Local properties of geometric graphs

Privacy preserving network publishing has been studied extensively in recent years. Although more works have adopted un-weighted graphs to model network relationships, weighted graph modeling can provide deeper analysis of the degree of relationships. Previous works on weighted graph privacy have concentrated on preserving the shortest path characteristic between pairs of vertices. Two common types of privacy have been proposed. One type of privacy tried to add random noise edge weights to the graph but still maintain the same shortest path. The other privacy, k-shortest path privacy, minimally perturbed edge weights so that there exists k shortest paths. However, the k-shortest path privacy only considers anonymizing same fixed number of shortest paths for all pairs of source and destination vertices. In this work, we present a new concept called [k1, k2]-shortest path privacy to allow different number of shortest paths for different pairs of vertices. A published network graph with [k1, k2]-shortest path privacy has at least k' indistinguishable shortest paths between the source and destination vertices, where k1 l k' l k2. A heuristic algorithm based on modifying only Non-Visited (NV) edges is proposed and experimental results showing the feasibility and characteristics of the proposed approach are presented.

The complete twisted graph of order n, denoted by $$T_n$$ , is a complete simple topological graph with vertices $$u_1, u_2, \ldots , u_n$$ such that two edges $$u_iu_j$$ and $$u_{i'}u_{j'}$$ cross if and only if $$ i<i'<j'<j $$ or $$ i'<i<j<j' $$ . The convex geometric complete graph of order n, denoted by $$G_n$$ , is a convex geometric graph with vertices $$v_1, v_2, \ldots , v_n$$ placed counterclockwise, in which every pair of vertices is adjacent. A biplanar tree of order n is a labeled tree with vertex set $$\{v_1,v_2,\ldots , v_n \}$$ having the property of being planar when embedded in both $$T_n$$ and $$G_n$$ . Given a connected graph G the (combinatorial) tree graph $$\mathcal {T}(G)$$ is the graph whose vertices are the spanning trees of G and two trees P and Q are adjacent in $$\mathcal {T}(G)$$ if there are edges $$e \in P$$ and $$f\in Q$$ such that $$Q=P - e + f$$ . For all positive integers n, we denote by $$\mathcal {T}(n)$$ the graph $$\mathcal {T}(K_n)$$ . The biplanar tree graph, $$\mathcal {B}(n)$$ , is the subgraph of $$\mathcal {T}(n)$$ induced by the biplanar trees of order n. In this paper we give a characterization of the biplanar trees and we study the structure, the radius and the diameter of the biplanar tree graph.

An approximate distance query data structure is a compact representation of a graph, and can be queried to approximate shortest paths between any pair of vertices. Any such data structure that retrieves stretch 2k Ω 1 paths must require space Ω(n1+1/k) for graphs of n nodes. The hard cases that enforce this lower bound are, however, rather dense graphs with average degree Ω(n1/k).

In this paper we consider two families of competing algorithms for finding the shortest paths between all pairs of vertices (APSP) in directed weighted large graphs with different edge densities: Dijkstra and Floyd-Warshall. For comparison, we have taken Dijkstra's algorithm with dynamically varying binary heap, which solves the APSP prob-lem purely in parallel by repeatedly executing on all vertices of the graph considered as source vertices, and we have taken blocked Floyd-Warshall algorithm, which is also well-parallelizable. It is known that in terms of computational complexity, the first algorithm is preferable on sparse graphs and the second algorithm is preferable on dense graphs. At the same time, it is not clear what are the ranges of graph densities at which the first algorithm will consume less CPU time than the second algorithm. This paper describes multithreaded implementations of parallel algorithms on multicore processors that make different usage of synchronization primitives such as mutex, conditional variable, lock-ing, and atomic operation. By conducting computational experiments on an 8-core Intel(R) Core(TM) i7-10700 CPU @ 2.90GHz, we found that each algorithm has a preferred graph density. In the case of multi-threaded parallel imple-mentation, the blocked Floyd-Warshall algorithm has lower running time than Dijkstra's algorithm if the graph densi-ty is greater than 0.5. Otherwise, Dijkstra's algorithm runs faster. In the case of single-threaded implementation, the split point is 0.43.

