Incidence Geometries and the Pass Complexity of Semi-Streaming Set Cover

Set cover, over a universe of size $n$, may be modelled as a data-streaming problem, where the $m$ sets that comprise the instance are to be read one by one. A semi-streaming algorithm is allowed only $O(n\, \mathrm{poly}\{\log n, \log m\})$ space to process this stream. For each $p \ge 1$, we give a very simple deterministic algorithm that makes $p$ passes over the input stream and returns an appropriately certified $(p+1)n^{1/(p+1)}$-approximation to the optimum set cover. More importantly, we proceed to show that this approximation factor is essentially tight, by showing that a factor better than $0.99\,n^{1/(p+1)}/(p+1)^2$ is unachievable for a $p$-pass semi-streaming algorithm, even allowing randomisation. In particular, this implies that achieving a $\Theta(\log n)$-approximation requires $\Omega(\log n/\log\log n)$ passes, which is tight up to the $\log\log n$ factor. These results extend to a relaxation of the set cover problem where we are allowed to leave an $\varepsilon$ fraction of the universe uncovered: the tight bounds on the best approximation factor achievable in $p$ passes turn out to be $\Theta_p(\min\{n^{1/(p+1)}, \varepsilon^{-1/p}\})$. Our lower bounds are based on a construction of a family of high-rank incidence geometries, which may be thought of as vast generalisations of affine planes. This construction, based on algebraic techniques, appears flexible enough to find other applications and is therefore interesting in its own right.Comment: 20 page

Streaming Verification of Graph Computations via Graph Structure

We give new algorithms in the annotated data streaming setting - also known as verifiable data stream computation - for certain graph problems. This setting is meant to model outsourced computation, where a space-bounded verifier limited to sequential data access seeks to overcome its computational limitations by engaging a powerful prover, without needing to trust the prover. As is well established, several problems that admit no sublinear-space algorithms under traditional streaming do allow protocols using a sublinear amount of prover/verifier communication and sublinear-space verification. We give algorithms for many well-studied graph problems including triangle counting, its generalization to subgraph counting, maximum matching, problems about the existence (or not) of short paths, finding the shortest path between two vertices, and testing for an independent set. While some of these problems have been studied before, our results achieve new tradeoffs between space and communication costs that were hitherto unknown. In particular, two of our results disprove explicit conjectures of Thaler (ICALP, 2016) by giving triangle counting and maximum matching algorithms for n-vertex graphs, using o(n) space and o(n^2) communication

Time-Space Tradeoffs for the Memory Game

A single-player game of Memory is played with $n$ distinct pairs of cards, with the cards in each pair bearing identical pictures. The cards are laid face-down. A move consists of revealing two cards, chosen adaptively. If these cards match, i.e., they bear the same picture, they are removed from play; otherwise, they are turned back to face down. The object of the game is to clear all cards while minimizing the number of moves. Past works have thoroughly studied the expected number of moves required, assuming optimal play by a player has that has perfect memory. In this work, we study the Memory game in a space-bounded setting. We prove two time-space tradeoff lower bounds on algorithms (strategies for the player) that clear all cards in $T$ moves while using at most $S$ bits of memory. First, in a simple model where the pictures on the cards may only be compared for equality, we prove that $ST = \Omega(n^2 \log n)$. This is tight: it is easy to achieve $ST = O(n^2 \log n)$ essentially everywhere on this tradeoff curve. Second, in a more general model that allows arbitrary computations, we prove that $ST^2 = \Omega(n^3)$. We prove this latter tradeoff by modeling strategies as branching programs and extending a classic counting argument of Borodin and Cook with a novel probabilistic argument. We conjecture that the stronger tradeoff $ST = \widetilde{\Omega}(n^2)$ in fact holds even in this general model

An Optimal Lower Bound on the Communication Complexity of Gap-Hamming-Distance

We prove an optimal $\Omega(n)$ lower bound on the randomized communication complexity of the much-studied Gap-Hamming-Distance problem. As a consequence, we obtain essentially optimal multi-pass space lower bounds in the data stream model for a number of fundamental problems, including the estimation of frequency moments. The Gap-Hamming-Distance problem is a communication problem, wherein Alice and Bob receive $n$-bit strings $x$ and $y$, respectively. They are promised that the Hamming distance between $x$ and $y$ is either at least $n/2+\sqrt{n}$ or at most $n/2-\sqrt{n}$, and their goal is to decide which of these is the case. Since the formal presentation of the problem by Indyk and Woodruff (FOCS, 2003), it had been conjectured that the naive protocol, which uses $n$ bits of communication, is asymptotically optimal. The conjecture was shown to be true in several special cases, e.g., when the communication is deterministic, or when the number of rounds of communication is limited. The proof of our aforementioned result, which settles this conjecture fully, is based on a new geometric statement regarding correlations in Gaussian space, related to a result of C. Borell (1985). To prove this geometric statement, we show that random projections of not-too-small sets in Gaussian space are close to a mixture of translated normal variables

