2 Search Results for "Lamprou, Ioannis"

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DOI: 10.4230/LIPIcs.AofA.2024.22

The Recurrence/Transience of Random Walks on a Bounded Grid in an Increasing Dimension

Authors: Shuma Kumamoto, Shuji Kijima, and Tomoyuki Shirai

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract

It is celebrated that a simple random walk on ℤ and ℤ² returns to the initial vertex v infinitely many times during infinitely many transitions, which is said recurrent, while it returns to v only finite times on ℤ^d for d ≥ 3, which is said transient. It is also known that a simple random walk on a growing region on ℤ^d can be recurrent depending on growing speed for any fixed d. This paper shows that a simple random walk on {0,1,…,N}ⁿ with an increasing n and a fixed N can be recurrent depending on the increasing speed of n. Precisely, we are concerned with a specific model of a random walk on a growing graph (RWoGG) and show a phase transition between the recurrence and transience of the random walk regarding the growth speed of the graph. For the proof, we develop a pausing coupling argument introducing the notion of weakly less homesick as graph growing (weakly LHaGG).

Cite as

Shuma Kumamoto, Shuji Kijima, and Tomoyuki Shirai. The Recurrence/Transience of Random Walks on a Bounded Grid in an Increasing Dimension. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kumamoto_et_al:LIPIcs.AofA.2024.22,
  author =	{Kumamoto, Shuma and Kijima, Shuji and Shirai, Tomoyuki},
  title =	{{The Recurrence/Transience of Random Walks on a Bounded Grid in an Increasing Dimension}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{22:1--22:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.22},
  URN =		{urn:nbn:de:0030-drops-204577},
  doi =		{10.4230/LIPIcs.AofA.2024.22},
  annote =	{Keywords: Random walk, dynamic graph, recurrence, transience, coupling}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2018.25

Maximum Rooted Connected Expansion

Authors: Ioannis Lamprou, Russell Martin, Sven Schewe, Ioannis Sigalas, and Vassilis Zissimopoulos

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract

Prefetching constitutes a valuable tool toward the goal of efficient Web surfing. As a result, estimating the amount of resources that need to be preloaded during a surfer's browsing becomes an important task. In this regard, prefetching can be modeled as a two-player combinatorial game [Fomin et al., Theoretical Computer Science 2014], where a surfer and a marker alternately play on a given graph (representing the Web graph). During its turn, the marker chooses a set of k nodes to mark (prefetch), whereas the surfer, represented as a token resting on graph nodes, moves to a neighboring node (Web resource). The surfer's objective is to reach an unmarked node before all nodes become marked and the marker wins. Intuitively, since the surfer is step-by-step traversing a subset of nodes in the Web graph, a satisfactory prefetching procedure would load in cache (without any delay) all resources lying in the neighborhood of this growing subset. Motivated by the above, we consider the following maximization problem to which we refer to as the Maximum Rooted Connected Expansion (MRCE) problem. Given a graph G and a root node v_0, we wish to find a subset of vertices S such that S is connected, S contains v_0 and the ratio |N[S]|/|S| is maximized, where N[S] denotes the closed neighborhood of S, that is, N[S] contains all nodes in S and all nodes with at least one neighbor in S. We prove that the problem is NP-hard even when the input graph G is restricted to be a split graph. On the positive side, we demonstrate a polynomial time approximation scheme for split graphs. Furthermore, we present a 1/6(1-1/e)-approximation algorithm for general graphs based on techniques for the Budgeted Connected Domination problem [Khuller et al., SODA 2014]. Finally, we provide a polynomial-time algorithm for the special case of interval graphs. Our algorithm returns an optimal solution for MRCE in O(n^3) time, where n is the number of nodes in G.

Cite as

Ioannis Lamprou, Russell Martin, Sven Schewe, Ioannis Sigalas, and Vassilis Zissimopoulos. Maximum Rooted Connected Expansion. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{lamprou_et_al:LIPIcs.MFCS.2018.25,
  author =	{Lamprou, Ioannis and Martin, Russell and Schewe, Sven and Sigalas, Ioannis and Zissimopoulos, Vassilis},
  title =	{{Maximum Rooted Connected Expansion}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{25:1--25:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.25},
  URN =		{urn:nbn:de:0030-drops-96076},
  doi =		{10.4230/LIPIcs.MFCS.2018.25},
  annote =	{Keywords: prefetching, domination, expansion, ratio}
}

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