17 Search Results for "Fox, Kyle"

Document
DOI: 10.4230/LIPIcs.ESA.2024.56
A Simple Deterministic Near-Linear Time Approximation Scheme for Transshipment with Arbitrary Positive Edge Costs

Authors: Emily Fox

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
We describe a simple deterministic near-linear time approximation scheme for uncapacitated minimum cost flow in undirected graphs with positive real edge weights, a problem also known as transshipment. Specifically, our algorithm takes as input a (connected) undirected graph G = (V, E), vertex demands b ∈ R^V such that ∑_{v ∈ V} b(v) = 0, positive edge costs c ∈ R_{> 0}^E, and a parameter ε > 0. In O(ε^{-2} m log^{O(1)} n) time, it returns a flow f such that the net flow out of each vertex is equal to the vertex’s demand and the cost of the flow is within a (1 + ε) factor of optimal. Our algorithm is combinatorial and has no running time dependency on the demands or edge costs. With the exception of a recent result presented at STOC 2022 for polynomially bounded edge weights, all almost- and near-linear time approximation schemes for transshipment relied on randomization to embed the problem instance into low-dimensional space. Our algorithm instead deterministically approximates the cost of routing decisions that would be made if the input were subject to a random tree embedding. To avoid computing the Ω(n²) vertex-vertex distances that an approximation of this kind suggests, we also take advantage of the clustering method used in the well-known Thorup-Zwick distance oracle.
Cite as

Emily Fox. A Simple Deterministic Near-Linear Time Approximation Scheme for Transshipment with Arbitrary Positive Edge Costs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fox:LIPIcs.ESA.2024.56,
  author =	{Fox, Emily},
  title =	{{A Simple Deterministic Near-Linear Time Approximation Scheme for Transshipment with Arbitrary Positive Edge Costs}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{56:1--56:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.56},
  URN =		{urn:nbn:de:0030-drops-211270},
  doi =		{10.4230/LIPIcs.ESA.2024.56},
  annote =	{Keywords: Transshipment, minimum cost flow, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.AFT.2024.25
Adaptive Curves for Optimally Efficient Market Making

Authors: Viraj Nadkarni, Sanjeev Kulkarni, and Pramod Viswanath

Published in: LIPIcs, Volume 316, 6th Conference on Advances in Financial Technologies (AFT 2024)

Abstract
Automated Market Makers (AMMs) are essential in Decentralized Finance (DeFi) as they match liquidity supply with demand. They function through liquidity providers (LPs) who deposit assets into liquidity pools. However, the asset trading prices in these pools often trail behind those in more dynamic, centralized exchanges, leading to potential arbitrage losses for LPs. This issue is tackled by adapting market maker bonding curves to trader behavior, based on the classical market microstructure model of Glosten and Milgrom. Our approach ensures a zero-profit condition for the market maker’s prices. We derive the differential equation that an optimal adaptive curve should follow to minimize arbitrage losses while remaining competitive. Solutions to this optimality equation are obtained for standard Gaussian and Lognormal price models using Kalman filtering. A key feature of our method is its ability to estimate the external market price without relying on price or loss oracles. We also provide an equivalent differential equation for the implied dynamics of canonical static bonding curves and establish conditions for their optimality. Our algorithms demonstrate robustness to changing market conditions and adversarial perturbations, and we offer an on-chain implementation using Uniswap v4 alongside off-chain AI co-processors.
Cite as

Viraj Nadkarni, Sanjeev Kulkarni, and Pramod Viswanath. Adaptive Curves for Optimally Efficient Market Making. In 6th Conference on Advances in Financial Technologies (AFT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 316, pp. 25:1-25:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{nadkarni_et_al:LIPIcs.AFT.2024.25,
  author =	{Nadkarni, Viraj and Kulkarni, Sanjeev and Viswanath, Pramod},
  title =	{{Adaptive Curves for Optimally Efficient Market Making}},
  booktitle =	{6th Conference on Advances in Financial Technologies (AFT 2024)},
  pages =	{25:1--25:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-345-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{316},
  editor =	{B\"{o}hme, Rainer and Kiffer, Lucianna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AFT.2024.25},
  URN =		{urn:nbn:de:0030-drops-209612},
  doi =		{10.4230/LIPIcs.AFT.2024.25},
  annote =	{Keywords: Automated market makers, Adaptive, Glosten-Milgrom, Decentralized Finance}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.15
Breaking the Barrier 2^k for Subset Feedback Vertex Set in Chordal Graphs

