Andrzej Lingas

On the complexity of constructing evolutionary trees

AbstractIn this paper we study a few important tree optimization problems with applications to computational biology. These problems ask for trees that are consistent with an as large part of the given data as possible. We show that the maximum homeomorphic agreement subtree problem cannot be approximated within a factor of

Ther Are Planar Graphs Almost as Good as the Complete Graphs and as Short as Minimum Spanning Trees

N^\varepsilon

Oblivious gossiping in ad-hoc radio networks

, where N is the input size, for any 0 ≤

On Approximation Behavior of the Greedy Triangulation for Convex Polygons

\varepsilon < \tfrac{1}{9}

Polynomial Time Approximation Schemes for MAX-BISECTION on Planar and Geometric Graphs

in polynomial time unless P = NP, even if all the given trees are of height 2. On the other hand, we present an O(N log N)-time heuristic for the restriction of this problem to instances with O(1) trees of height O(1) yielding solutions within a constant factor of the optimum. We prove that the maximum inferred consensus tree problem is NP-complete, and provide a simple, fast heuristic for it yielding solutions within one third of the optimum. We also present a more specialized polynomial-time heuristic for the maximum inferred local consensus tree problem.

A fast algorithm for optimal alignment between similar ordered trees

We study deterministic algorithms for the gossiping problem in ad hoc radio networks under the assumption that each combined message contains at most b(n) single messages or bits of auxiliary information, where b is an integer function and n is the number of nodes in the network.We term such a restricted gossiping problem b(n)-gossiping. We show that ?n-gossiping in an ad hoc radio network on n nodes can be done deterministically in time O(n3/2) which asymptotically matches the best known upper bound on the time complexity of unrestricted deterministic gossiping. Our upper bound on ?n-gossiping is tight up to a poly-logarithmic factor and it implies similarly tight upper bounds on b(n)-gossiping where function b is computable and 1 ? b(n) ? ?n holds. For symmetric ad hoc radio networks, we show that even 1-gossiping can be done deterministically in time O(n3/2). We also demonstrate that O(nt)-gossiping in a symmetric ad hoc radio network on n nodes can be done in time O(n2-t). Note that the latter upper bound is o(n3/2) when the size of a combined message is ?(n1/2). Furthermore, by adopting known results on repeated randomized broadcasting in symmetric ad hoc radio networks, we derive a randomized protocol for 1-gossiping in these networks running in time O(n) on the average. Finally, we observe that when a collision detection mechanism is available, even deterministic 1-gossiping in symmetric ad hoc radio networks can be performed in time O(n).