Luca Versari

Sublinear-space bounded-delay enumeration for massive network analytics: Maximal cliques

Due to the sheer size of real-world networks, delay and space become quite relevant measures for the cost of enumeration in network analytics. This paper presents efficient algorithms for listing maximum cliques in networks, providing the first sublinear-space bounds with guaranteed delay per enumerated clique, thus comparing favorably with the known literature.

Measuring the clustering effect of BWT via RLE

Abstract The Burrows–Wheeler Transform (BWT) is a reversible transformation on which are based several text compressors and many other tools used in Bioinformatics and Computational Biology. The BWT is not actually a compressor, but a transformation that performs a context-dependent permutation of the letters of the input text that often create runs of equal letters (clusters) longer than the ones in the original text, usually referred to as the “clustering effect” of BWT. In particular, from a combinatorial point of view, great attention has been given to the case in which the BWT produces the fewest number of clusters (cf. [5] , [16] , [21] , [23] ). In this paper we are concerned about the cases when the clustering effect of the BWT is not achieved. For this purpose we introduce a complexity measure that counts the number of equal-letter runs of a word. This measure highlights that there exist many words for which BWT gives an “un-clustering effect”, that is BWT produce a great number of short clusters. More in general we show that the application of BWT to any word at worst doubles the number of equal-letter runs. Moreover, we prove that this bound is tight by exhibiting some families of words where such upper bound is always reached. We also prove that for binary words the case in which the BWT produces the maximal number of clusters is related to the very well known Artins conjecture on primitive roots. The study of some combinatorial properties underlying this transformation could be useful for improving indexing and compression strategies.

A Fast Algorithm for Large Common Connected Induced Subgraphs

We present a fast algorithm for finding large common subgraphs, which can be exploited for detecting structural and functional relationships between biological macromolecules. Many fast algorithms exist for finding a single maximum common subgraph. We show with an example that this gives limited information, motivating the less studied problem of finding many large common subgraphs covering different areas. As the latter is also hard, we give heuristics that improve performance by several orders of magnitude. As a case study, we validate our findings experimentally on protein graphs with thousands of atoms.

Directing Road Networks by Listing Strong Orientations

A connected road network with N nodes and L edges has \(K \le L\) edges identified as one-way roads. In a feasible direction, these one-way roads are assigned a direction each, so that every node can reach any other [Robbins ’39]. Using O(L) preprocessing time and space usage, it is shown that all feasible directions can be found in O(K) amortized time each. To do so, we give a new algorithm that lists all the strong orientations of an undirected connected graph with m edges in O(m) amortized time each, using O(m) space. The cost can be deamortized to obtain O(m) delay with \(O(m^2)\) preprocessing time and space.

Tight Lower Bounds for the Number of Inclusion-Minimal st-Cuts.

We study the number of inclusion-minimal cuts in an undirected connected graph G, also called \(st\)-cuts, for any two distinct nodes s and t: the \(st\)-cuts are in one-to-one correspondence with the partitions \(S \cup T\) of the nodes of G such that \(S \cap T = \emptyset \), \(s \in S\), \(t \in T\), and the subgraphs induced by S and T are connected. It is easy to find an exponential upper bound to the number of \(st\)-cuts (e.g. if G is a clique) and a constant lower bound. We prove that there is a more interesting lower bound on this number, namely, \(\varOmega (m)\), for undirected m-edge graphs that are biconnected or triconnected (2- or 3-node-connected). The wheel graphs show that this lower bound is the best possible asymptotically.

Efficient Algorithms for Listing k Disjoint st-Paths in Graphs

Given a connected graph G of m edges and n vertices, we consider the basic problem of listing all the choices of k vertex-disjoint st-paths, for any two input vertices s, t of G and a positive integer k. Our algorithm takes O(m) time per solution, using O(m) space and requiring \(O(F_k(G))\) setup time, where \(F_k(G) = O(m \min \{k, n^{2/3} \log n, \sqrt{m} \log n\} )\) is the cost of running a max-flow algorithm on G to compute a flow of size k. The proposed techniques are simple and apply to other related listing problems discussed in the paper.

D2K: Scalable Community Detection in Massive Networks via Small-Diameter k-Plexes

This paper studies k-plexes, a well known pseudo-clique model for network communities. In a k-plex, each node can miss at most k-1 links. Our goal is to detect large communities in todays real-world graphs which can have hundreds of millions of edges. While many have tried, this task has been elusive so far due to its computationally challenging nature: k-plexes and other pseudo-cliques are harder to find and more numerous than cliques, a well known hard problem. We present D2K, which is the first algorithm able to find large k-plexes of very large graphs in just a few minutes. The good performance of our algorithm follows from a combination of graph-theoretical concepts, careful algorithm engineering and a high-performance implementation. In particular, we exploit the low degeneracy of real-world graphs, and the fact that large enough k-plexes have diameter 2. We validate a sequential and a parallel/distributed implementation of D2K on real graphs with up to half a billion edges.

Finding Maximal Common Subgraphs via Time-Space Efficient Reverse Search

For any two given graphs, we study the problem of finding isomorphisms that correspond to inclusion-maximal common induced subgraphs that are connected. While common (induced or not) subgraphs can be easily listed using some well known reduction and state-of-the-art algorithms, they are not guaranteed to be connected. To meet the connectivity requirement, we propose an algorithm that revisits the paradigm of reverse search and guarantees polynomial time per solution (delay) and linear space, on top of showing good practical performance.

Listing Maximal Independent Sets with Minimal Space and Bounded Delay

An independent set is a set of nodes in a graph such that no two of them are adjacent. It is maximal if there is no node outside the independent set that may join it. Listing maximal independent sets in graphs can be applied, for example, to sample nodes belonging to different communities or clusters in network analysis and document clustering. The problem has a rich history as it is related to maximal cliques, dominance sets, vertex covers and 3-colorings in graphs. We are interested in reducing the delay, which is the worst-case time between any two consecutively output solutions, and the memory footprint, which is the additional working space behind the read-only input graph.