Document

**Published in:** OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

We consider a problem of searching for an unknown target vertex t in a (possibly edge-weighted) graph. Each vertex-query points to a vertex v and the response either admits that v is the target or provides any neighbor s of v that lies on a shortest path from v to t. This model has been introduced for trees by Onak and Parys [FOCS 2006] and for general graphs by Emamjomeh-Zadeh et al. [STOC 2016]. In the latter, the authors provide algorithms for the error-less case and for the independent noise model (where each query independently receives an erroneous answer with known probability p<1/2 and a correct one with probability 1-p).
We study this problem both with adversarial errors and independent noise models. First, we show an algorithm that needs at most (log_2 n)/(1 - H(r)) queries in case of adversarial errors, where the adversary is bounded with its rate of errors by a known constant r<1/2. Our algorithm is in fact a simplification of previous work, and our refinement lies in invoking an amortization argument. We then show that our algorithm coupled with a Chernoff bound argument leads to a simpler algorithm for the independent noise model and has a query complexity that is both simpler and asymptotically better than the one of Emamjomeh-Zadeh et al. [STOC 2016].
Our approach has a wide range of applications. First, it improves and simplifies the Robust Interactive Learning framework proposed by Emamjomeh-Zadeh and Kempe [NIPS 2017]. Secondly, performing analogous analysis for edge-queries (where a query to an edge e returns its endpoint that is closer to the target) we actually recover (as a special case) a noisy binary search algorithm that is asymptotically optimal, matching the complexity of Feige et al. [SIAM J. Comput. 1994]. Thirdly, we improve and simplify upon an algorithm for searching of unbounded domains due to Aslam and Dhagat [STOC 1991].

Dariusz Dereniowski, Stefan Tiegel, Przemyslaw Uznanski, and Daniel Wolleb-Graf. A Framework for Searching in Graphs in the Presence of Errors. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 4:1-4:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dereniowski_et_al:OASIcs.SOSA.2019.4, author = {Dereniowski, Dariusz and Tiegel, Stefan and Uznanski, Przemyslaw and Wolleb-Graf, Daniel}, title = {{A Framework for Searching in Graphs in the Presence of Errors}}, booktitle = {2nd Symposium on Simplicity in Algorithms (SOSA 2019)}, pages = {4:1--4:17}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-099-6}, ISSN = {2190-6807}, year = {2019}, volume = {69}, editor = {Fineman, Jeremy T. and Mitzenmacher, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.4}, URN = {urn:nbn:de:0030-drops-100305}, doi = {10.4230/OASIcs.SOSA.2019.4}, annote = {Keywords: graph algorithms, noisy binary search, query complexity, reliability} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Computing the distance between a given pattern of length n and a text of length m is defined as calculating, for every m-substring of the text, the distance between the pattern and the substring. This naturally generalizes the standard notion of exact pattern matching to incorporate dissimilarity score. For both Hamming and L_{1} distance only relatively slow O~(n sqrt{m}) solutions are known for this generalization. This can be overcome by relaxing the question. For Hamming distance, the usual relaxation is to consider the k-bounded variant, where distances exceeding k are reported as infty, while for L_{1} distance asking for a (1 +/- epsilon)-approximation seems more natural. For k-bounded Hamming distance, Amir et al. [J. Algorithms 2004] showed an O~(n sqrt{k}) time algorithm, and Clifford et al. [SODA 2016] designed an O~((m+k^{2})* n/m) time solution. We provide a smooth time trade-off between these bounds by exhibiting an O~((m+k sqrt{m})* n/m) time algorithm. We complement the trade-off with a matching conditional lower bound, showing that a significantly faster combinatorial algorithm is not possible, unless the combinatorial matrix multiplication conjecture fails. We also exhibit a series of reductions that together allow us to achieve essentially the same complexity for k-bounded L_1 distance. Finally, for (1 +/- epsilon)-approximate L_1 distance, the running time of the best previously known algorithm of Lipsky and Porat [Algorithmica 2011] was O(epsilon^{-2} n). We improve this to O~(epsilon^{-1}n), thus essentially matching the complexity of the best known algorithm for (1 +/- epsilon)-approximate Hamming distance.

