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Documents authored by Nimbhorkar, Prajakta


Document
Popular Edges with Critical Nodes

Authors: Kushagra Chatterjee and Prajakta Nimbhorkar

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)


Abstract
In the popular edge problem, the input is a bipartite graph G = (A ∪ B,E) where A and B denote a set of men and a set of women respectively, and each vertex in A∪ B has a strict preference ordering over its neighbours. A matching M in G is said to be popular if there is no other matching M' such that the number of vertices that prefer M' to M is more than the number of vertices that prefer M to M'. The goal is to determine, whether a given edge e belongs to some popular matching in G. A polynomial-time algorithm for this problem appears in [Cseh and Kavitha, 2018]. We consider the popular edge problem when some men or women are prioritized or critical. A matching that matches all the critical nodes is termed as a feasible matching. It follows from [Telikepalli Kavitha, 2014; Kavitha, 2021; Nasre et al., 2021; Meghana Nasre and Prajakta Nimbhorkar, 2017] that, when G admits a feasible matching, there always exists a matching that is popular among all feasible matchings. We give a polynomial-time algorithm for the popular edge problem in the presence of critical men or women. We also show that an analogous result does not hold in the many-to-one setting, which is known as the Hospital-Residents Problem in literature, even when there are no critical nodes.

Cite as

Kushagra Chatterjee and Prajakta Nimbhorkar. Popular Edges with Critical Nodes. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{chatterjee_et_al:LIPIcs.ISAAC.2022.54,
  author =	{Chatterjee, Kushagra and Nimbhorkar, Prajakta},
  title =	{{Popular Edges with Critical Nodes}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{54:1--54:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.54},
  URN =		{urn:nbn:de:0030-drops-173399},
  doi =		{10.4230/LIPIcs.ISAAC.2022.54},
  annote =	{Keywords: Matching, Stable Matching, Popular feasible Matching}
}
Document
Popular Matchings in the Hospital-Residents Problem with Two-Sided Lower Quotas

Authors: Meghana Nasre, Prajakta Nimbhorkar, Keshav Ranjan, and Ankita Sarkar

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)


Abstract
We consider the hospital-residents problem where both hospitals and residents can have lower quotas. The input is a bipartite graph G = (ℛ∪ℋ,E), each vertex in ℛ∪ℋ has a strict preference ordering over its neighbors. The sets ℛ and ℋ denote the sets of residents and hospitals respectively. Each hospital has an upper and a lower quota denoting the maximum and minimum number of residents that can be assigned to it. Residents have upper quota equal to one, however, there may be a requirement that some residents must not be left unassigned in the output matching. We call this as the residents' lower quota. We show that whenever the set of matchings satisfying all the lower and upper quotas is non-empty, there always exists a matching that is popular among the matchings in this set. We give a polynomial-time algorithm to compute such a matching.

Cite as

Meghana Nasre, Prajakta Nimbhorkar, Keshav Ranjan, and Ankita Sarkar. Popular Matchings in the Hospital-Residents Problem with Two-Sided Lower Quotas. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 30:1-30:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{nasre_et_al:LIPIcs.FSTTCS.2021.30,
  author =	{Nasre, Meghana and Nimbhorkar, Prajakta and Ranjan, Keshav and Sarkar, Ankita},
  title =	{{Popular Matchings in the Hospital-Residents Problem with Two-Sided Lower Quotas}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{30:1--30:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.30},
  URN =		{urn:nbn:de:0030-drops-155419},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.30},
  annote =	{Keywords: Matching, Popularity, Lower quota, Preferences}
}
Document
How Good Are Popular Matchings?

Authors: Krishnapriya A M, Meghana Nasre, Prajakta Nimbhorkar, and Amit Rawat

Published in: LIPIcs, Volume 103, 17th International Symposium on Experimental Algorithms (SEA 2018)


