Graph Reconstruction with a Betweenness Oracle

Graph reconstruction algorithms seek to learn a hidden graph by repeatedly querying a black-box oracle for information about the graph structure. Perhaps the most well studied and applied version of the problem uses a distance oracle, which can report the shortest path distance between any pair of nodes. We introduce and study the betweenness oracle, where bet(a, m, z) is true iff m lies on a shortest path between a and z. This oracle is strictly weaker than a distance oracle, in the sense that a betweenness query can be simulated by a constant number of distance queries, but not vice versa. Despite this, we are able to develop betweenness reconstruction algorithms that match the current state of the art for distance reconstruction, and even improve it for certain types of graphs. We obtain the following algorithms: (1) Reconstruction of general graphs in O(n^2) queries, (2) Reconstruction of degree-bounded graphs in ~O(n^{3/2}) queries, (3) Reconstruction of geodetic degree-bounded graphs in ~O(n) queries In addition to being a fundamental graph theoretic problem with some natural applications, our new results shed light on some avenues for progress in the distance reconstruction problem

Tight query complexity bounds for learning graph partitions

Given a partition of a graph into connected components, the membership oracle asserts whether any two vertices of the graph lie in the same component or not. We prove that for $n\ge k\ge 2$, learning the components of an $n$-vertex hidden graph with $k$ components requires at least $(k-1)n-\binom k2$ membership queries. Our result improves on the best known information-theoretic bound of $\Omega(n\log k)$ queries, and exactly matches the query complexity of the algorithm introduced by [Reyzin and Srivastava, 2007] for this problem. Additionally, we introduce an oracle, with access to which one can learn the number of components of $G$ in asymptotically fewer queries than learning the full partition, thus answering another question posed by the same authors. Lastly, we introduce a more applicable version of this oracle, and prove asymptotically tight bounds of $\widetilde\Theta(m)$ queries for both learning and verifying an $m$-edge hidden graph $G$ using it.Comment: Accepted for presentation at the 35th Annual Conference of Learning Theory, 202

A Simple Algorithm for Graph Reconstruction

How efficiently can we find an unknown graph using distance queries between its vertices? We assume that the unknown graph is connected, unweighted, and has bounded degree. The goal is to find every edge in the graph. This problem admits a reconstruction algorithm based on multi-phase Voronoi-cell decomposition and using $\tilde O(n^{3/2})$ distance queries. In our work, we analyze a simple reconstruction algorithm. We show that, on random $\Delta$-regular graphs, our algorithm uses $\tilde O(n)$ distance queries. As by-products, we can reconstruct those graphs using $O(\log^2 n)$ queries to an all-distances oracle or $\tilde O(n)$ queries to a betweenness oracle, and we bound the metric dimension of those graphs by $\log^2 n$. Our reconstruction algorithm has a very simple structure, and is highly parallelizable. On general graphs of bounded degree, our reconstruction algorithm has subquadratic query complexity

Bipartite Graph Pre-training for Unsupervised Extractive Summarization with Graph Convolutional Auto-Encoders

Pre-trained sentence representations are crucial for identifying significant sentences in unsupervised document extractive summarization. However, the traditional two-step paradigm of pre-training and sentence-ranking, creates a gap due to differing optimization objectives. To address this issue, we argue that utilizing pre-trained embeddings derived from a process specifically designed to optimize cohensive and distinctive sentence representations helps rank significant sentences. To do so, we propose a novel graph pre-training auto-encoder to obtain sentence embeddings by explicitly modelling intra-sentential distinctive features and inter-sentential cohesive features through sentence-word bipartite graphs. These pre-trained sentence representations are then utilized in a graph-based ranking algorithm for unsupervised summarization. Our method produces predominant performance for unsupervised summarization frameworks by providing summary-worthy sentence representations. It surpasses heavy BERT- or RoBERTa-based sentence representations in downstream tasks.Comment: Accepted by the 2023 Conference on Empirical Methods in Natural Language Processing (EMNLP 2023