Authors: Tian Bai and Mingyu Xiao

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
The Subset Feedback Vertex Set problem (SFVS) is to delete k vertices from a given graph such that in the remaining graph, any vertex in a subset T of vertices (called a terminal set) is not in a cycle. The famous Feedback Vertex Set problem is the special case of SFVS with T being the whole set of vertices. In this paper, we study exact algorithms for SFVS in Split Graphs (SFVS-S) and SFVS in Chordal Graphs (SFVS-C). SFVS-S generalizes the minimum vertex cover problem and the prize-collecting version of the maximum independent set problem in hypergraphs (PCMIS), and SFVS-C further generalizes SFVS-S. Both SFVS-S and SFVS-C are implicit 3-Hitting Set problems. However, it is not easy to solve them faster than 3-Hitting Set. In 2019, Philip, Rajan, Saurabh, and Tale (Algorithmica 2019) proved that SFVS-C can be solved in 𝒪^*(2^k) time, slightly improving the best result 𝒪^*(2.0755^k) for 3-Hitting Set. In this paper, we break the "2^k-barrier" for SFVS-S and SFVS-C by introducing an 𝒪^*(1.8192^k)-time algorithm. This achievement also indicates that PCMIS can be solved in 𝒪^*(1.8192ⁿ) time, marking the first exact algorithm for PCMIS that outperforms the trivial 𝒪^*(2ⁿ) threshold. Our algorithm uses reduction and branching rules based on the Dulmage-Mendelsohn decomposition and a divide-and-conquer method.
Cite as

Tian Bai and Mingyu Xiao. Breaking the Barrier 2^k for Subset Feedback Vertex Set in Chordal Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bai_et_al:LIPIcs.MFCS.2024.15,
  author =	{Bai, Tian and Xiao, Mingyu},
  title =	{{Breaking the Barrier 2^k for Subset Feedback Vertex Set in Chordal Graphs}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{15:1--15:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.15},
  URN =		{urn:nbn:de:0030-drops-205711},
  doi =		{10.4230/LIPIcs.MFCS.2024.15},
  annote =	{Keywords: Subset Feedback Vertex Set, Prize-Collecting Maximum Independent Set, Parameterized Algorithms, Split Graphs, Chordal Graphs, Dulmage-Mendelsohn Decomposition}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.38
The Group Access Bounds for Binary Search Trees

Authors: Parinya Chalermsook, Manoj Gupta, Wanchote Jiamjitrak, Akash Pareek, and Sorrachai Yingchareonthawornchai

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
The access lemma (Sleator and Tarjan, JACM 1985) is a property of binary search trees (BSTs) that implies interesting consequences such as static optimality, static finger, and working set property on any access sequence X = (x_1,x_2,… ,x_m). However, there are known corollaries of the dynamic optimality that cannot be derived via the access lemma, such as the dynamic finger, and any o(log n)-competitive ratio to the optimal BST where n is the number of keys. In this paper, we introduce the group access bound that can be defined with respect to a reference group access tree. Group access bounds generalize the access lemma and imply properties that are far stronger than those implied by the classical access lemma. For each of the following results, there is a group access tree whose group access bound 1) Is O(√{log n})-competitive to the optimal BST. 2) Achieves the k-finger bound with an additive term of O(m log k log log n) (randomized) when the reference tree is an almost complete binary tree. 3) Satisfies the unified bound with an additive term of O(m log log n). 4) Matches the unified bound with a time window k with an additive term of O(m log k log log n) (randomized). Furthermore, we prove the simulation theorem: For every group access tree, there is an online BST algorithm that is O(1)-competitive with its group access bound. In particular, any new group access bound will automatically imply a new BST algorithm achieving the same bound. Thereby, we obtain an improved k-finger bound (reference tree is an almost complete binary tree), an improved unified bound with a time window k, and matching the best-known bound for Unified bound in the BST model. Since any dynamically optimal BST must achieve the group access bounds, we believe our results provide a new direction towards proving o(log n)-competitiveness of the Splay tree and Greedy, two prime candidates for the dynamic optimality conjecture.
Cite as