Pawel Gawrychowski and Przemyslaw Uznanski. Towards Unified Approximate Pattern Matching for Hamming and L_1 Distance. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 62:1-62:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gawrychowski_et_al:LIPIcs.ICALP.2018.62, author = {Gawrychowski, Pawel and Uznanski, Przemyslaw}, title = {{Towards Unified Approximate Pattern Matching for Hamming and L\underline1 Distance}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {62:1--62:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.62}, URN = {urn:nbn:de:0030-drops-90669}, doi = {10.4230/LIPIcs.ICALP.2018.62}, annote = {Keywords: approximate pattern matching, conditional lower bounds, L\underline1 distance, Hamming distance} }

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Brief Announcement

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We show that a broad class of (+, diamond) vector products (for binary integer functions diamond) are equivalent under one-to-polylog reductions to the computation of the Hamming distance. Examples include: the dominance product, the threshold product and l_{2p+1} distances for constant p. Our results imply equivalence (up to poly log n factors) between complexity of computation of All Pairs: Hamming Distances, l_{2p+1} Distances, Dominance Products and Threshold Products. As a consequence, Yuster's (SODA'09) algorithm improves not only Matousek's (IPL'91), but also the results of Indyk, Lewenstein, Lipsky and Porat (ICALP'04) and Min, Kao and Zhu (COCOON'09). Furthermore, our reductions apply to the pattern matching setting, showing equivalence (up to poly log n factors) between pattern matching under Hamming Distance, l_{2p+1} Distance, Dominance Product and Threshold Product, with current best upperbounds due to results of Abrahamson (SICOMP'87), Amir and Farach (Ann. Math. Artif. Intell.'91), Atallah and Duket (IPL'11), Clifford, Clifford and Iliopoulous (CPM'05) and Amir, Lipsky, Porat and Umanski (CPM'05). The resulting algorithms for l_{2p+1} Pattern Matching and All Pairs l_{2p+1}, for 2p+1 = 3,5,7,... are new.
Additionally, we show that the complexity of AllPairsHammingDistances (and thus of other aforementioned AllPairs- problems) is within poly log n from the time it takes to multiply matrices n x (n * d) and (n * d) x n, each with (n * d) non-zero entries. This means that the current upperbounds by Yuster (SODA'09) cannot be improved without improving the sparse matrix multiplication algorithm by Yuster and Zwick (ACM TALG'05) and vice versa.

Daniel Graf, Karim Labib, and Przemyslaw Uznanski. Brief Announcement: Hamming Distance Completeness and Sparse Matrix Multiplication. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 109:1-109:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{graf_et_al:LIPIcs.ICALP.2018.109, author = {Graf, Daniel and Labib, Karim and Uznanski, Przemyslaw}, title = {{Brief Announcement: Hamming Distance Completeness and Sparse Matrix Multiplication}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {109:1--109:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.109}, URN = {urn:nbn:de:0030-drops-91134}, doi = {10.4230/LIPIcs.ICALP.2018.109}, annote = {Keywords: fine-grained complexity, matrix multiplication, high dimensional geometry, pattern matching} }

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Brief Announcement

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Depth first search is a natural algorithmic technique for constructing a closed route that visits all vertices of a graph. The length of such route equals, in an edge-weighted tree, twice the total weight of all edges of the tree and this is asymptotically optimal over all exploration strategies. This paper considers a variant of such search strategies where the length of each route is bounded by a positive integer B (e.g. due to limited energy resources of the searcher). The objective is to cover all the edges of a tree T using the minimum number of routes, each starting and ending at the root and each being of length at most B. To this end, we analyze the following natural greedy tree traversal process that is based on decomposing a depth first search traversal into a sequence of limited length routes. Given any arbitrary depth first search traversal R of the tree T, we cover R with routes R_1,...,R_l, each of length at most B such that: R_i starts at the root, reaches directly the farthest point of R visited by R_{i-1}, then R_i continues along the path R as far as possible, and finally R_i returns to the root. We call the above algorithm piecemeal-DFS and we prove that it achieves the asymptotically minimal number of routes l, regardless of the choice of R. Our analysis also shows that the total length of the traversal (and thus the traversal time) of piecemeal-DFS is asymptotically minimum over all energy-constrained exploration strategies. The fact that R can be chosen arbitrarily means that the exploration strategy can be constructed in an online fashion when the input tree T is not known in advance. Each route R_i can be constructed without any knowledge of the yet unvisited part of T. Surprisingly, our results show that depth first search is efficient for energy constrained exploration of trees, even though it is known that the same does not hold for energy constrained exploration of arbitrary graphs.

