On the Complexity of Hub Labeling

Algorithmica, 2014

Given an unlabeled, unweighted, and undirected graph with n vertices and small (but not necessarily constant) treewidth k, we consider the problem of preprocessing the graph to build space-efficient encodings (oracles) to perform various queries efficiently. We assume the word RAM model where the size of a word is Ω (log n) bits. The first oracle, we present, is the navigation oracle which facilitates primitive navigation operations of adjacency, neighborhood, and degree queries. By way of an enumerate argument, which is of independent interest, we show the space requirement of the oracle is optimal to within lower order terms for all treewidths. The oracle supports the mentioned queries all in constant worst-case time. The second oracle, we present, is an exact distance oracle which facilitates distance queries between any pair of vertices (i.e., an all-pair shortest-path oracle). The space requirement of the oracle is also optimal to within lower order terms. Moreover, the distance queries perform in O k 2 log 3 k time. Particularly, for the class of graphs of our interest, graphs of bounded treewidth (where k is constant), the distances are reported in constant worst-case time.

… Colloquium on Automata …, 2008

Thorup and Zwick, in the seminal paper [Journal of ACM, 52(1), 2005, pp 1-24], showed that a weighted undirected graph on n vertices can be preprocessed in subcubic time to design a data structure which occupies only subquadratic space, and yet, for any pair of vertices, can answer distance query approximately in constant time. The data structure is termed as approximate distance oracle. Subsequently, there has been improvement in their preprocessing time, and presently the best known algorithms [4, 3] achieve expected O(n 2 ) preprocessing time for these oracles. For a class of graphs, these algorithms indeed run in Θ(n 2 ) time. In this paper, we are able to break this quadratic barrier at the expense of introducing a (small) constant additive error for unweighted graphs. In achieving this goal, we have been able to preserve the optimal size-stretch trade offs of the oracles. One of our algorithms can be extended to weighted graphs, where the additive error becomes 2 · wmax(u, v) -here wmax(u, v) is the heaviest edge in the shortest path between vertices u, v.

World Wide Web, 2021

Shortest distance computation is one of the widely researched areas in theoretical computer science and graph databases. Distance labeling are well-known for improving the performance of shortest distance queries. One of the best distance labeling approaches is Pruned Landmark Labeling (PLL). PLL is a 2-hop distance labeling which prunes a lot of unnecessary labels while doing breadth-first-search. Another well-known 2-hop labeling is Pruned Highway Labeling (PHL) which is designed for undirected road networks. Both PLL and PHL suffer from the problem of large index size. In this paper, we propose two approaches to address the problem, one is to compress the PLL index as well as the graph for directed graphs; the other is to compress undirected road networks using linear sets, which are essentially maximal-length non-branching paths. Our aim is to reduce the index size and index construction time without significantly compromising query performance. Extensive experiments with real world datasets confirm the effectiveness of our approaches.

GeoInformatica, 2017

Shortest-path computation on graphs is one of the most well-studied problems in algorithmic theory. An aspect that has only recently attracted attention is the use of databases in combination with graph algorithms, socalled distance oracles, to compute shortest-path queries on large graphs. To this purpose, we propose a novel, efficient, pure-SQL framework for answering exact distance queries on large-scale graphs, implemented entirely on an open-source database engine. Our COLD framework (COmpressed Labels on the Database) may answer multiple distance queries (vertex-to-vertex, one-tomany, k-Nearest Neighbors, Reverse k-Nearest Neighbors, Reverse k-Farthest Neighbors and Top-k Range) not handled by previous methods, rendering it a complete database solution for a variety of practical large-scale graph applications. Our experimentation shows that COLD outperforms existing approaches (including popular graph databases) in terms of query time and efficiency, while requiring significantly less storage space than these methods.

We study an approximation algorithm with a performance guarantee to solve a new NP-hard optimization problem on planar graphs. The problem, which is referred to as the minimum hub cover problem, has recently been introduced to the literature to improve query processing over large graph databases. Planar graphs also arise in various graph query processing applications, such as; biometric identification, image classification, object recognition, and so on. Our algorithm is based on a well-known graph decomposition technique that partitions the graph into a set of outerplanar graphs and provides an approximate solution with a proven performance ratio. We conduct a comprehensive computational experiment to investigate the empirical performance of the algorithm. Computational results demonstrate that the empirical performance of the algorithm surpasses its guaranteed performance. We also apply the same decomposition approach to develop a decomposition-based heuristic, which is much more efficient than the approximation algorithm in terms of computation time. Computational results also indicate that the efficacy of the decomposition-based heuristic in terms of solution quality is comparable to that of the approximation algorithm.