We prove that, for fixed k ≥ 3, the following classes of labeled n-vertex graphs are asymptotically equicardinal: graphs of diameter k, connected graphs of diameter at least k, and (not necessarily connected) graphs with a shortest path of length at least k. An asymptotically exact approximation of the number of such n-vertex graphs is obtained, and an explicit error estimate in the approximation is found. Thus, the estimates are improved for the asymptotic approximation of the number of n-vertex graphs of fixed diameter k earlier obtained by Füredi and Kim. It is shown that almost all graphs of diameter k have a unique pair of diametrical vertices but almost all graphs of diameter 2 have more than one pair of such vertices.

Neighborhood unions and hamiltonian properties in graphs

Estimating the size of the maximum matching is a canonical problem in graph analysis, and one that has attracted extensive study over a range of different computational models. We present improved streaming algorithms for approximating the size of maximum matching with sparse (bounded arboricity) graphs. &#13;\n&#13;\n* (Insert-Only Streams) We present a one-pass algorithm that takes O(alpha log n) space and approximates the size of the maximum matching in graphs with arboricity alpha within a factor of O(alpha). This improves significantly upon the state-of-the-art tilde{O}(alpha n^{2/3})-space streaming algorithms, and is the first poly-logarithmic space algorithm for this problem. &#13;\n&#13;\n* (Dynamic Streams) Given a dynamic graph stream (i.e., inserts and deletes) of edges of an underlying alpha-bounded arboricity graph, we present an one-pass algorithm that uses space tilde{O}(alpha^{10/3}n^{2/3}) and returns an O(alpha)-estimator for the size of the maximum matching on the condition that the number edge deletions in the stream is bounded by O(alpha n). For this class of inputs, our algorithm improves the state-of-the-art tilde{O}(\\alpha n^{4/5})-space algorithms, where the \\tilde{O}(.) notation hides logarithmic in n dependencies.&#13;\n&#13;\nIn contrast to prior work, our results take more advantage of the streaming access to the input and characterize the matching size based on the ordering of the edges in the stream in addition to the degree distributions and structural properties of the sparse graphs.

We describe algorithms, based on Avis and Fukuda's reverse search paradigm, for listing all maximal independent sets in a sparse graph in polynomial time and delay per output. For bounded degree graphs, our algorithms take constant time per set generated; for minor-closed graph families, the time is O ( n ) per set, and for more general sparse graph families we achieve subquadratic time per set. We also describe new data structures for maintaining a dynamic vertex set S in a sparse or minor-closed graph family, and querying the number of vertices not dominated by S ; for minor-closed graph families the time per update is constant, while it is sublinear for any sparse graph family. We can also maintain a dynamic vertex set in an arbitrary m -edge graph and test the independence of the maintained set in time O (√m) per update. We use the domination data structures as part of our enumeration algorithms.

We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph in order to find a subgraph which possesses some target property with high probability. In this paper we focus on finding long paths in when for some fixed constant . This random graph is known to have typically linearly long paths. To have edges with high probability in one clearly needs to query at least pairs of vertices. Can we find a path of length economically, i.e., by querying roughly that many pairs? We argue that this is not possible and one needs to query significantly more pairs. We prove that any randomised algorithm which finds a path of length with at least constant probability in with must query at least pairs of vertices. This is tight up to the factor. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 71–85, 2017

