Erasure-Resilient Sublinear-Time Graph Algorithms

Modern large-scale graph processing faces a critical challenge: conventional adjacency lists incur excessive L3 cache misses due to irregular memory access. We introduce GraphTwin, a hybrid graph representation system combining: (1) Cache-optimized k -bit vectors (termed GT-vectors, 64 bits per vertex), where each bit indicates vertex membership in a precomputed independent set; and (2) Memory-resident adjacency lists for exact verification of queries unresolved by GT-vectors. This dual-component design enables 95% of negative edge queries, which are dominant in sparse graphs, to be resolved in 1 CPU cycle via in-cache bitwise-AND operations, reducing latency from 54ns (adjacency list) to 18ns per query. Unresolved queries delegate to adjacency lists, guaranteeing zero false positives/negatives. Crucially, GT-vectors scales linearly with vertex count ( k|V| bits ), decoupling the space overhead from edge density and minimizing cache dependency. For example, GT-vectors for a graph with |V|=10 7 vertices occupy 80MB, fitting entirely within modern CPU caches (e.g., AMD Ryzen 7 9800X3D's 96MB L3). We formalize the GT-vectors construction as an NP-hard and submodular optimization problem and introduce GTWICE, a linear-time heuristic algorithm that iteratively extracts diversified maximal independent sets to maximize non-edge coverage. Experiments on 15 graphs show that GraphTwin reduces L3 cache misses by 63% on average, achieves 6.2× speedup for edge queries and accelerates triangle counting and set inclusion by 1.7× and 5.4×, respectively. By optimizing cache residency and accelerating foundational primitives, GraphTwin addresses cache inefficiencies in graph processing, enabling fast graph queries without sacrificing exactness.

In standard property testing, the task is to distinguish between objects that have a property 𝒫 and those that are ε‐far from 𝒫, for some ε > 0. In this setting, it is perfectly acceptable for the tester to provide a negative answer for every input object that does not satisfy 𝒫. This implies that property testing in and of itself cannot be expected to yield any information whatsoever about the distance from the object to the property. We address this problem in this paper, restricting our attention to monotonicity testing. A function f : {1,…,n} ↦ R is at distance εf from being monotone if it can (and must) be modified at εfn places to become monotone. For any fixed δ > 0, we compute, with probability at least 2/3, an interval [(1/2 − δ)ε,ε] that encloses εf. The running time of our algorithm is O(εf−1 log log εf− 1 log n), which is optimal within a factor of loglog εf−1 and represents a substantial improvement over previous work. We give a second algorithm with an expected running time of O(εf−1 log nlog log log n). Finally, we extend our results to multivariate functions. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007

Given a group $\mathcal {Q}$ of query points and a set $\mathcal {P}$ of points of interest (POIs), aggregate nearest neighbor (ANN) queries find a POI p from $\mathcal {P}$ that achieves the smallest aggregate distance. Specifically, the aggregate distance is defined over the set of distances between p and all query points in $\mathcal {Q}$ . Existing studies on ANN query mainly consider the spatial proximity, whereas the textual similarity has received considerable attention recently. In this work, we utilize user-specified query keywords to capture textual similarity. We study the aggregate keyword nearest neighbor (AKNN) queries, finding the POI that has the smallest aggregate distance and covers all query keywords. Nevertheless, existing methods on ANN query are either inapplicable or inefficient when applied to the AKNN query. To answer our query efficiently, we first develop a dual-granularity (DG) indexing schema. It preserves abstracts of the road network by a tree structure, and preserves detailed network information by an extended adjacency list. Then, we propose a minimal first search (MFS) algorithm. It traverses the tree and explores the node with the minimal aggregate distance iteratively. This method suffers from false hits arising from keyword tests. Thus, we propose the collaborative filtering technique, which performs keywords test by multiple keyword bitmaps collectively rather than by only one. Extensive experiments on both real and synthetic datasets demonstrate the superiority of our algorithms and optimizing strategies.

In the subgraph counting problem, we are given a (large) input graph G(V, E) and a (small) target graph H (e.g., a triangle); the goal is to estimate the number of occurrences of H in G. Our focus here is on designing sublinear-time algorithms for approximately computing number of occurrences of H in G in the setting where the algorithm is given query access to G. This problem has been studied in several recent papers which primarily focused on specific families of graphs H such as triangles, cliques, and stars. However, not much is known about approximate counting of arbitrary graphs H in the literature. This is in sharp contrast to the closely related subgraph enumeration problem that has received significant attention in the database community as the database join problem. The AGM bound shows that the maximum number of occurrences of any arbitrary subgraph H in a graph G with m edges is O(m^{rho(H)}), where rho(H) is the fractional edge-cover of H, and enumeration algorithms with matching runtime are known for any H. We bridge this gap between subgraph counting and subgraph enumeration by designing a simple sublinear-time algorithm that can estimate the number of occurrences of any arbitrary graph H in G, denoted by #H, to within a (1 +/- epsilon)-approximation with high probability in O(m^{rho(H)}/#H) * poly(log(n),1/epsilon) time. Our algorithm is allowed the standard set of queries for general graphs, namely degree queries, pair queries and neighbor queries, plus an additional edge-sample query that returns an edge chosen uniformly at random. The performance of our algorithm matches those of Eden et al. [FOCS 2015, STOC 2018] for counting triangles and cliques and extend them to all choices of subgraph H under the additional assumption of edge-sample queries.