Algorithms - ESA 2015, 2015

In the last decade, there has been a substantial amount of research in finding routing algorithms designed specifically to run on real-world graphs. In 2010, Abraham et al. showed upper bounds on the query time in terms of a graph's highway dimension and diameter for the current fastest routing algorithms, including contraction hierarchies, transit node routing, and hub labeling. In this paper, we show corresponding lower bounds for the same three algorithms. We also show how to improve a result by Milosavljević which lower bounds the number of shortcuts added in the preprocessing stage for contraction hierarchies. We relax the assumption of an optimal contraction order (which is NP-hard to compute), allowing the result to be applicable to real-world instances. Finally, we give a proof that optimal preprocessing for hub labeling is NP-hard. Hardness of optimal preprocessing is known for most routing algorithms, and was suspected to be true for hub labeling.

2007

We present an experimental evaluation of an approximate distance oracle recently suggested by Thorup [1] for undirected planar graphs. The oracle uses the existence of graph separators for planar graphs, discovered by Lipton and Tarjan [2], in order to divide the graph into smaller subgraphs. For a planar graph with n nodes, the algorithmic variant considered uses O(n(log n) 3 / ) preprocessing time and O(n(log n) 2 / ) space to answer factor (1 + ) distance queries in O((log n) 2 / ) time. By performing experiments on randomly generated planar graphs and on planar graphs derived from real world road networks, we investigate some key characteristics of the oracle, such as preprocessing time, query time, precision, and characteristics related to the underlying data structure, including space consumption. For graphs with one million nodes, the average query time is less than 20μs.

Electronic Journal of Graph Theory and Applications

In this paper we consider node labelings c of an undirected connected graph G = (V, E) with labels {1, 2, ..., |V |}, which induce a list distance c(u, v) = |c(v) − c(u)| besides the usual graph distance d(u, v). Our main aim is to find a labeling c so c(u, v) is as close to d(u, v) as possible. For any graph we specify algorithms to find a distance-consistent labeling, which is a labeling c that minimize u,v∈V (c(u, v) − d(u, v)) 2. Such labeliings may provide structure for very large graphs. Furthermore, we define a labeling c fulfilling d(u 1 , v 1) < d(u 2 , v 2) ⇒ c(u 1 , v 1) ≤ c(u 2 , v 2) for all node pairs u 1 , v 1 and u 2 , v 2 as a list labeling, and a graph that has a list labeling is a list graph. We prove that list graphs exist for all n = |V | and all k = |E| : n − 1 ≤ k ≤ n(n − 1)/2, and establish basic properties. List graphs are Hamiltonian, and show weak versions of properties of path graphs.

2014 Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments (ALENEX), 2013

We propose a new labeling method for shortest-path and distance queries on road networks. We present a new framework (i.e. data structure and query algorithm) referred to as highway-based labelings and a preprocessing algorithm for it named pruned highway labeling. Our proposed method has several appealing features from different aspects in the literature. Indeed, we take advantages of theoretical analysis of the seminal result by Thorup for distance oracles, more detailed structures of real road networks, and the pruned labeling algorithm that conducts pruned Dijkstra's algorithm. The experimental results show that the proposed method is comparable to the previous state-of-the-art labeling method in both query time and in data size, while our main improvement is that the preprocessing time is much faster.

Proceedings of the 2012 ACM SIGMOD International Conference on Management of Data, 2012

The distance query, which asks the length of the shortest path from a vertex u to another vertex v, has applications ranging from link analysis, semantic web and other ontology processing, to social network operations. Here, we propose a novel labeling scheme, referred to as Highway-Centric Labeling, for answering distance queries in a large sparse graph. It empowers the distance labeling with a highway structure and leverages a novel bipartite set cover framework/algorithm. Highway-centric labeling provides better labeling size than the state-of-the-art 2-hop labeling, theoretically and empirically. It also offers both exact distance and approximate distance with bounded accuracy. A detailed experimental evaluation on both synthetic and real datasets demonstrates that highwaycentric labeling can outperform the state-of-the-art distance computation approaches in terms of both index size and query time.