Shantanu Das, Dariusz Dereniowski, and Przemyslaw Uznanski. Brief Announcement: Energy Constrained Depth First Search. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 165:1-165:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{das_et_al:LIPIcs.ICALP.2018.165, author = {Das, Shantanu and Dereniowski, Dariusz and Uznanski, Przemyslaw}, title = {{Brief Announcement: Energy Constrained Depth First Search}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {165:1--165:5}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.165}, URN = {urn:nbn:de:0030-drops-91693}, doi = {10.4230/LIPIcs.ICALP.2018.165}, annote = {Keywords: DFS traversal, distributed algorithm, graph exploration, piecemeal exploration, online exploration} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

In the 2-reachability problem we are given a directed graph G and we wish to determine if there are two (edge or vertex) disjoint paths from u to v, for given pair of vertices u and v. In this paper, we present an algorithm that computes 2-reachability information for all pairs of vertices in O(n^w log n) time, where n is the number of vertices and w is the matrix multiplication exponent. Hence, we show that the running time of all-pairs 2-reachability is only within a log factor of transitive closure. Moreover, our algorithm produces a witness (i.e., a separating edge or a separating vertex) for all pair of vertices where 2-reachability does not hold. By processing these witnesses, we can compute all the edge- and vertex-dominator trees of G in O(n^2) additional time, which in turn enables us to answer various connectivity queries in O(1) time. For instance, we can test in constant time if there is a path from u to v avoiding an edge e, for any pair of query vertices u and v, and any query edge e, or if there is a path from u to v avoiding a vertex w, for any query vertices u, v, and w.

Loukas Georgiadis, Daniel Graf, Giuseppe F. Italiano, Nikos Parotsidis, and Przemyslaw Uznanski. All-Pairs 2-Reachability in O(n^w log n) Time. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 74:1-74:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{georgiadis_et_al:LIPIcs.ICALP.2017.74, author = {Georgiadis, Loukas and Graf, Daniel and Italiano, Giuseppe F. and Parotsidis, Nikos and Uznanski, Przemyslaw}, title = {{All-Pairs 2-Reachability in O(n^w log n) Time}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {74:1--74:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.74}, URN = {urn:nbn:de:0030-drops-74510}, doi = {10.4230/LIPIcs.ICALP.2017.74}, annote = {Keywords: 2-reachability, All Dominator Trees, Directed Graphs, Boolean Matrix Multiplication} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We consider the following generalization of the binary search problem. A search strategy is required to locate an unknown target node t in a given tree T. Upon querying a node v of the tree, the strategy receives as a reply an indication of the connected component of T\{v} containing the target t. The cost of querying each node is given by a known non-negative weight function, and the considered objective is to minimize the total query cost for a worst-case choice of the target.
Designing an optimal strategy for a weighted tree search instance is known to be strongly NP-hard, in contrast to the unweighted variant of the problem which can be solved optimally in linear time. Here, we show that weighted tree search admits a quasi-polynomial time approximation scheme (QPTAS): for any 0 < epsilon < 1, there exists a (1+epsilon)-approximation strategy with a computation time of n^O(log n / epsilon^2). Thus, the problem is not APX-hard, unless NP is contained in DTIME(n^O(log n)). By applying a generic reduction, we obtain as a corollary that the studied problem admits a polynomial-time O(sqrt(log n))-approximation.
This improves previous tilde-O(log n)-approximation approaches, where the tilde-O-notation disregards O(poly log log n)-factors.

Dariusz Dereniowski, Adrian Kosowski, Przemyslaw Uznanski, and Mengchuan Zou. Approximation Strategies for Generalized Binary Search in Weighted Trees. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 84:1-84:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dereniowski_et_al:LIPIcs.ICALP.2017.84, author = {Dereniowski, Dariusz and Kosowski, Adrian and Uznanski, Przemyslaw and Zou, Mengchuan}, title = {{Approximation Strategies for Generalized Binary Search in Weighted Trees}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {84:1--84:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.84}, URN = {urn:nbn:de:0030-drops-74507}, doi = {10.4230/LIPIcs.ICALP.2017.84}, annote = {Keywords: Approximation Algorithm, Adaptive Algorithm, Graph Search, Binary Search, Vertex Ranking, Trees} }

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**Published in:** LIPIcs, Volume 54, 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)

We consider computing a longest palindrome in the streaming model, where the symbols arrive one-by-one and we do not have random access to the input. While computing the answer exactly using sublinear space is not possible in such a setting, one can still hope for a good approximation guarantee. Our contribution is twofold. First, we provide lower bounds on the space requirements for randomized approximation algorithms processing inputs of length n. We rule out Las Vegas algorithms, as they cannot achieve sublinear space complexity. For Monte Carlo algorithms, we prove a lower bounds of Omega(M log min {|Sigma|, M}) bits of memory; here M=n/E for approximating the answer with additive error E, and M= log n / log (1 + epsilon) for approximating the answer with multiplicative error (1 + epsilon). Second, we design three real-time algorithms for this problem. Our Monte Carlo approximation algorithms for both additive and multiplicative versions of the problem use O(M) words of memory. Thus the obtained lower bounds are asymptotically tight up to a logarithmic factor. The third algorithm is deterministic and finds a longest palindrome exactly if it is short. This algorithm can be run in parallel with a Monte Carlo algorithm to obtain better results in practice. Overall, both the time and space complexity of finding a longest palindrome in a stream are essentially settled.