We introduce a model for differentially private analysis of weighted graphs in which the graph topology (υ,e) is assumed to be public and the private information consists only of the edge weights ω : e → R+. This can express hiding congestion patterns in a known system of roads. Differential privacy requires that the output of an algorithm provides little advantage, measured by privacy parameters e and δ, for distinguishing between neighboring inputs, which are thought of as inputs that differ on the contribution of one individual. In our model, two weight functions w,w' are considered to be neighboring if they have l1 distance at most one.We study the problems of privately releasing a short path between a pair of vertices and of privately releasing approximate distances between all pairs of vertices. We are concerned with the approximation error, the difference between the length of the released path or released distance and the length of the shortest path or actual distance.For the problem of privately releasing a short path between a pair of vertices, we prove a lower bound of Ω(|υ|) on the additive approximation error for fixed privacy parameters e,δ. We provide a differentially private algorithm that matches this error bound up to a logarithmic factor and releases paths between all pairs of vertices, not just a single pair. The approximation error achieved by our algorithm can be bounded by the number of edges on the shortest path, so we achieve better accuracy than the worst-case bound for pairs of vertices that are connected by a low-weight path consisting of o(|υ|) vertices.For the problem of privately releasing all-pairs distances, we show that for trees we can release all-pairs distances with approximation error $O(log2.5|υ|) for fixed privacy parameters. For arbitrary bounded-weight graphs with edge weights in [0,M] we can brelease all distances with approximation error O(√>(|υ|M).

This paper presents a general recursive formula to estimate the number of labeled graphs as well as details to evaluate the formula for the following graph properties: number of edges (graph density), degree sequence, degree distribution, classification mixing, and degree mixing, i.e., the formula estimates the number of labeled graphs that have given values for graph properties. The proposed approach can be extended to additional graph properties (e.g., number of triangles) as well as properties of bipartite graphs. For special settings in which formulas exist from previous research, simulation studies demonstrate the validity of the proposed approach. In addition, we demonstrate how our approach can be used to quantify the level of variability in values of a graph property in the subset of graphs that hold a specified value of a different graph property (or properties) constant.

Spanners in sparse graphs

A geometric graph in the plane is angle-monotone of width $\gamma$ if every pair of vertices is connected by an angle-monotone path of width $\gamma$, a path such that the angles of any two edges in the path differ by at most $\gamma$. Angle-monotone graphs have good spanning properties. We prove that every point set in the plane admits an angle-monotone graph of width $90^\circ$, hence with spanning ratio $\sqrt 2$, and a subquadratic number of edges. This answers an open question posed by Dehkordi, Frati and Gudmundsson. We show how to construct, for any point set of size $n$ and any angle $\alpha$, $0 &lt; \alpha &lt; 45^\circ$, an angle-monotone graph of width $(90^\circ+\alpha)$ with $O(\frac{n}{\alpha})$ edges. Furthermore, we give a local routing algorithm to find angle-monotone paths of width $(90^\circ+\alpha)$ in these graphs. The \emph{routing ratio}, which is the ratio of path length to Euclidean distance, is at most $1/\cos(45^\circ + \frac{\alpha}{2})$, i.e., ranging from $\sqrt 2 \approx 1.414$ to $2.613$. For the special case $\alpha = 30^\circ$, we obtain the full-$\Theta_6$-graph and our routing algorithm achieves the known routing ratio 2 while finding angle-monotone paths of width $120^\circ$.

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 regularity lemma of Szemeredi turned out to be the most powerful tool for studying the testability of graph properties in the dense graph model. In fact, as we argue in this paper, this lemma can be used in order to prove (essentially) all the previous results in this area. More precisely, a barrier for obtaining an efficient testing algorithm for a graph property P was having an efficient regularity lemma for graphs satisfying P. The problem is that for many natural graph properties (e.g. triangle freeness) it is known that a graph can satisfy P and still only have regular partitions of tower-type size. This means that there was no viable path for obtaining reasonable bounds on the query complexity of testing such properties. In this paper we consider the property of being induced C_4-free, which also suffers from the fact that a graph might satisfy this property but still have only regular partitions of tower-type size. By developing a new approach for this problem we manage to overcome this barrier and thus obtain a merely exponential bound for testing this property. This is the first substantial progress on a problem raised by Alon in 2001, and more recently by Alon, Conlon and Fox. We thus obtain the first example of an efficient testing algorithm that cannot be derived from an efficient version of the regularity lemma.