Michal Forišek

Advice complexity of online coloring for paths

In online graph coloring a graph is revealed to an online algorithm one vertex at a time, and the algorithm must color the vertices as they appear. This paper starts to investigate the advice complexity of this problem --- the amount of oracle information an online algorithm needs in order to make optimal choices. We also consider a more general problem --- a trade-off between online and offline graph coloring. In the paper we prove that precisely ⌈n/2 ⌉−1 bits of advice are needed when the vertices on a path are presented for coloring in arbitrary order. The same holds in the more general case when just a subset of the vertices is colored online. However, the problem turns out to be non-trivial for the case where the online algorithm is guaranteed that the vertices it receives form a subset of a path and are presented in the order in which they lie on the path. For this variant we prove that its advice complexity is βn+O(logn) bits, where β≈0.406 is a fixed constant (we give its closed form). This suggests that the generalized problem will be challenging for more complex graph classes.

Computational complexity of two-dimensional platform games

We analyze the computational complexity of various two-dimensional platform games. We state and prove several meta-theorems that identify a class of these games for which the set of solvable levels is NP-hard, and another class for which the set is even PSPACE-hard. Notably COMMANDERKEEN is shown to be NP-hard, and PRINCE OF PERSIA is shown to be PSPACE-complete. We then analyze the related game Lemmings, where we construct a set of instances which only have exponentially long solutions. This shows that an assumption by Cormode in [3] is false and invalidates the proof that the general version of the LEMMINGS decision problem is in NP. We then augment our construction to only include one entrance, which makes our instances perfectly natural within the context of the original game.

Metaphors and analogies for teaching algorithms

In this paper we explore the topic of using metaphors and analogies in teaching algorithms. We argue their importance in the teaching process. We present a selection of metaphors we successfully used when teaching algorithms to secondary school students. We also discuss the suitability of several commonly used metaphors, and in several cases we show significant weaknesses of these metaphors.

Online bandwidth allocation

The paper investigates a version of the resource allocation problem arising in the wireless networking, namely in the OVSF code reallocation process. In this setting a complete binary tree of a given height n is considered, together with a sequence of requests which have to be served in an online manner. The requests are of two types: an insertion request requires to allocate a complete subtree of a given height, and a deletion request frees a given allocated subtree. In order to serve an insertion request it might be necessary to move some already allocated subtrees to other locations in order to free a large enough subtree. We are interested in the worst case average number of such reallocations needed to serve a request. In [4] the authors delivered bounds on the competitive ratio of online algorithm solving this problem, and showed that the ratio is between 1.5 and O(n). We partially answer their question about the exact value by giving an O(1)-competitive online algorithm. In [3], authors use the same model in the context of memory management systems, and analyze the number of reallocations needed to serve a request in the worst case. In this setting, our result is a corresponding amortized analysis.

The Difficulty of Programming Contests Increases

In this paper we give a detailed quantitative and qualitative analysis of the difficulty of programming contests in past years. We analyze task topics in past competition tasks, and also analyze an entire problem set in terms of required algorithm efficiency. We provide both subjective and objective data on how contestants are getting better over the years and how the tasks are getting harder. We use an exact, formal method based on Item Response Theory to analyze past contest results.

Didactic Games for Teaching Information Theory

We developed a set of didactic games and activities that can be used to illustrate and teach various concepts from Information Theory. For each of the games and activities we list the topics it covers, give its rules and related information, describe our practical experiences and give an overview of its scientific background. We also discuss the proper ways to integrate these games into the knowledge acquisition process.

Strings and Sequences

In the last main chapter we deal with algorithms on strings and sequences. In the first section we consider two elementary data structures: the stack and the queue. We discuss the abundance of flawed metaphors used in education for the queue data structure, and propose better metaphors that should be used for this purpose. In the second section, we give our original exposition of the one-dimensional facility location problem and its solution. Our metaphor easily generalizes to more complicated variants of the problem. Finally, we present our original metaphor that illuminates the inner workings of the Knuth-Morris-Pratt substring search algorithm. This metaphor also leads to a clean implementation that easily avoids off-by-one errors (for which other implementations of KMP are well known).

THE UNIFORM MINIMUM-ONES 2SAT PROBLEM AND ITS APPLICATION TO HAPLOTYPE CLASSIFICATION

Analyzing genomic data for finding those gene variations which are responsible for hereditary diseases is one of the great challenges in modern bioinformatics. In many living beings (including the human), every gene is present in two copies, inherited from the two parents, the so-called haplotypes . In this paper, we propose a simple combinatorial model for classifying the set of haplotypes in a population according to their responsibility for a certain genetic disease. This model is based on the minimum-ones 2SAT problem with uniform clauses. The minimum-ones 2SAT problem asks for a satisfying assignment to a satisfiable formula in 2CNF which sets a minimum number of variables to true. This problem is well-known to be -hard, even in the case where all clauses are uniform, i.e. , do not contain a positive and a negative literal. We analyze the approximability and present the first non-trivial exact algorithm for the uniform minimum-ones 2SAT problem with a running time of (1.21061 n ) on a 2SAT formula with n variables. We also show that the problem is fixed-parameter tractable by showing that our algorithm can be adapted to verify in (2 k ) time whether an assignment with at most k true variables exists.