(log n) space. Therefore, the ...
متن کاملCs-621 Theory Gems
That is, we want the hyperplane corresponding to (w, θ) to separate the positive examples from the negative ones. As we already argued previously, wlog we can constrain ourselves to the case when θ = 0 (i.e., the hyperplane passes through the origin) and there are only positive examples (i.e., l = 1, for all j). Last time we presented a simple algorithm for this problem called Perceptron algori...
متن کاملCs-621 Theory Gems
In Lecture 10, we introduced a fundamental object of spectral graph theory: the graph Laplacian, and established some of its basic properties. We then focused on the task of estimating the value of eigenvalues of Laplacians. In particular, we proved the Courant-Fisher theorem that is instrumental in obtaining upper-bounding estimates on eigenvalues. Today, we continue by showing a technique – s...
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ژورنال
عنوان ژورنال: Advances in Mathematics
سال: 1999
ISSN: 0001-8708
DOI: 10.1006/aima.1998.1796