Metrics for the Adaptation of Site Structure

This paper presents an overview of metrics for web site structure and user navigation paths. Particular attention will be paid to the question what these metrics really say about a site and its usage, and how they can be applied for adapting navigation support to the mobile context

PlanE: Representation Learning over Planar Graphs

Graph neural networks are prominent models for representation learning over graphs, where the idea is to iteratively compute representations of nodes of an input graph through a series of transformations in such a way that the learned graph function is isomorphism invariant on graphs, which makes the learned representations graph invariants. On the other hand, it is well-known that graph invariants learned by these class of models are incomplete: there are pairs of non-isomorphic graphs which cannot be distinguished by standard graph neural networks. This is unsurprising given the computational difficulty of graph isomorphism testing on general graphs, but the situation begs to differ for special graph classes, for which efficient graph isomorphism testing algorithms are known, such as planar graphs. The goal of this work is to design architectures for efficiently learning complete invariants of planar graphs. Inspired by the classical planar graph isomorphism algorithm of Hopcroft and Tarjan, we propose PlanE as a framework for planar representation learning. PlanE includes architectures which can learn complete invariants over planar graphs while remaining practically scalable. We empirically validate the strong performance of the resulting model architectures on well-known planar graph benchmarks, achieving multiple state-of-the-art results.Comment: Proceedings of the Thirty-Seventh Annual Conference on Advances in Neural Information Processing Systems (NeurIPS 2023). Code and data available at: https://github.com/ZZYSonny/Plan

Faster Ray Tracing through Hierarchy Cut Code

We propose a novel ray reordering technique to accelerate the ray tracing process by encoding and sorting rays prior to traversal. Instead of spatial coordinates, our method encodes rays according to the cuts of the hierarchical acceleration structure, which is called the hierarchy cut code. This approach can better adapt to the acceleration structure and obtain a more reliable encoding result. We also propose a compression scheme to decrease the sorting overhead by a shorter sorting key. In addition, based on the phenomenon of boundary drift, we theoretically explain the reason why existing reordering methods cannot achieve better performance by using longer sorting keys. The experiment demonstrates that our method can accelerate secondary ray tracing by up to 1.81 times, outperforming the existing methods. Such result proves the effectiveness of hierarchy cut code, and indicate that the reordering technique can achieve greater performance improvement, which worth further research

Characterizing and recognizing exact-distance squares of graphs

For a graph $G=(V,E)$, its exact-distance square, $G^{[\sharp 2]}$, is the graph with vertex set $V$ and with an edge between vertices $x$ and $y$ if and only if $x$ and $y$ have distance (exactly) $2$ in $G$. The graph $G$ is an exact-distance square root of $G^{[\sharp 2]}$. We give a characterization of graphs having an exact-distance square root, our characterization easily leading to a polynomial-time recognition algorithm. We show that it is NP-complete to recognize graphs with a bipartite exact-distance square root. These two results strongly contrast known results on (usual) graph squares. We then characterize graphs having a tree as an exact-distance square root, and from this obtain a polynomial-time recognition algorithm for these graphs. Finally, we show that, unlike for usual square roots, a graph might have (arbitrarily many) non-isomorphic exact-distance square roots which are trees.Comment: 15 pages, 6 figure

FACES OF MATCHING POLYHEDRA

Let G = (V, E, ~) be a finite loopless graph, let b=(bi:ieV) be a vector of positive integers. A feasible matching is a vector X = (x.: j e: E) J of nonnegative integers such that for each node i of G, the sum of the over the edges j of G incident with i is no greater than bi. The matching polyhedron P(G, b) is the convex hull of the set of feasible matchings. In Chapter 3 we describe a version of Edmonds' blossom algorithm which solves the problem of maximizing C • X over P (G, b) where c =. (c.: j e: E) J is an arbitrary real vector. This algorithm proves a theorem of Edmonds which gives a set of linear inequalities sufficient to define P(G, b). In Chapter 4 we prescribe the unique subset of these inequalities which are necessary to define P(G, b), that is, we characterize the facets of P(G, b). We also characterize the vertices of P(G, b), thus describing the structure possessed by the members of the minimal set X of feasible matchings of G such that for any real vector c = (c.: j e: E), c • x is maximized over P(G, b) J member of X. by a In Chapter 5 we present a generalization of the blossom algorithm which solves the problem: maximize c • x over a face F of P(G, b) for any real vector c = (c.: j e: E). J In other words, we find a feasible matching x of G which satisfies the constraints obtained by replacing an arbitrary subset of the inequalities which define P(G, b) by equations and which maximizes c • x subject to this restriction. We also describe an application of this algorithm to matching problems having a hierarchy of objective functions, so called ''multi-optimization'' problems. In Chapter 6 we show how the blossom algorithm can be combined with relatively simple initialization algorithms to give an algorithm which solves the following postoptimality problem. Given that we know a matching 0 x £ P(G, b) maximizes c · x over P(G, b), we wish to utilize 0 X which to find a feasible matching x' £ P(G, b') which maximizes c • x over P(G, b'), where b' = (b!: i £ V) ]_ vector of positive integers and arbitrary real vector. c=(c.:j£E) J is a is an In Chapter 7 we describe a computer implementation of the blossom algorithm described herein