Abstract
In this paper, we consider the Hospital Residents problem (HR) and the Hospital Residents problem with Lower Quotas (HRLQ). In this model with two sided preferences, stability is a well accepted notion of optimality. However, in the presence of lower quotas, a stable and feasible matching need not exist. For the HRLQ problem, our goal therefore is to output a good feasible matching assuming that a feasible matching exists. Computing matchings with minimum number of blocking pairs (Min-BP) and minimum number of blocking residents (Min-BR) are known to be NP-Complete. The only approximation algorithms for these problems work under severe restrictions on the preference lists. We present an algorithm which circumvents this restriction and computes a popular matching in the HRLQ instance. We show that on data-sets generated using various generators, our algorithm performs very well in terms of blocking pairs and blocking residents. Yokoi [Yokoi, 2017] recently studied envy-free matchings for the HRLQ problem. We propose a simple modification to Yokoi's algorithm to output a maximal envy-free matching. We observe that popular matchings outperform envy-free matchings on several parameters of practical importance, like size, number of blocking pairs, number of blocking residents. In the absence of lower quotas, that is, in the Hospital Residents (HR) problem, stable matchings are guaranteed to exist. Even in this case, we show that popularity is a practical alternative to stability. For instance, on synthetic data-sets generated using a particular model, as well as on real world data-sets, a popular matching is on an average 8-10% larger in size, matches more number of residents to their top-choice, and more residents prefer the popular matching as compared to a stable matching. Our comprehensive study reveals the practical appeal of popular matchings for the HR and HRLQ problems. To the best of our knowledge, this is the first study on the empirical evaluation of popular matchings in this setting.

Cite as

Krishnapriya A M, Meghana Nasre, Prajakta Nimbhorkar, and Amit Rawat. How Good Are Popular Matchings?. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{am_et_al:LIPIcs.SEA.2018.9,
  author =	{A M, Krishnapriya and Nasre, Meghana and Nimbhorkar, Prajakta and Rawat, Amit},
  title =	{{How Good Are Popular Matchings?}},
  booktitle =	{17th International Symposium on Experimental Algorithms (SEA 2018)},
  pages =	{9:1--9:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-070-5},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{103},
  editor =	{D'Angelo, Gianlorenzo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2018.9},
  URN =		{urn:nbn:de:0030-drops-89440},
  doi =		{10.4230/LIPIcs.SEA.2018.9},
  annote =	{Keywords: bipartite graphs, hospital residents, lower-quotas, popular matchings}
}
Document
Popular Matchings with Lower Quotas

Authors: Meghana Nasre and Prajakta Nimbhorkar

Published in: LIPIcs, Volume 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)


Abstract
We consider the well-studied Hospital Residents (HR) problem in the presence of lower quotas (LQ). The input instance consists of a bipartite graph G = (R U H, E) where R and H denote sets of residents and hospitals, respectively. Every vertex has a preference list that imposes a strict ordering on its neighbors. In addition, each hospital has an associated upper-quota and a lower-quota. A matching M in G is an assignment of residents to hospitals, and M is said to be feasible if every resident is assigned to at most one hospital and a hospital is assigned at least its lower-quota many residents and at most its upper-quota many residents. Stability is a de-facto notion of optimality in a model where both sets of vertices have preferences. A matching is stable if no unassigned pair has an incentive to deviate from it. It is well-known that an instance of the HRLQ problem need not admit a feasible stable matching. In this paper, we consider the notion of popularity for the HRLQ problem. A matching M is popular if no other matching M' gets more votes than M when vertices vote between M and M'. When there are no lower quotas, there always exists a stable matching and it is known that every stable matching is popular. We show that in an HRLQ instance, although a feasible stable matching need not exist, there is always a matching that is popular in the set of feasible matchings. We give an efficient algorithm to compute a maximum cardinality matching that is popular amongst all the feasible matchings in an HRLQ instance.

Cite as

Meghana Nasre and Prajakta Nimbhorkar. Popular Matchings with Lower Quotas. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{nasre_et_al:LIPIcs.FSTTCS.2017.44,
  author =	{Nasre, Meghana and Nimbhorkar, Prajakta},
  title =	{{Popular Matchings with Lower Quotas}},
  booktitle =	{37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)},
  pages =	{44:1--44:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-055-2},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{93},
  editor =	{Lokam, Satya and Ramanujam, R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.44},
  URN =		{urn:nbn:de:0030-drops-83937},
  doi =		{10.4230/LIPIcs.FSTTCS.2017.44},
  annote =	{Keywords: bipartite matchings, preferences, hospital residents, lower-quota, popular matchings}
}
Document
Computing the Maximum using (min, +) Formulas

Authors: Meena Mahajan, Prajakta Nimbhorkar, and Anuj Tawari

Published in: LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)


Abstract
We study computation by formulas over (min,+). We consider the computation of max{x_1,...,x_n} over N as a difference of (min,+) formulas, and show that size n + n \log n is sufficient and necessary. Our proof also shows that any (min,+) formula computing the minimum of all sums of n-1 out of n variables must have n \log n leaves; this too is tight. Our proofs use a complexity measure for (min,+) functions based on minterm-like behaviour and on the entropy of an associated graph.

