Dana Pardubská

Measuring the problem-relevant information in input

We propose a new way of characterizing the complexity of online problems. Instead of measuring the degradation of the output quality caused by the ignorance of the future we choose to quantify the amount of additional global information needed for an online algorithm to solve the problem optimally. In our model, the algorithm cooperates with an oracle that can see the whole input. We define the advice complexity of the problem to be the minimal number of bits (normalized per input request, and minimized over all algorithm-oracle pairs) communicated by the algorithm to the oracle in order to solve the problem optimally. Hence, the advice complexity measures the amount of problem-relevant information contained in the input. We introduce two modes of communication between the algorithm and the oracle based on whether the oracle offers an advice spontaneously (helper) or on request (answerer). We analyze the Paging and DiffServ problems in terms of advice complexity and deliver upper and lower bounds in both communication modes; in the case of DiffServ problem in helper mode the bounds are tight.

How much information about the future is needed

We propose a new way of characterizing the complexity of online problems. Instead of measuring the degradation of output quality caused by the ignorance of the future we choose to quantify the amount of additional global information needed for an online algorithm to solve the problem optimally. In our model, the algorithm cooperates with an oracle that can see the whole input. We define the advice complexity of the problem to be the minimal number of bits (normalized per input request, and minimized over all algorithm-oracle pairs) communicated between the algorithm and the oracle in order to solve the problem optimally. Hence, the advice complexity measures the amount of problem-relevant information contained in the input. We introduce two modes of communication between the algorithm and the oracle based on whether the oracle offers an advice spontaneously (helper) or on request (answerer). We analyze the Paging and DiffServ problems in terms of advice complexity and deliver tight bounds in both communication modes.

Antibandwidth and Cyclic Antibandwidth of Hamming Graphs

Abstract The antibandwidth problem is to label vertices of graph G ( V , E ) bijectively by integers 0 , 1 , … , | V | − 1 in such a way that the minimal difference of labels of adjacent vertices is maximised. In this paper we study the antibandwidth of Hamming graphs. We provide labeling algorithms and tight upper bounds for general Hamming graphs ∏ k = 1 d K n k . We have exact values for special choices of n i s and equality between antibandwidth and cyclic antibandwidth values.

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.

Two Lower Bounds on Distributive Generation of Languages

The lower bounds on communication complexity measures of language generation by Parallel Communicating Grammar Systems (PCGS) are investigated. The first result shows that there exists a language that can be generated by some dag-PCGS (PCGS with communication structures realizable by directed acyclic graphs) consisting of 3 grammars, but by no PCGS with tree communication structure. The second result shows that dag-PCGS have their communication complexity of language generation either constant or linear.

Effective systolic algorithms for gossiping in cycles and two-dimensional grids

The complexity of systolic dissemination of information in one-way (telegraph) and two-way (telephone) communication mode is investigated. The following main results are established: (i) tight lower and upper bounds on the complexity of one-way systolic gossip in cycles for any length of the systolic period, (ii) optimal one-way and two-way systolic gossip algorithms in 2-dimensional grids whose complexity (the number of rounds) meets the trivial lower bound (the sum of the sizes of its dimensions).

On Minimalism of Analysis by Reduction by Restarting Automata

The paper provides linguistic observations as a motivation for a formal study of an analysis by reduction. It concentrates on a study of the whole mechanism through a class of restarting automata with meta-instructions using pebbles, with delete and shift operations DS-automata. Four types of infinite sets defined by these automata are considered as linguistically relevant: basic languages on word forms marked with grammatical categories, proper languages on unmarked word forms, categorial languages on grammatical categories, and sets of reductions reduction languages. The equivalence of proper languages is considered for a weak equivalence of DS-automata, and the equivalence of reduction languages for a strong equivalence of DS-automata. The complexity of a language is naturally measured by the number of pebbles, the number of deletions, and the number of word order shifts used in a single reduction step. We have obtained unbounded hierarchies scales for all four types of classes of finite languages considered here, as well as for Chomskys classes of infinite languages. The scales make it possible to estimate relevant complexity issues of analysis by reduction for natural languages.

UNARY CODED NP-COMPLETE LANGUAGES IN ASPACE(log log n)

We shall show that (i) there exists a binary NP-complete language such that its unary coded version is in ASPACE(log log n), (ii) if P ≠ NP, there exists a binary language such that its unary version is in ASPACE(log log n), while the language itself is not in ASPACE(log n). As a consequence, under assumption that P ≠ NP, the standard space translation between unary and binary languages does not work for alternating machines with small space; the equivalence is valid only if s(n) ∈ Ω(n). This is quite different from deterministic and nondeterministic machines, for which the corresponding equivalence holds for each s(n) ∈ Ω(log n), and hence for s(log n) ∈ Ω(log log n). Under assumption that NP ≠ co-NP, we also show that binary versions of unary languages in ASPACE(log log n) form a complexity class not contained in NP.

On Parallel Communicating Grammar Systems and Correctness Preserving Restarting Automata

This paper contributes to the study of Freely Rewriting Re-starting Automata (FRR-automata) and Parallel Communicating Grammar Systems (PCGS) as formalizations of the linguistic method of analysis by reduction . For PCGS we study two complexity measures called generation complexity and distribution complexity , and we prove that a PCGS *** , for which both these complexity measures are bounded by constants, can be simulated by a freely rewriting restarting automaton of a very restricted form. From this characterization it follows that the language L (*** ) is semi-linear, that its characteristic analysis is of polynomial size, and that this analysis can be computed in polynomial time.

Factoring and Testing Primes in Small Space

We discuss how much space is sufficient to decide whether a unary number n is a prime. We show that O(log log n) space is sufficient for a deterministic Turing machine, if it is equipped with an additional pebble movable along the input tape, and also for an alternating machine, if the space restriction applies only to its accepting computation subtrees. That is, un-Primes is in pebble-DSPACE(log log n) and also in accept-ASPACE(log log n), where un-primes={1 n :n is a prime}. Moreover, if the given n is composite, such machines are able to find a divisor of n. Since O(log log n) space is too small to write down a divisor which might require Ω(log n) bits, the witness divisor is indicated by the input head position at the moment when the machine halts.