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Register a new userHow to Store a Random Walk
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Omri Weinstein
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Motivated by storage applications, we study the following data structure problem: An encoder wishes to store a collection of jointly-distributed files $overline{X}:=(X_1,X_2,ldots, X_n) sim mu$ which are emph{correlated} ($H_mu(overline{X}) ll sum_i H_mu(X_i)$), using as little (expected) memory as possible, such that each individual file $X_i$ can be recovered quickly with few (ideally constant) memory accesses. In the case of independent random files, a dramatic result by Pat (FOCS08) and subsequently by Dodis, Pat and Thorup (STOC10) shows that it is possible to store $overline{X}$ using just a emph{constant} number of extra bits beyond the information-theoretic minimum space, while at the same time decoding each $X_i$ in constant time. However, in the (realistic) case where the files are correlated, much weaker results are known, requiring at least $Omega(n/polylg n)$ extra bits for constant decoding time, even for simple joint distributions $mu$. We focus on the natural case of compressingemph{Markov chains}, i.e., storing a length-$n$ random walk on any (possibly directed) graph $G$. Denoting by $kappa(G,n)$ the number of length-$n$ walks on $G$, we show that there is a succinct data structure storing a random walk using $lg_2 kappa(G,n) + O(lg n)$ bits of space, such that any vertex along the walk can be decoded in $O(1)$ time on a word-RAM. For the harder task of matching the emph{point-wise} optimal space of the walk, i.e., the empirical entropy $sum_{i=1}^{n-1} lg (deg(v_i))$, we present a data structure with $O(1)$ extra bits at the price of $O(lg n)$ decoding time, and show that any improvement on this would lead to an improved solution on the long-standing Dictionary problem. All of our data structures support the emph{online} version of the problem with constant update and query time.
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In this paper, we study the average case complexity of the Unique Games problem. We propose a natural semi-random model, in which a unique game instance is generated in several steps. First an adversary selects a completely satisfiable instance of Unique Games, then she chooses an epsilon-fraction of all edges, and finally replaces (corrupts) the constraints corresponding to these edges with new constraints. If all steps are adversarial, the adversary can obtain any (1-epsilon) satisfiable instance, so then the problem is as hard as in the worst case. In our semi-random model, one of the steps is random, and all other steps are adversarial. We show that known algorithms for unique games (in particular, all algorithms that use the standard SDP relaxation) fail to solve semi-random instances of Unique Games. We present an algorithm that with high probability finds a solution satisfying a (1-delta) fraction of all constraints in semi-random instances (we require that the average degree of the graph is Omega(log k). To this end, we consider a new non-standard SDP program for Unique Games, which is not a relaxation for the problem, and show how to analyze it. We present a new rounding scheme that simultaneously uses SDP and LP solutions, which we believe is of independent interest. Our result holds only for epsilon less than some absolute constant. We prove that if epsilon > 1/2, then the problem is hard in one of the models, the result assumes the 2-to-2 conjecture. Finally, we study semi-random instances of Unique Games that are at most (1-epsilon) satisfiable. We present an algorithm that with high probability, distinguishes between the case when the instance is a semi-random instance and the case when the instance is an (arbitrary) (1-delta) satisfiable instance if epsilon > c delta.
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