Cite as

Meena Mahajan, Prajakta Nimbhorkar, and Anuj Tawari. Computing the Maximum using (min, +) Formulas. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 74:1-74:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{mahajan_et_al:LIPIcs.MFCS.2017.74,
  author =	{Mahajan, Meena and Nimbhorkar, Prajakta and Tawari, Anuj},
  title =	{{Computing the Maximum using (min, +) Formulas}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{74:1--74:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.74},
  URN =		{urn:nbn:de:0030-drops-80706},
  doi =		{10.4230/LIPIcs.MFCS.2017.74},
  annote =	{Keywords: Formulas, Circuits, Lower bounds, Tropical semiring}
}
Document
Log-space Algorithms for Paths and Matchings in k-trees

Authors: Bireswar Das, Samir Datta, and Prajakta Nimbhorkar

Published in: LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)


Abstract
Reachability and shortest path problems are \NLC\ for general graphs. They are known to be in \Log\ for graphs of tree-width $2$ \cite{JT07}. However, for graphs of tree-width larger than $2$, no bound better than \NL\ is known. In this paper, we improve these bounds for $k$-trees, where $k$ is a constant. In particular, the main results of our paper are log-space algorithms for reachability in directed $k$-trees, and for computation of shortest and longest paths in directed acyclic $k$-trees. Besides the path problems mentioned above, we consider the problem of deciding whether a $k$-tree has a perfect macthing (decision version), and if so, finding a perfect matching (search version), and prove that these problems are \Log-complete. These problems are known to be in \Ptime\ and in \RNC\ for general graphs, and in \SPL\ for planar bipartite graphs \cite{DKR08}. Our results settle the complexity of these problems for the class of $k$-trees. The results are also applicable for bounded tree-width graphs, when a tree-decomposition is given as input. The technique central to our algorithms is a careful implementation of divide-and-conquer approach in log-space, along with some ideas from \cite{JT07} and \cite{LMR07}.

Cite as

Bireswar Das, Samir Datta, and Prajakta Nimbhorkar. Log-space Algorithms for Paths and Matchings in k-trees. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 215-226, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)


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@InProceedings{das_et_al:LIPIcs.STACS.2010.2456,
  author =	{Das, Bireswar and Datta, Samir and Nimbhorkar, Prajakta},
  title =	{{Log-space Algorithms for Paths and Matchings in k-trees}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{215--226},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Marion, Jean-Yves and Schwentick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2456},
  URN =		{urn:nbn:de:0030-drops-24563},
  doi =		{10.4230/LIPIcs.STACS.2010.2456},
  annote =	{Keywords: k-trees, reachability, matching, log-space}
}
Document
Planar Graph Isomorphism is in Log-Space

Authors: Samir Datta, Nutan Limaye, Prajakta Nimbhorkar, Thomas Thierauf, and Fabian Wagner

Published in: Dagstuhl Seminar Proceedings, Volume 9421, Algebraic Methods in Computational Complexity (2010)


Abstract
Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. There is a significant gap between extant lower and upper bounds for planar graphs as well. We bridge the gap for this natural and important special case by presenting an upper bound that matches the known log-space hardness [JKMT03]. In fact, we show the formally stronger result that planar graph canonization is in log-space. This improves the previously known upper bound of AC1 [MR91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to log-space reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in log-space by [DLN08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell’s algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph.