Pawel Gawrychowski, Oleg Merkurev, Arseny Shur, and Przemyslaw Uznanski. Tight Tradeoffs for Real-Time Approximation of Longest Palindromes in Streams. In 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 54, pp. 18:1-18:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{gawrychowski_et_al:LIPIcs.CPM.2016.18, author = {Gawrychowski, Pawel and Merkurev, Oleg and Shur, Arseny and Uznanski, Przemyslaw}, title = {{Tight Tradeoffs for Real-Time Approximation of Longest Palindromes in Streams}}, booktitle = {27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)}, pages = {18:1--18:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-012-5}, ISSN = {1868-8969}, year = {2016}, volume = {54}, editor = {Grossi, Roberto and Lewenstein, Moshe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2016.18}, URN = {urn:nbn:de:0030-drops-60765}, doi = {10.4230/LIPIcs.CPM.2016.18}, annote = {Keywords: streaming algorithms, space lower bounds, real-time algorithms, palin- dromes, Monte Carlo algorithms} }

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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

Given n colored balls, we want to detect if more than n/2 of them have the same color, and if so find one ball with such majority color. We are only allowed to choose two balls and compare their colors, and the goal is to minimize the total number of such operations. A well-known exercise is to show how to find such a ball with only 2n comparisons while using only a logarithmic number of bits for bookkeeping. The resulting algorithm is called the Boyer-Moore majority vote algorithm. It is known that any deterministic method needs 3n/2-2 comparisons in the worst case, and this is tight. However, it is not clear what is the required number of comparisons if we allow randomization. We construct a randomized algorithm which always correctly finds a ball of the majority color (or detects that there is none) using, with high probability, only 7n/6+o(n) comparisons. We also prove that the expected number of comparisons used by any such randomized method is at least 1.019n.

Pawel Gawrychowski, Jukka Suomela, and Przemyslaw Uznanski. Randomized Algorithms for Finding a Majority Element. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{gawrychowski_et_al:LIPIcs.SWAT.2016.9, author = {Gawrychowski, Pawel and Suomela, Jukka and Uznanski, Przemyslaw}, title = {{Randomized Algorithms for Finding a Majority Element}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {9:1--9:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.9}, URN = {urn:nbn:de:0030-drops-60273}, doi = {10.4230/LIPIcs.SWAT.2016.9}, annote = {Keywords: majority, randomized algorithms, lower bounds} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

The rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, a set of k identical walkers is deployed in parallel, starting from a chosen subset of nodes, and moving around the graph in synchronous steps. During the process, each node maintains a cyclic ordering of its outgoing arcs, and successively propagates walkers which visit it along its outgoing arcs in round-robin fashion, according to the fixed ordering.
We consider the cover time of such a system, i.e., the number of steps after which each node has been visited by at least one walk, regardless of the starting locations of the walks. In the case of k=1, [Yanovski et al., 2003] and [Bampas et al., 2009] showed that a single walk achieves a cover time of exactly Theta(mD) for any n-node graph with m edges and diameter D, and that the walker eventually stabilizes to a traversal of an Eulerian circuit on the set of all directed edges of the graph. For k>1 parallel walks, no similar structural behaviour can be observed.
In this work we provide tight bounds on the cover time of k parallel rotor walks in a graph. We show that this cover time is at most (mD/log(k)) and at least Theta(mD/k) for any graph, which corresponds to a speedup of between Theta(log(k)) and Theta(k) with respect to the cover time of a single walk. Both of these extremal values of speedup are achieved for some graph classes. Our results hold for up to a polynomially large number of walks, k=O(poly(n)).

Dariusz Dereniowski, Adrian Kosowski, Dominik Pajak, and Przemyslaw Uznanski. Bounds on the Cover Time of Parallel Rotor Walks. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 263-275, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{dereniowski_et_al:LIPIcs.STACS.2014.263, author = {Dereniowski, Dariusz and Kosowski, Adrian and Pajak, Dominik and Uznanski, Przemyslaw}, title = {{Bounds on the Cover Time of Parallel Rotor Walks}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {263--275}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.263}, URN = {urn:nbn:de:0030-drops-44637}, doi = {10.4230/LIPIcs.STACS.2014.263}, annote = {Keywords: Distributed graph exploration, Rotor-Router, Collaborative robots, Parallel random walks, Derandomization} }

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