We study a scale‐free random graph process in which the number of edges added at each step increases. This differs from the standard model in which a fixed number,m, of edges are added at each step.Letf(t) be the number of edges added at stept. In the standard scale‐free model,f(t) =mconstant, whereas in this paper we considerf(t) = [tc],c> 0. Such a graph process, in which the number of edges grows non‐linearly with the number of vertices is said to have accelerating growth.We analyze both an undirected and a directed process. The power law of the degree sequence of these processes exhibits widely differing behavior.For the undirected process, the terminal vertex of each edge is chosen by preferential attachment based on vertex degree. Whenf(t) =mconstant, this is the standard scale‐free model, and the power law of the degree sequence is 3. Whenf(t) = [tc],c< 1, the degree sequence of the process exhibits a power law with parameterx= (3 −c)/(1 −c). Asc→ 0,x→ 3, which gives a value ofx= 3, as in standard scale‐free model. Thus no more slowly growing monotone functionf(t) alters the power law of this model away fromx= 3. Whenc= 1, so thatf(t) =t, the expected degree of all vertices ist, the vertex degree is concentrated, and the degree sequence does not have a power law.For the directed process, the terminal vertex is chosen proportional to in‐degree plus an additive constant, to allow the selection of vertices of in‐degree zero. For this process whenf(t) =mis constant, the power law of the degree sequence isx= 2 + 1/m. Whenf(t) = [tc],c> 0, the power law becomesx= 1 + 1/(1 +c), which naturally extends the power law to [1,2]. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 396–421, 2011

A graph consists of a finite set of vertices and edges. Graphs are used to represent a significant number of real life applications. For example, in computer science, graphs are employed for the representation of networks of communication, organisation of data, flow of computation, computational devices, etc. Several data structures have been proposed for representing graphs among which the adjacency matrix, adjacency list and edge list are the most important and widely used ones. The choice of a graph representation is mainly situation-specific and depends on the type of operations required to be performed on a given graph as well as the ease of use. In this research, a specialised graph representation is proposed, specifically designed for use when coping with graph-based optimisation problems (e.g., graph layout problems) through heuristic search methods with the aim of speeding up the search. The results of numerical experiments show that for the purpose of this study, the proposed approach performs extremely well compared to well-known graph representation approaches.

This article documents two programmatic comments regarding testing graph properties, which I made during the Dagstuhl workshop on Sublinear-Time Algorithms (July 2005). The first comment advocates paying more attention to the dependence of the tester’s complexity on the proximity parameter. The second comment advocates paying more attention to the question of testing general graphs (rather than dense or bounded-degree ones). In addition, this article includes a suggestion to view and discuss property testing within the framework of promise problems.KeywordsProperty testinggraph properties

Graph compression based on transitivity for neighborhood query

Mixed-protocol multi-party computation framework towards complex computation tasks with malicious security

BackgroundThe need for bone graft substitutes including those being developed to be applied together with new strategies of bone regeneration such as tissue engineering and cell-based approaches is growing. No large animal model of bone regeneration has been accepted as a standard testing model. Standardization may be the key to moving systematically towards better bone regeneration. This study aimed to establish a model of bone regeneration in the sheep that lends itself to strict standardization and in which a number of substances can be tested within the same animal. To this end the caudal border of the ovine scapula was used as a consistent bed of mineralized tissue that provided sufficient room for a serial alignment of multiple experimental drill holes.ResultsThe findings show that for the sake of standardization, surgery should be restricted to the middle part of the caudal margin, an area at least 80 mm proximal from the Glenoid cavity, but not more than 140 mm away from it, in the adult female Land Merino sheep. A distance of 5 mm from the caudal margin should also be observed.ConclusionsThis standardized model with defined uniform defects and defect sites results in predictable and reproducible bone regeneration processes. Defects are placed unilaterally in only one limb of the animal, avoiding morbidity in multiple limbs. The fact that five defects per animal can be evaluated is conducive to intra-animal comparisons and reduces the number of animals that have to be subject to experimentation.

