We study the space complexity of sketching cuts and Laplacian quadratic forms of graphs. We show that any data structure which approximately stores the sizes of all cuts in an undirected graph on @math vertices up to a @math error must use @math bits of space in the worst case, improving the @math bound of and matching the best known upper bound achieved by spectral sparsifiers. Our proof is based on a rigidity phenomenon for cut (and spectral) approximation which may be of independent interest: any two @math regular graphs which approximate each other's cuts significantly better than a random graph approximates the complete graph must overlap in a constant fraction of their edges.
The paper @cite_1 also studied the problem of producing a for each'' sketching algorithm, which has the property that for any particular query @math , @math approximates @math with constant probability. They showed that in this weaker model it is possible to obtain @math cut sketches of size @math and @math -spectral sketches of size @math . The latter bound was recently improved to @math and generalized to include queries to the pseudoinverse in @cite_4 . In contrast, in this paper we study only the for all'' model, in which the sketch is required to work for all queries simultaneously.
@cite_1 @cite_4
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This paper resolves one of the longest standing basic problems in the streaming computational model. Namely, optimal construction of quantile sketches. An e approximate quantile sketch receives a stream of items x1,·,xn and allows one to approximate the rank of any query item up to additive error e n with probability at least 1-δ.The rank of a query x is the number of stream items such that xi ≤ x. The minimal sketch size required for this task is trivially at least 1 e.Felber and Ostrovsky obtain a O((1 e)log(1 e)) space sketch for a fixed δ.Without restrictions on the nature of the stream or the ratio between e and n, no better upper or lower bounds were known to date. This paper obtains an O((1 e)log log (1 δ)) space sketch and a matching lower bound. This resolves the open problem and proves a qualitative gap between randomized and deterministic quantile sketching for which an Ω((1 e)log(1 e)) lower bound is known. One of our contributions is a novel representation and modification of the widely used merge-and-reduce construction. This modification allows for an analysis which is both tight and extremely simple. The same technique was reported, in private communications, to be useful for improving other sketching objectives and geometric coreset constructions. We undertake a systematic study of sketching a quadratic form: given an @math matrix @math , create a succinct sketch @math which can produce (without further access to @math ) a multiplicative @math -approximation to @math for any desired query @math . While a general matrix does not admit non-trivial sketches, positive semi-definite (PSD) matrices admit sketches of size @math , via the Johnson-Lindenstrauss lemma, achieving the "for each" guarantee, namely, for each query @math , with a constant probability the sketch succeeds. (For the stronger "for all" guarantee, where the sketch succeeds for all @math 's simultaneously, again there are no non-trivial sketches.) We design significantly better sketches for the important subclass of graph Laplacian matrices, which we also extend to symmetric diagonally dominant matrices. A sequence of work culminating in that of Batson, Spielman, and Srivastava (SIAM Review, 2014), shows that by choosing and reweighting @math edges in a graph, one achieves the "for all" guarantee. Our main results advance this front. @math For the "for all" guarantee, we prove that 's bound is optimal even when we restrict to "cut queries" @math . In contrast, previous lower bounds showed the bound only for spectral-sparsifiers . @math For the "for each" guarantee, we design a sketch of size @math bits for "cut queries" @math . We prove a nearly-matching lower bound of @math bits. For general queries @math , we construct sketches of size @math bits. In this article, we show several results obtained by combining the use of stable distributions with pseudorandom generators for bounded space. In particular:---We show that, for any p ∈ (0, 2], one can maintain (using only O(log n e2) words of storage) a sketch C(q) of a point q ∈ lnp under dynamic updates of its coordinates. The sketch has the property that, given C(q) and C(s), one can estimate Vq − sVp up to a factor of (1 p e) with large probability. This solves the main open problem of [1999].---We show that the aforementioned sketching approach directly translates into an approximate algorithm that, for a fixed linear mapping A, and given x ∈ ℜn and y ∈ ℜm, estimates VAx − yVp in O(n p m) time, for any p ∈ (0, 2]. This generalizes an earlier algorithm of Wasserman and Blum [1997] which worked for the case p e 2.---We obtain another sketch function C′ which probabilistically embeds ln1 into a normed space lm1. The embedding guarantees that, if we set m e log(1 Δ)O(1 e), then for any pair of points q, s ∈ ln1, the distance between q and s does not increase by more than (1 p e) with constant probability, and it does not decrease by more than (1 − e) with probability 1 − Δ. This is the only known dimensionality reduction theorem for the l1 norm. In fact, stronger theorems of this type (i.e., that guarantee very low probability of expansion as well as of contraction) cannot exist [Brinkman and Charikar 2003].---We give an explicit embedding of ln2 into lnO(log n)1 with distortion (1 p 1 nΘ(1)). Motivated by the problem of querying and communicating bidders' valuations in combinatorial auctions, we study how well different classes of set functions can be sketched. More formally, let f be a function mapping subsets of some ground set [n] to the non-negative real numbers. We say that f' is an α-sketch of f if for every set S, the value f'(S) lies between f(S) α and f(S), and f' can be specified by poly(n) bits. We show that for every subadditive function f there exists an α-sketch where α = n1 2. O(polylog(n)). Furthermore, we provide an algorithm that finds these sketches with a polynomial number of demand queries. This is essentially the best we can hope for since: 1. We show that there exist subadditive functions (in fact, XOS functions) that do not admit an o(n1 2) sketch. (Balcan and Harvey [3] previously showed that there exist functions belonging to the class of substitutes valuations that do not admit an O(n1 3) sketch.) 2. We prove that every deterministic algorithm that accesses the function via value queries only cannot guarantee a sketching ratio better than n1−e. We also show that coverage functions, an interesting subclass of submodular functions, admit arbitrarily good sketches. Finally, we show an interesting connection between sketching and learning. We show that for every class of valuations, if the class admits an α-sketch, then it can be α-approximately learned in the PMAC model of Balcan and Harvey. The bounds we prove are only information-theoretic and do not imply the existence of computationally efficient learning algorithms in general.