Karim Douïeb

π/2-Angle Yao Graphs are Spanners

We show that the Yao graph Y 4 in the L 2 metric is a spanner with stretch factor \(8\sqrt{2}(29+23\sqrt{2})\).

Near-Entropy Hotlink Assignments

AbstractConsider a rooted tree T of arbitrary maximum degree d representing a collection of n web pages connected via a set of links, all reachable from a source home page represented by the root of T. Each web page i carries a probability pi representative of the frequency with which it is visited. By adding hotlinks—shortcuts from a node to one of its descendents—we wish to minimize the expected number of steps l needed to visit pages from the home page, expressed as a function of the entropy H(p) of the access probabilities p. This paper introduces several new strategies for effectively assigning hotlinks in a tree. For assigning exactly one hotlink per node, our method guarantees an upper bound on l of 1.141H(p)+1 if d>2 and 1.08H(p)+2/3 if d=2. We also present the first efficient general methods for assigning at most k hotlinks per node in trees of arbitrary maximum degree, achieving bounds on l of at most

Dynamic hotlinks

\frac{2H(p)}{\log(k+1)}+1

Fast local searches and updates in bounded universes

and

Near-entropy hotlink assignments

\frac{H(p)}{\log(k+d)-\log d}+1

Layered Working-Set Trees

, respectively. All our methods are strong, i.e., they provide the same guarantees on all subtrees after the assignment. We also present an algorithm implementing these methods in O(nlog n) time, an improvement over the previous O(n2) time algorithms. Finally we prove a Ω(nlog n) lower bound on the running time of any strong method that guarantee an average access time strictly better than 2H(p).

Common unfoldings of polyominoes and polycubes

Consider a directed rooted tree T=(V,E) representing a collection V of n web pages connected via a set E of links all reachable from a source home page, represented by the root of T. Each web page i carries a weight wi representative of the frequency with which it is visited. By adding hotlinks, shortcuts from a node to one of its descendents, we are interested in minimizing the expected number of steps needed to visit pages from the home page. We give the first linear time algorithm for assigning hotlinks so that the number of steps to accede to a page i from the root of the tree reaches the entropy bound, i.e. is at most

Efficient Construction of Near-Optimal Binary and Multiway Search Trees

O({\rm log} \frac{W}{w_i})

Dynamic Hotlinks

where W=∑i∈Twi. The best previously known algorithm for this task runs in time O(n2). We also give the first efficient data structure for maintaining hotlinks when nodes are added, deleted or their weights modified, in amortized time

The Power and Limitations of Static Binary Search Trees with Lazy Finger

O({\rm log} \frac{W}{w_i})