We study the power of four query models in the context of property testing in general graphs (i.e., with arbitrary edge densities), where our main case study is the problem of testing k-colorability. Two query types, which have been studied extensively in the past, are pair queries and neighbor queries. The former corresponds to asking whether there is an edge between any particular pair of vertices, and the latter to asking for the i’th neighbor of a particular vertex. We show that while for pair queries, testing k-colorability requires a number of queries that is a monotone decreasing function in the average degree d, the query complexity in the case of neighbor queries remains roughly the same for every density and for large values of k. We also consider a combined model that allows both types of queries, and we propose a new, stronger, query model, related to the field of Group Testing. We give one-sided error upper and lower bounds for all the models, where the bounds are nearly tight for three of the models. In some of the cases our lower bounds extend to two-sided error algorithms.The problem of testing k-colorability was previously studied in the contexts of dense and sparse graphs, and in our proofs we unify approaches from those cases, and also provide some new tools and techniques that may be of independent interest.Keywordspair queriesneighbor queriesgroup queriesk-colorability

The article is devoted to the problem of representing graphs in the form that is most suitable for their recording in relational databases and for subsequent efficient extracting and processing. The article analyzes various ways to describe graphs, such as adjacency and, incidence matrices, and adjacency lists. Each of them is reviewed from the point of view of their compliance with normal forms to assess the possibility of using a particular method when developing databases for storing graphs. It is shown that for such a task, each of these methods has a large number of disadvantages that lead to low efficiency of both data storing and processing. The article suggests the way to represent graphs in the form of a relational list of edges corresponding to the third normal form and allowing to eliminate the disadvantages of other methods.

Static or dynamic graphs are typically visualized by either node-link diagrams, adjacency matrices, adjacency lists, orhybrids thereof. In particular, for the case of a changing graphstructure a viewer wishes to be able to visually compare thegraphs in a sequence. Doing such a comparison task rapidlyand reliably can give support to visually analyze the dynamicgraph for certain dynamic patterns. In this paper we describe a novel dynamic graph visualization that is based on the concept of smooth density fields generated by first splatting the links of a given graph in a certain layout. To further visually enhance the time-varying graph structures we add user-adaptable isolines to the resulting dynamic graph representation. The computed visual encoding of the dynamic graph is aesthetically appealing due to its smooth curves and can additionally be used to do comparisons in a long graph sequence, i.e., from an information visualization perspective it serves as an overview representation supporting to start more detailed analyses processes. To demonstrate theusefulness of the technique we explore real-world dynamic graph data by taking into account visual parameters like node-link layouts, smoothing iterations, number of isolines, and different color codings.

We investigate the relationship between the sizes of the sum and difference sets attached to a subset of {0,1,…,N}, chosen randomly according to a binomial model with parameter p(N), with N−1 = o(p(N)). We show that the random subset is almost surely difference dominated, as N → ∞, for any choice of p(N) tending to zero, thus confirming a conjecture of Martin and O'Bryant. The proofs use recent strong concentration results.Furthermore, we exhibit a threshold phenomenon regarding the ratio of the size of the difference to the sumset. If p(N) = o(N−1/2) then almost all sums and differences in the random subset are almost surely distinct and, in particular, the difference set is almost surely about twice as large as the sumset. If N−1/2 = o(p(N)) then both the sum and difference sets almost surely have size (2N + 1) − O(p(N)−2), and so the ratio in question is almost surely very close to one. If p(N) = c · N−1/2 then as c increases from zero to infinity (i.e., as the threshold is crossed), the same ratio almost surely decreases continuously from two to one according to an explicitly given function of c.We also extend our results to the comparison of the generalized difference sets attached to an arbitrary pair of binary linear forms. For certain pairs of forms f and g, we show that there in fact exists a sharp threshold at cf,g · N−1/2, for some computable constant cf,g, such that one form almost surely dominates below the threshold and the other almost surely above it.The heart of our approach involves using different tools to obtain strong concentration of the sizes of the sum and difference sets about their mean values, for various ranges of the parameter p. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009

We study the computational complexity of approximately counting the number of independent sets of a graph with maximum degree Δ. More generally, for an input graph and an activity , we are interested in the quantity defined as the sum over independent sets I weighted as . In statistical physics, is the partition function for the hard‐core model, which is an idealized model of a gas where the particles have non‐negligible size. Recently, an interesting phase transition was shown to occur for the complexity of approximating the partition function. Weitz showed an FPAS for the partition function for any graph of maximum degree Δ when Δ is constant and . The quantity is the critical point for the so‐called uniqueness threshold on the infinite, regular tree of degree Δ. On the other side, Sly proved that there does not exist efficient (randomized) approximation algorithms for , unless , for some function . We remove the upper bound in the assumptions of Sly's result for , that is, we show that there does not exist efficient randomized approximation algorithms for all for and . Sly's inapproximability result uses a clever reduction, combined with a second‐moment analysis of Mossel, Weitz and Wormald which prove torpid mixing of the Glauber dynamics for sampling from the associated Gibbs distribution on almost every regular graph of degree Δ for the same range of λ as in Sly's result. We extend Sly's result by improving upon the technical work of Mossel et al., via a more detailed analysis of independent sets in random regular graphs. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 78–110, 2014