Sublinear Communication Protocols for Multi-Party Pointer Jumping and a Related Lower Bound

We study the one-way number-on-the-forehead (NOF) communication complexity of the $k$-layer pointer jumping problem with $n$ vertices per layer. This classic problem, which has connections to many aspects of complexity theory, has seen a recent burst of research activity, seemingly preparing the ground for an $\Omega(n)$ lower bound, for constant $k$. Our first result is a surprising sublinear -- i.e., $o(n)$ -- upper bound for the problem that holds for $k \ge 3$, dashing hopes for such a lower bound. A closer look at the protocol achieving the upper bound shows that all but one of the players involved are collapsing, i.e., their messages depend only on the composition of the layers ahead of them. We consider protocols for the pointer jumping problem where all players are collapsing. Our second result shows that a strong $n - O(\log n)$ lower bound does hold in this case. Our third result is another upper bound showing that nontrivial protocols for (a non-Boolean version of) pointer jumping are possible even when all players are collapsing. Our lower bound result uses a novel proof technique, different from those of earlier lower bounds that had an information-theoretic flavor. We hope this is useful in further study of the problem

When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks

Let $G=(V,E)$ be a supply graph and $H=(V,F)$ a demand graph defined on the same set of vertices. An assignment of capacities to the edges of $G$ and demands to the edges of $H$ is said to satisfy the \emph{cut condition} if for any cut in the graph, the total demand crossing the cut is no more than the total capacity crossing it. The pair $(G,H)$ is called \emph{cut-sufficient} if for any assignment of capacities and demands that satisfy the cut condition, there is a multiflow routing the demands defined on $H$ within the network with capacities defined on $G$. We prove a previous conjecture, which states that when the supply graph $G$ is series-parallel, the pair $(G,H)$ is cut-sufficient if and only if $(G,H)$ does not contain an \emph{odd spindle} as a minor; that is, if it is impossible to contract edges of $G$ and delete edges of $G$ and $H$ so that $G$ becomes the complete bipartite graph $K_{2,p}$, with $p\geq 3$ odd, and $H$ is composed of a cycle connecting the $p$ vertices of degree 2, and an edge connecting the two vertices of degree $p$. We further prove that if the instance is \emph{Eulerian} --- that is, the demands and capacities are integers and the total of demands and capacities incident to each vertex is even --- then the multiflow problem has an integral solution. We provide a polynomial-time algorithm to find an integral solution in this case. In order to prove these results, we formulate properties of tight cuts (cuts for which the cut condition inequality is tight) in cut-sufficient pairs. We believe these properties might be useful in extending our results to planar graphs.Comment: An extended abstract of this paper will be published at the 44th Symposium on Theory of Computing (STOC 2012

Spatio-temporal correlations in Wigner molecules

The dynamical response of Coulomb-interacting particles in nano-clusters are analyzed at different temperatures characterizing their solid- and liquid-like behavior. Depending on the trap-symmetry, both the spatial and temporal correlations undergo slow, stretched exponential relaxations at long times, arising from spatially correlated motion in string-like paths. Our results indicate that the distinction between the `solid' and `liquid' is soft: While particles in a `solid' flow producing dynamic heterogeneities, motion in `liquid' yields unusually long tail in the distribution of particle-displacements. A phenomenological model captures much of the subtleties of our numerical simulations.Comment: 5 pages, 4 figures, includes supplementary material

A Note on Randomized Streaming Space Bounds for the Longest Increasing Subsequence Problem

The deterministic space complexity of approximating the length of the longest increasing subsequence of a stream of N integers is known to be Theta~(sqrt N). However, the randomized complexity is wide open. We show that the technique used in earlier work to establish the Omega(sqrt N) deterministic lower bound fails strongly under randomization: specifically, we show that the communication problems on which the lower bound is based have very efficient randomized protocols. The purpose of this note is to guide and alert future researchers working on this very interesting problem