Time and Space Efficient Discovery of Maximal Geometric Graphs

Abstract

Abstract.A geometric graph is a labeled graph whose vertices are points in the 2D plane with an isomorphism invariant under geometric transformations such as translation, rotation, and scaling. While Kuramochi and Karypis (ICDM2002) extensively studied the frequent pattern mining problem for geometric subgraphs, the maximal graph mining has not been considered so far. In this paper, we study the maximal (or closed) graph mining problem for the general class of geometric graphs in the 2D plane by extending the framework of Kuramochi and Karypis. Combining techniques of canonical encoding and a depth-first search tree for the class of maximal patterns, we present a polynomial delay and polynomial space algorithm, MaxGeo, that enumerates all maximal subgraphs in a given input geometric graph without duplicates. This is the first result establishing the outputsensitive complexity of closed graph mining for geometric graphs. We also show that the frequent graph mining problem is also solvable in polynomial delay and polynomial time.

Polynomial-Delay and Polynomial-Space Algorithms for Mining Closed Sequences, Graphs, and Pictures in Accessible Set Systems

“…In this paper, we mainly studied a group of pattern classes called rigid patterns [3,4,6,7,12,17,18,22], with closure mappings C. On the other hand, although classes of flexible patterns [5,28,27] have no notion of closure mappings C, they still allow efficient depth-first search. Thus, it an interesting future problem to extend the approach of this paper to these classes.…”

“…The problem of closed set mining (CIM) is one of the most extensively studied topics for years that attracts many researchers from both practical and theoretical views [3,4,5,6,7,11,12,17,18,22,25,26,28,27,31]. A closed itemset (a closed sets, for short) is a representative among an equivalence class of itemsets that have the same set of occurrences in common in a given transaction database D. The collection C of all closed sets in D contains the same information as the original collection F of frequent itemsets, while the size of C is possibly much smaller than F. Hence, from the engineering point of view, CIM is considered useful for increasing the comprehensibility and the efficiency of FIM [26,31].…”

“…On the other hand, some authors show that enumeration of closed patterns is non-trivial task than that of frequent patterns by proving the #P -hardness of the enumeration problem [7,30]. Furthermore, extensions of CIM for various subclasses of semi-structured data such as sequences, trees, and graphs are ongoing [3,4,5,6,7,12,17,18,22,28,27]. …”

GriMa: A Grid Mining Algorithm for Bag-of-Grid-Based Classification

“…GriMA enumerates all frequent patterns in O(kn 2 .|P | 2 ) = O(kn 4 ) time per pattern P . This is a significant improvement over FreqGeo and MaxGeo [1] which have a time complexity of O(k 2 n 4 . ln n) per pattern.…”

“…Grid graphs have already been introduced in [14], but authors did not take into account the rigid topology of the grid and, in particular, the angle information that we use to define our grid mining problem. Finally, grid graphs are special cases of geometric graphs, for which mining algorithms have been proposed in [1]. These algorithms (FreqGeo and MaxGeo) may be seen as a generalization of our new algorithm GriMA but, as such, they are not optimized for cases where the graph is known to be a grid and we show in Sec.…”