Cite as

Samir Datta, Nutan Limaye, Prajakta Nimbhorkar, Thomas Thierauf, and Fabian Wagner. Planar Graph Isomorphism is in Log-Space. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 9421, pp. 1-32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)


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@InProceedings{datta_et_al:DagSemProc.09421.6,
  author =	{Datta, Samir and Limaye, Nutan and Nimbhorkar, Prajakta and Thierauf, Thomas and Wagner, Fabian},
  title =	{{Planar Graph Isomorphism is in Log-Space}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--32},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{9421},
  editor =	{Manindra Agrawal and Lance Fortnow and Thomas Thierauf and Christopher Umans},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09421.6},
  URN =		{urn:nbn:de:0030-drops-24169},
  doi =		{10.4230/DagSemProc.09421.6},
  annote =	{Keywords: Planar Graphs, Graph Isomorphism, Logspace}
}
Document
Graph Isomorphism for K_{3,3}-free and K_5-free graphs is in Log-space

Authors: Samir Datta, Prajakta Nimbhorkar, Thomas Thierauf, and Fabian Wagner

Published in: LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)


Abstract
Graph isomorphism is an important and widely studied computational problem with a yet unsettled complexity. However, the exact complexity is known for isomorphism of various classes of graphs. Recently, \cite{DLNTW09} proved that planar isomorphism is complete for log-space. We extend this result %of \cite{DLNTW09} further to the classes of graphs which exclude $K_{3,3}$ or $K_5$ as a minor, and give a log-space algorithm. Our algorithm decomposes $K_{3,3}$ minor-free graphs into biconnected and those further into triconnected components, which are known to be either planar or $K_5$ components \cite{Vaz89}. This gives a triconnected component tree similar to that for planar graphs. An extension of the log-space algorithm of \cite{DLNTW09} can then be used to decide the isomorphism problem. For $K_5$ minor-free graphs, we consider $3$-connected components. These are either planar or isomorphic to the four-rung mobius ladder on $8$ vertices or, with a further decomposition, one obtains planar $4$-connected components \cite{Khu88}. We give an algorithm to get a unique decomposition of $K_5$ minor-free graphs into bi-, tri- and $4$-connected components, and construct trees, accordingly. Since the algorithm of \cite{DLNTW09} does not deal with four-connected component trees, it needs to be modified in a quite non-trivial way.

Cite as

Samir Datta, Prajakta Nimbhorkar, Thomas Thierauf, and Fabian Wagner. Graph Isomorphism for K_{3,3}-free and K_5-free graphs is in Log-space. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 145-156, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)


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@InProceedings{datta_et_al:LIPIcs.FSTTCS.2009.2314,
  author =	{Datta, Samir and Nimbhorkar, Prajakta and Thierauf, Thomas and Wagner, Fabian},
  title =	{{Graph Isomorphism for K\underline\{3,3\}-free and K\underline5-free graphs is in Log-space}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{145--156},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-13-2},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{4},
  editor =	{Kannan, Ravi and Narayan Kumar, K.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2314},
  URN =		{urn:nbn:de:0030-drops-23144},
  doi =		{10.4230/LIPIcs.FSTTCS.2009.2314},
  annote =	{Keywords: Graph isomorphism, K\underline\{3,3\}-free graphs, K\underline5-free graphs, log-space}
}
Document
3-connected Planar Graph Isomorphism is in Log-space

Authors: Samir Datta, Nutan Limaye, and Prajakta Nimbhorkar

Published in: LIPIcs, Volume 2, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2008)


Abstract
We consider the isomorphism and canonization problem for $3$-connected planar graphs. The problem was known to be \Log-hard and in \ULcoUL\ \cite{TW07}. In this paper, we give a deterministic log-space algorithm for $3$-connected planar graph isomorphism and canonization. This gives an \Log-completeness result, thereby settling its complexity. \par The algorithm uses the notion of universal exploration sequences from \cite{koucky01} and \cite{Rei05}. To our knowledge, this is a completely new approach to graph canonization.

Cite as

Samir Datta, Nutan Limaye, and Prajakta Nimbhorkar. 3-connected Planar Graph Isomorphism is in Log-space. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 2, pp. 155-162, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)


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@InProceedings{datta_et_al:LIPIcs.FSTTCS.2008.1749,
  author =	{Datta, Samir and Limaye, Nutan and Nimbhorkar, Prajakta},
  title =	{{3-connected Planar Graph Isomorphism is in Log-space}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{155--162},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-08-8},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{2},
  editor =	{Hariharan, Ramesh and Mukund, Madhavan and Vinay, V},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2008.1749},
  URN =		{urn:nbn:de:0030-drops-17491},
  doi =		{10.4230/LIPIcs.FSTTCS.2008.1749},
  annote =	{Keywords: Planar graph isomorphism, three connected graphs, logspace, universal exploration sequence}
}
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