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Documents authored by Liaee, Mehraneh

Document
DOI: 10.4230/LIPIcs.STACS.2023.23
Online Paging with Heterogeneous Cache Slots

Authors: Marek Chrobak, Samuel Haney, Mehraneh Liaee, Debmalya Panigrahi, Rajmohan Rajaraman, Ravi Sundaram, and Neal E. Young

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
It is natural to generalize the online k-Server problem by allowing each request to specify not only a point p, but also a subset S of servers that may serve it. To initiate a systematic study of this generalization, we focus on uniform and star metrics. For uniform metrics, the problem is equivalent to a generalization of Paging in which each request specifies not only a page p, but also a subset S of cache slots, and is satisfied by having a copy of p in some slot in S. We call this problem Slot-Heterogenous Paging. In realistic settings only certain subsets of cache slots or servers would appear in requests. Therefore we parameterize the problem by specifying a family 𝒮 ⊆ 2^[k] of requestable slot sets, and we establish bounds on the competitive ratio as a function of the cache size k and family 𝒮. If all request sets are allowed (𝒮 = 2^[k]), the optimal deterministic and randomized competitive ratios are exponentially worse than for standard Paging (𝒮 = {[k]}). As a function of |𝒮| and k, the optimal deterministic ratio is polynomial: at most O(k²|𝒮|) and at least Ω(√{|𝒮|}). For any laminar family {𝒮} of height h, the optimal ratios are O(hk) (deterministic) and O(h²log k) (randomized). The special case that we call All-or-One Paging extends standard Paging by allowing each request to specify a specific slot to put the requested page in. For All-or-One Paging the optimal competitive ratios are Θ(k) (deterministic) and Θ(log k) (randomized), while the offline problem is NP-hard. We extend the deterministic upper bound to the weighted variant of All-or-One Paging (a generalization of standard Weighted Paging), showing that it is also Θ(k). Some results for the laminar case are shown via a reduction to the generalization of Paging in which each request specifies a set P of pages, and is satisfied by fetching any page from P into the cache. The optimal ratios for the latter problem (with laminar family of height h) are at most hk (deterministic) and hH_k (randomized).
Cite as

Marek Chrobak, Samuel Haney, Mehraneh Liaee, Debmalya Panigrahi, Rajmohan Rajaraman, Ravi Sundaram, and Neal E. Young. Online Paging with Heterogeneous Cache Slots. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 23:1-23:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chrobak_et_al:LIPIcs.STACS.2023.23,
  author =	{Chrobak, Marek and Haney, Samuel and Liaee, Mehraneh and Panigrahi, Debmalya and Rajaraman, Rajmohan and Sundaram, Ravi and Young, Neal E.},
  title =	{{Online Paging with Heterogeneous Cache Slots}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{23:1--23:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.23},
  URN =		{urn:nbn:de:0030-drops-176759},
  doi =		{10.4230/LIPIcs.STACS.2023.23},
  annote =	{Keywords: Caching and paging algorithms, k-server, weighted paging, laminar family}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.70
Retracting Graphs to Cycles

Authors: Samuel Haney, Mehraneh Liaee, Bruce M. Maggs, Debmalya Panigrahi, Rajmohan Rajaraman, and Ravi Sundaram

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
We initiate the algorithmic study of retracting a graph into a cycle in the graph, which seeks a mapping of the graph vertices to the cycle vertices so as to minimize the maximum stretch of any edge, subject to the constraint that the restriction of the mapping to the cycle is the identity map. This problem has its roots in the rich theory of retraction of topological spaces, and has strong ties to well-studied metric embedding problems such as minimum bandwidth and 0-extension. Our first result is an O(min{k, sqrt{n}})-approximation for retracting any graph on n nodes to a cycle with k nodes. We also show a surprising connection to Sperner’s Lemma that rules out the possibility of improving this result using certain natural convex relaxations of the problem. Nevertheless, if the problem is restricted to planar graphs, we show that we can overcome these integrality gaps by giving an optimal combinatorial algorithm, which is the technical centerpiece of the paper. Building on our planar graph algorithm, we also obtain a constant-factor approximation algorithm for retraction of points in the Euclidean plane to a uniform cycle.
Cite as

Samuel Haney, Mehraneh Liaee, Bruce M. Maggs, Debmalya Panigrahi, Rajmohan Rajaraman, and Ravi Sundaram. Retracting Graphs to Cycles. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 70:1-70:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{haney_et_al:LIPIcs.ICALP.2019.70,
  author =	{Haney, Samuel and Liaee, Mehraneh and Maggs, Bruce M. and Panigrahi, Debmalya and Rajaraman, Rajmohan and Sundaram, Ravi},
  title =	{{Retracting Graphs to Cycles}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{70:1--70:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.70},
  URN =		{urn:nbn:de:0030-drops-106462},
  doi =		{10.4230/LIPIcs.ICALP.2019.70},
  annote =	{Keywords: Graph algorithms, Graph embedding, Planar graphs, Approximation algorithms}
}
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