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1. Local Access to Huge Random Objects Through Partial Sampling

   Biswas, Amartya Shankha ; Rubinfeld, Ronitt ; Yodpinyanee, Anak ( January 2020 , 11th Innovations in Theoretical Computer Science Conference, ITCS 2020)

   null (Ed.)

   Consider an algorithm performing a computation on a huge random object (for example a random graph or a "long" random walk). Is it necessary to generate the entire object prior to the computation, or is it possible to provide query access to the object and sample it incrementally "on-the-fly" (as requested by the algorithm)? Such an implementation should emulate the random object by answering queries in a manner consistent with an instance of the random object sampled from the true distribution (or close to it). This paradigm is useful when the algorithm is sub-linear and thus, sampling the entire object up front would ruin its efficiency. Our first set of results focus on undirected graphs with independent edge probabilities, i.e. each edge is chosen as an independent Bernoulli random variable. We provide a general implementation for this model under certain assumptions. Then, we use this to obtain the first efficient local implementations for the Erdös-Rényi G(n,p) model for all values of p, and the Stochastic Block model. As in previous local-access implementations for random graphs, we support Vertex-Pair and Next-Neighbor queries. In addition, we introduce a new Random-Neighbor query. Next, we give the first local-access implementation for All-Neighbors queries in the (sparse and directed) Kleinberg’s Small-World model. Our implementations require no pre-processing time, and answer each query using O(poly(log n)) time, random bits, and additional space. Next, we show how to implement random Catalan objects, specifically focusing on Dyck paths (balanced random walks on the integer line that are always non-negative). Here, we support Height queries to find the location of the walk, and First-Return queries to find the time when the walk returns to a specified location. This in turn can be used to implement Next-Neighbor queries on random rooted ordered trees, and Matching-Bracket queries on random well bracketed expressions (the Dyck language). Finally, we introduce two features to define a new model that: (1) allows multiple independent (and even simultaneous) instantiations of the same implementation, to be consistent with each other without the need for communication, (2) allows us to generate a richer class of random objects that do not have a succinct description. Specifically, we study uniformly random valid q-colorings of an input graph G with maximum degree Δ. This is in contrast to prior work in the area, where the relevant random objects are defined as a distribution with O(1) parameters (for example, n and p in the G(n,p) model). The distribution over valid colorings is instead specified via a "huge" input (the underlying graph G), that is far too large to be read by a sub-linear time algorithm. Instead, our implementation accesses G through local neighborhood probes, and is able to answer queries to the color of any given vertex in sub-linear time for q ≥ 9Δ, in a manner that is consistent with a specific random valid coloring of G. Furthermore, the implementation is memory-less, and can maintain consistency with non-communicating copies of itself. 

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2. Local Access to Random Walks , January 2022

   Biswas, A.S. ; Pyne, E. ; Rubinfeld, R. ( January 2022 , Innovations in Theoretical Computer Science (ITCS 2022))

   For a graph G on n vertices, naively sampling the position of a random walk of at time t requires work Ω(t). We desire local access algorithms supporting positionG(t) queries, which return the position of a random walk from some fixed start vertex s at time t, where the joint distribution of returned positions is 1/ poly(n) close to those of a uniformly random walk in ℓ1 distance. We first give an algorithm for local access to random walks on a given undirected d-regular graph with eO( 1 1−λ √ n) runtime per query, where λ is the second-largest eigenvalue of the random walk matrix of the graph in absolute value. Since random d-regular graphs G(n, d) are expanders with high probability, this gives an eO(√ n) algorithm for a graph drawn from G(n, d) whp, which improves on the naive method for small numbers of queries. We then prove that no algorithm with subconstant error given probe access to an input d-regular graph can have runtime better than Ω(√ n/ log(n)) per query in expectation when the input graph is drawn from G(n, d), obtaining a nearly matching lower bound. We further show an Ω(n1/4) 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 polylog(n) per query. This also allows for efficient local access to walks on polylog degree expanders. We show that our techniques apply to graphs with high degree by extending or results to graphs constructed using the tensor product (giving fast local access to walks on degree nϵ graphs for any ϵ ∈ (0, 1]) and Cartesian product. 

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3. Sublinear Time Hypergraph Sparsification via Cut and Edge Sampling Queries

   https://doi.org/10.4230/LIPIcs.ICALP.2021.53  

   Chen, Yu ; Khanna, Sanjeev ; Nagda, Sanjeev ( July 2021 , 48th International Colloquium on Automata, Languages, and Programming)

   null (Ed.)

