Top-Down Skiplists

We describe todolists (top-down skiplists), a variant of skiplists (Pugh 1990) that can execute searches using at most $\log_{2-\varepsilon} n + O(1)$ binary comparisons per search and that have amortized update time $O(\varepsilon^{-1}\log n)$. A variant of todolists, called working-todolists, can execute a search for any element $x$ using $\log_{2-\varepsilon} w(x) + o(\log w(x))$ binary comparisons and have amortized search time $O(\varepsilon^{-1}\log w(w))$. Here, $w(x)$ is the "working-set number" of $x$. No previous data structure is known to achieve a bound better than $4\log_2 w(x)$ comparisons. We show through experiments that, if implemented carefully, todolists are comparable to other common dictionary implementations in terms of insertion times and outperform them in terms of search times.Comment: 18 pages, 5 figure

Planar Visibility: Testing and Counting

In this paper we consider query versions of visibility testing and visibility counting. Let $S$ be a set of $n$ disjoint line segments in $\R^2$ and let $s$ be an element of $S$. Visibility testing is to preprocess $S$ so that we can quickly determine if $s$ is visible from a query point $q$. Visibility counting involves preprocessing $S$ so that one can quickly estimate the number of segments in $S$ visible from a query point $q$. We present several data structures for the two query problems. The structures build upon a result by O'Rourke and Suri (1984) who showed that the subset, $V_S(s)$, of $\R^2$ that is weakly visible from a segment $s$ can be represented as the union of a set, $C_S(s)$, of $O(n^2)$ triangles, even though the complexity of $V_S(s)$ can be $\Omega(n^4)$. We define a variant of their covering, give efficient output-sensitive algorithms for computing it, and prove additional properties needed to obtain approximation bounds. Some of our bounds rely on a new combinatorial result that relates the number of segments of $S$ visible from a point $p$ to the number of triangles in $\bigcup_{s\in S} C_S(s)$ that contain $p$.Comment: 22 page

The Fresh-Finger Property

The unified property roughly states that searching for an element is fast when the current access is close to a recent access. Here, "close" refers to rank distance measured among all elements stored by the dictionary. We show that distance need not be measured this way: in fact, it is only necessary to consider a small working-set of elements to measure this rank distance. This results in a data structure with access time that is an improvement upon those offered by the unified property for many query sequences