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For a graph $G$ on $n$ vertices, naively sampling the position of a random walk of at time $t$ requires work $Omega(t)$. We desire local access algorithms supporting $text{position}(G,s,t)$ queries, which return the position of a random walk from some start vertex $s$ at time $t$, where the joint distribution of returned positions is $1/text{poly}(n)$ close to the uniform distribution over such walks in $ell_1$ distance. We first give an algorithm for local access to walks on undirected regular graphs with $widetilde{O}(frac{1}{1-lambda}sqrt{n})$ runtime per query, where $lambda$ is the second-largest eigenvalue in absolute value. Since random $d$-regular graphs are expanders with high probability, this gives an $widetilde{O}(sqrt{n})$ algorithm for $G(n,d)$, which improves on the naive method for small numbers of queries. We then prove that no that algorithm with sub-constant error given probe access to random $d$-regular graphs can have runtime better than $Omega(sqrt{n}/log(n))$ per query in expectation, obtaining a nearly matching lower bound. We further show an $Omega(n^{1/4})$ runtime per query lower bound even with an oblivious adversary (i.e. when the query sequence is fixed in advance). We then show that for families of graphs with additional group theoretic structure, dramatically better results can be achieved. We give local access to walks on small-degree abelian Cayley graphs, including cycles and hypercubes, with runtime $text{polylog}(n)$ per query. This also allows for efficient local access to walks on $text{polylog}$ degree expanders. We extend our results to graphs constructed using the tensor product (giving local access to walks on degree $n^epsilon$ graphs for any $epsilon in (0,1]$) and Cartesian product.
In this paper, we introduce the on-line Viterbi algorithm for decoding hidden Markov models (HMMs) in much smaller than linear space. Our analysis on two-state HMMs suggests that the expected maximum memory used to decode sequence of length $n$ with
Under the Strong Exponential Time Hypothesis, an integer linear program with $n$ Boolean-valued variables and $m$ equations cannot be solved in $c^n$ time for any constant $c < 2$. If the domain of the variables is relaxed to $[0,1]$, the associated
We prove that the local time process of a planar simple random walk, when time is scaled logarithmically, converges to a non-degenerate pure jump process. The convergence takes place in the Skorokhod space with respect to the $M1$ topology and fails to hold in the $J1$ topology.
The aim of this paper is the study of the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite dimensional generating function G and a maximum principle which, we
Lecture Notes. Minicourse given at the workshop Activated Random Walks, DLA, and related topics at IMeRA-Marseille, March 2015.