Schnyder woods for higher genus triangulated surfaces, with applications to encoding

Schnyder woods are a well-known combinatorial structure for plane triangulations, which yields a decomposition into 3 spanning trees. We extend here definitions and algorithms for Schnyder woods to closed orientable surfaces of arbitrary genus. In particular, we describe a method to traverse a triangulation of genus $g$ and compute a so-called $g$-Schnyder wood on the way. As an application, we give a procedure to encode a triangulation of genus $g$ and $n$ vertices in $4n+O(g \log(n))$ bits. This matches the worst-case encoding rate of Edgebreaker in positive genus. All the algorithms presented here have execution time $O((n+g)g)$, hence are linear when the genus is fixed.Comment: 27 pages, to appear in a special issue of Discrete and Computational Geometr

Succinct Representations of Dynamic Strings

The rank and select operations over a string of length n from an alphabet of size $\sigma$ have been used widely in the design of succinct data structures. In many applications, the string itself need be maintained dynamically, allowing characters of the string to be inserted and deleted. Under the word RAM model with word size $w=\Omega(\lg n)$, we design a succinct representation of dynamic strings using $nH_0 + o(n)\lg\sigma + O(w)$ bits to support rank, select, insert and delete in $O(\frac{\lg n}{\lg\lg n}(\frac{\lg \sigma}{\lg\lg n}+1))$ time. When the alphabet size is small, i.e. when \sigma = O(\polylog (n)), including the case in which the string is a bit vector, these operations are supported in $O(\frac{\lg n}{\lg\lg n})$ time. Our data structures are more efficient than previous results on the same problem, and we have applied them to improve results on the design and construction of space-efficient text indexes

Compact Binary Relation Representations with Rich Functionality

Binary relations are an important abstraction arising in many data representation problems. The data structures proposed so far to represent them support just a few basic operations required to fit one particular application. We identify many of those operations arising in applications and generalize them into a wide set of desirable queries for a binary relation representation. We also identify reductions among those operations. We then introduce several novel binary relation representations, some simple and some quite sophisticated, that not only are space-efficient but also efficiently support a large subset of the desired queries.Comment: 32 page

The Representation and Analysis of Dynamic Networks

Many real-world phenomena, such as article citations and social interactions, can be viewed in terms of a set of entities connected by relationships. Utilizing this abstraction, the system can be represented as a network. This universal nature of networks, combined with the rapid growth in scope of data collection, has caused significant focus to be placed on techniques for mining these networks of high-level information. However, despite the strong temporal dependencies present in many of these systems, such as social networks, substantially less is understood about the evolution of their structures. A dynamic network representation captures this additional dimension by containing a series of static network “snapshots.” The complexity and scale of such a representation poses several challenges regarding storage and analysis. This research explores a novel bit-vector representation of node interactions, which offers advantages in its ability to be compressed and manipulated through established methods from the fields of digital signal processing and information theory. The results have demonstrated high-level similarity between the considered datasets, giving insights into efficient representations. By way of the discrete Fourier transform, this research has also revealed underlying behavioral patterns, particularly in the social network realm. These approaches offer improved characterization and predictive capacity over that gained from analyzing the network as a static system, and the extent of this descriptive power obtainable through the bit-vector representation is a question which this research aims to address.National Science FoundationNo embarg