   The problem of sparsifying a graph or a hypergraph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Benczúr and Karger (1996) showed that given any n-vertex undirected weighted graph G and a parameter ε ∈ (0,1), there is a near-linear time algorithm that outputs a weighted subgraph G' of G of size Õ(n/ε²) such that the weight of every cut in G is preserved to within a (1 ± ε)-factor in G'. The graph G' is referred to as a (1 ± ε)-approximate cut sparsifier of G. Subsequent recent work has obtained a similar result for the more general problem of hypergraph cut sparsifiers. However, all known sparsification algorithms require Ω(n + m) time where n denotes the number of vertices and m denotes the number of hyperedges in the hypergraph. Since m can be exponentially large in n, a natural question is if it is possible to create a hypergraph cut sparsifier in time polynomial in n, independent of the number of edges. We resolve this question in the affirmative, giving the first sublinear time algorithm for this problem, given appropriate query access to the hypergraph. Specifically, we design an algorithm that constructs a (1 ± ε)-approximate cut sparsifier of a hypergraph H(V,E) in polynomial time in n, independent of the number of hyperedges, when given access to the hypergraph using the following two queries: 1) given any cut (S, ̄S), return the size |δ_E(S)| (cut value queries); and 2) given any cut (S, ̄S), return a uniformly at random edge crossing the cut (cut edge sample queries). Our algorithm outputs a sparsifier with Õ(n/ε²) edges, which is essentially optimal. We then extend our results to show that cut value and cut edge sample queries can also be used to construct hypergraph spectral sparsifiers in poly(n) time, independent of the number of hyperedges. We complement the algorithmic results above by showing that any algorithm that has access to only one of the above two types of queries can not give a hypergraph cut sparsifier in time that is polynomial in n. Finally, we show that our algorithmic results also hold if we replace the cut edge sample queries with a pair neighbor sample query that for any pair of vertices, returns a random edge incident on them. In contrast, we show that having access only to cut value queries and queries that return a random edge incident on a given single vertex, is not sufficient. 

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4. Talya Eden, Saleet Mossel, Ronitt Rubinfeld, Sampling Multiple Edges Efficiently

   Eden, T. ; Mossel, S. ; Rubinfeld, R. ( January 2021 , RANDOM 2021)

   We present a sublinear time algorithm that allows one to sample multiple edges from a distribution that is pointwise ε-close to the uniform distribution, in an amortized-efficient fashion. We consider the adjacency list query model, where access to a graph G is given via degree and neighbor queries. The problem of sampling a single edge in this model has been raised by Eden and Rosenbaum (SOSA 18). Let n and m denote the number of vertices and edges of G, respectively. Eden and Rosenbaum provided upper and lower bounds of Θ^*(n/√ m) for sampling a single edge in general graphs (where O^*(⋅) suppresses poly(1/ε) and poly(log n) dependencies). We ask whether the query complexity lower bound for sampling a single edge can be circumvented when multiple samples are required. That is, can we get an improved amortized per-sample cost if we allow a preprocessing phase? We answer in the affirmative. We present an algorithm that, if one knows the number of required samples q in advance, has an overall cost that is sublinear in q, namely, O^*(√ q ⋅(n/√ m)), which is strictly preferable to O^*(q⋅ (n/√ m)) cost resulting from q invocations of the algorithm by Eden and Rosenbaum. Subsequent to a preliminary version of this work, Tětek and Thorup (arXiv, preprint) proved that this bound is essentially optimal. 

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5. Sampling Multiple Edges Efficiently

   Eden, T. ; Mossel, S. ; Rubinfeld, R. ( January 2021 , Random)

   We present a sublinear time algorithm that allows one to sample multiple edges from a distribution that is pointwise ϵ-close to the uniform distribution, in an amortized-efficient fashion. We consider the adjacency list query model, where access to a graph G is given via degree and neighbor queries. The problem of sampling a single edge in this model has been raised by Eden and Rosenbaum (SOSA 18). Let n and m denote the number of vertices and edges of G, respectively. Eden and Rosenbaum provided upper and lower bounds of Θ∗(n/ √ m) for sampling a single edge in general graphs (where O ∗(·) suppresses poly(1/ϵ) and poly(log n) dependencies). We ask whether the query complexity lower bound for sampling a single edge can be circumvented when multiple samples are required. That is, can we get an improved amortized per-sample cost if we allow a preprocessing phase? We answer in the affirmative. We present an algorithm that, if one knows the number of required samples q in advance, has an overall cost that is sublinear in q, namely, O∗(√ q · (n/ √ m)), which is strictly preferable to O∗(q · (n/ √ m)) cost resulting from q invocations of the algorithm by Eden and Rosenbaum. Subsequent to a preliminary version of this work, Tětek and Thorup (arXiv, preprint) proved that this bound is essentially optimal. 

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