Optimal distance query reconstruction for graphs without long induced cycles

Let $G=(V,E)$ be an $n$-vertex connected graph of maximum degree $\Delta$. Given access to $V$ and an oracle that given two vertices $u,v\in V$, returns the shortest path distance between $u$ and $v$, how many queries are needed to reconstruct $E$? We give a simple deterministic algorithm to reconstruct trees using $\Delta n\log_\Delta n+(\Delta+2)n$ distance queries and show that even randomised algorithms need to use at least $\frac1{100} \Delta n\log_\Delta n$ queries in expectation. The best previous lower bound was an information-theoretic lower bound of $\Omega(n\log n/\log \log n)$. Our lower bound also extends to related query models including distance queries for phylogenetic trees, membership queries for learning partitions and path queries in directed trees. We extend our deterministic algorithm to reconstruct graphs without induced cycles of length at least $k$ using $O_{\Delta,k}(n\log n)$ queries, which includes various graph classes of interest such as chordal graphs, permutation graphs and AT-free graphs. Since the previously best known randomised algorithm for chordal graphs uses $O_{\Delta}(n\log^2 n)$ queries in expectation, we both get rid off the randomness and get the optimal dependency in $n$ for chordal graphs and various other graph classes. Finally, we build on an algorithm of Kannan, Mathieu, and Zhou [ICALP, 2015] to give a randomised algorithm for reconstructing graphs of treelength $k$ using $O_{\Delta,k}(n\log^2n)$ queries in expectation.Comment: 35 page

Mapping Networks via Parallel kth-Hop Traceroute Queries

?(v,w), which return the name of the kth vertex on a shortest path from v to w, where ?(v,w) is the distance between v and w, that is, the number of edges in a shortest-path from v to w. The traceroute command is often used for network mapping applications, the study of the connectivity of networks, and it has been studied theoretically with respect to biases it introduces for network mapping when only a subset of nodes in the network can be the source of traceroute queries. In this paper, we provide efficient network mapping algorithms, that are based on kth-hop traceroute queries. Our results include an algorithm that runs in a constant number of parallel rounds with a subquadratic number of queries under reasonable assumptions about the sampling coverage of the nodes that may issue kth-hop traceroute queries. In addition, we introduce a number of new algorithmic techniques, including a high-probability parametric parallelization of a graph clustering technique of Thorup and Zwick, which may be of independent interest

Enhanced Community-Based Routing for Low-Capacity Pocket Switched Networks

Sensor devices and the emergent networks that they enable are capable of transmitting information between data sources and a permanent data sink. Since these devices have low-power and intermittent connectivity, latency of the data may be tolerated in an effort to save energy for certain classes of data. The BUBBLE routing algorithm developed by Hui et al. in 2008 provides consistent routing by employing a model which computes individual nodes popularity from sets of nodes and then uses these popularity values for forwarding decisions. This thesis considers enhancements to BUBBLE based on the hypothesis that nodes do form groups and certain centrality values of nodes within these groups can be used to improve routing decisions further. Built on this insight, there are two algorithms proposed in this thesis. First is the Community-Based- Forwarding (CBF), which uses pairwise group interactions and pairwise node-to-group interactions as a measure of popularity for routing messages. By having a different measure of popularity than BUBBLE, as an additional factor in determining message forwarding, CBF is a more conservative routing scheme than BUBBLE. Thus, it provides consistently superior message transmission and delivery performance at an acceptable delay cost in resource constrained environments. To overcome this drawback, the concept of unique interaction pattern within groups of nodes is introduced in CBF and it is further renewed into an enhanced algorithm known as Hybrid-Community-Based- Forwarding (HCBF). Utilizing this factor will channel messages along the entire path with consideration for higher probability of contact with the destination group and the destination node. Overall, the major contribution of this thesis is to design and evaluate an enhanced social based routing algorithm for resource-constrained Pocket Switched Networks (PSNs), which will optimize energy consumption related to data transfer. It will do so by explicitly considering features of communities in order to reduce packet loss while maintaining high delivery ratio and reduced delay