In the communication problem @math (universal relation) [KRW95], Alice and Bob respectively receive @math with the promise that @math . The last player to receive a message must output an index @math such that @math . We prove that the randomized one-way communication complexity of this problem in the public coin model is exactly @math for failure probability @math . Our lower bound holds even if promised @math . As a corollary, we obtain optimal lower bounds for @math -sampling in strict turnstile streams for @math , as well as for the problem of finding duplicates in a stream. Our lower bounds do not need to use large weights, and hold even if promised @math at all points in the stream. We give two different proofs of our main result. The first proof demonstrates that any algorithm @math solving sampling problems in turnstile streams in low memory can be used to encode subsets of @math of certain sizes into a number of bits below the information theoretic minimum. Our encoder makes adaptive queries to @math throughout its execution, but done carefully so as to not violate correctness. This is accomplished by injecting random noise into the encoder's interactions with @math , which is loosely motivated by techniques in differential privacy. Our second proof is via a novel randomized reduction from Augmented Indexing [MNSW98] which needs to interact with @math adaptively. To handle the adaptivity we identify certain likely interaction patterns and union bound over them to guarantee correct interaction on all of them. To guarantee correctness, it is important that the interaction hides some of its randomness from @math in the reduction.
The question of whether @math -sampling is possible in low memory in turnstile streams was first asked in @cite_0 @cite_8 . The work @cite_8 applied @math -sampling as a subroutine in approximating the cost of the Euclidean minimum spanning tree of a subset @math of a discrete geometric space subject to insertions and deletions. The algorithm given there used space @math bits to achieve failure probability @math (though it is likely that the space could be improved to @math with a worse failure probability, by replacing a subroutine used there with a more recent @math -estimation algorithm of @cite_22 ). As mentioned, the currently best known upper bound solves @math -sampling @math using @math bits @cite_31 , which Theorem shows is tight.
@cite_0 @cite_31 @cite_22 @cite_8
2150516767 2089135543 2103318769 2950256099
We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph's edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for problems involving cuts in graphs. We present fast randomized (Monte Carlo and Las Vegas) algorithms for approximating and exactly finding minimum cuts and maximum flows in unweighted, undirected graphs. Our cut-approximation algorithms extend unchanged to weighted graphs while our weighted-graph flow algorithms are somewhat slower. Our approach gives a general paradigm with potential applications to any packing problem. It has since been used in a near-linear time algorithm for finding minimum cuts, as well as faster cut and flow algorithms. Our sampling theorems also yield faster algorithms for several other cut-based problems, including approximating the best balanced cut of a graph, finding a k-connected orientation of a 2k-connected graph, and finding integral multicommodity flows in graphs with a great deal of excess capacity. Our methods also improve the efficiency of some parallel cut and flow algorithms. Our methods also apply to the network design problem, where we wish to build a network satisfying certain connectivity requirements between vertices. We can purchase edges of various costs and wish to satisfy the requirements at minimum total cost. Since our sampling theorems apply even when the sampling probabilities are different for different edges, we can apply randomized rounding to solve network design problems. This gives approximation algorithms that guarantee much better approximations than previous algorithms whenever the minimum connectivity requirement is large. As a particular example, we improve the best approximation bound for the minimum k-connected subgraph problem from 1.85 to [math not displayed]. We give efficient algorithms for volume sampling, i.e., for picking @math -subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). In other words, we can efficiently sample @math -subsets of @math with probabilities proportional to the corresponding @math by @math principal minors of any given @math by @math positive semi definite matrix. This solves an open problem from the monograph on spectral algorithms by Kannan and Vempala (see Section @math of KV , also implicit in BDM, DRVW ). Our first algorithm for volume sampling @math -subsets of rows from an @math -by- @math matrix runs in @math arithmetic operations (where @math is the exponent of matrix multiplication) and a second variant of it for @math -approximate volume sampling runs in @math arithmetic operations, which is almost linear in the size of the input (i.e., the number of entries) for small @math . Our efficient volume sampling algorithms imply the following results for low-rank matrix approximation: (1) Given @math , in @math arithmetic operations we can find @math of its rows such that projecting onto their span gives a @math -approximation to the matrix of rank @math closest to @math under the Frobenius norm. This improves the @math -approximation of Boutsidis, Drineas and Mahoney BDM and matches the lower bound shown in DRVW . The method of conditional expectations gives a algorithm with the same complexity. The running time can be improved to @math at the cost of losing an extra @math in the approximation factor. (2) The same rows and projection as in the previous point give a @math -approximation to the matrix of rank @math closest to @math under the spectral norm. In this paper, we show an almost matching lower bound of @math , even for @math . [17] proved that a small sample of rows of a given matrix A contains a low-rank approximation D that minimizes ||A - D||F to within small additive error, and the sampling can be done efficiently using just two passes over the matrix [12]. In this paper, we generalize this result in two ways. First, we prove that the additive error drops exponentially by iterating the sampling in an adaptive manner. Using this result, we give a pass-efficient algorithm for computing low-rank approximation with reduced additive error. Our second result is that using a natural distribution on subsets of rows (called volume sampling), there exists a subset of k rows whose span contains a factor (k + 1) relative approximation and a subset of k + k(k + 1) e rows whose span contains a 1+e relative approximation. The existence of such a small certificate for multiplicative low-rank approximation leads to a PTAS for the following projective clustering problem: Given a set of points P in Rd, and integers k, j, find a set of j subspaces F 1 , . . ., F j , each of dimension at most k, that minimize Σ p∈P min i d(p, F i )2. We consider the unweighted bipartite maximum matching problem in the one-pass turnstile streaming model where the input stream consists of edge insertions and deletions. In the insertion-only model, a one-pass @math -approximation streaming algorithm can be easily obtained with space @math , where @math denotes the number of vertices of the input graph. We show that no such result is possible if edge deletions are allowed, even if space @math is granted, for every @math . Specifically, for every @math , we show that in the one-pass turnstile streaming model, in order to compute a @math -approximation, space @math is required for constant error randomized algorithms, and, up to logarithmic factors, space @math is sufficient. Our lower bound result is proved in the simultaneous message model of communication and may be of independent interest.