Parinya Chalermsook, Manoj Gupta, Wanchote Jiamjitrak, Akash Pareek, and Sorrachai Yingchareonthawornchai. The Group Access Bounds for Binary Search Trees. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 38:1-38:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chalermsook_et_al:LIPIcs.ICALP.2024.38,
  author =	{Chalermsook, Parinya and Gupta, Manoj and Jiamjitrak, Wanchote and Pareek, Akash and Yingchareonthawornchai, Sorrachai},
  title =	{{The Group Access Bounds for Binary Search Trees}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{38:1--38:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.38},
  URN =		{urn:nbn:de:0030-drops-201817},
  doi =		{10.4230/LIPIcs.ICALP.2024.38},
  annote =	{Keywords: Dynamic Optimality, Binary Search Tree, Online Algorithm}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.30
Õptimal Dynamic Time Warping on Run-Length Encoded Strings

Authors: Itai Boneh, Shay Golan, Shay Mozes, and Oren Weimann

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Dynamic Time Warping (DTW) distance is the optimal cost of matching two strings when extending runs of letters is for free. Therefore, it is natural to measure the time complexity of DTW in terms of the number of runs n (rather than the string lengths N). In this paper, we give an Õ(n²) time algorithm for computing the DTW distance. This matches (up to log factors) the known (conditional) lower bound, and should be compared with the previous fastest O(n³) time exact algorithm and the Õ(n²) time approximation algorithm. Our method also immediately implies an Õ(nk) time algorithm when the distance is bounded by k. This should be compared with the previous fastest O(n²k) and O(Nk) time exact algorithms and the Õ(nk) time approximation algorithm.
Cite as

Itai Boneh, Shay Golan, Shay Mozes, and Oren Weimann. Õptimal Dynamic Time Warping on Run-Length Encoded Strings. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{boneh_et_al:LIPIcs.ICALP.2024.30,
  author =	{Boneh, Itai and Golan, Shay and Mozes, Shay and Weimann, Oren},
  title =	{{\~{O}ptimal Dynamic Time Warping on Run-Length Encoded Strings}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{30:1--30:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.30},
  URN =		{urn:nbn:de:0030-drops-201730},
  doi =		{10.4230/LIPIcs.ICALP.2024.30},
  annote =	{Keywords: Dynamic time warping, Fr\'{e}chet distance, edit distance, run-length encoding}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.95
Fully Dynamic Strongly Connected Components in Planar Digraphs

Authors: Adam Karczmarz and Marcin Smulewicz

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
In this paper we consider maintaining strongly connected components (SCCs) of a directed planar graph subject to edge insertions and deletions. We show a data structure maintaining an implicit representation of the SCCs within Õ(n^{6/7}) worst-case time per update. The data structure supports, in O(log²{n}) time, reporting vertices of any specified SCC (with constant overhead per reported vertex) and aggregating vertex information (e.g., computing the maximum label) over all the vertices of that SCC. Furthermore, it can maintain global information about the structure of SCCs, such as the number of SCCs, or the size of the largest SCC. To the best of our knowledge, no fully dynamic SCCs data structures with sublinear update time have been previously known for any major subclass of digraphs. Our result should be contrasted with the n^{1-o(1)} amortized update time lower bound conditional on SETH, which holds even for dynamically maintaining whether a general digraph has more than two SCCs.
Cite as

Adam Karczmarz and Marcin Smulewicz. Fully Dynamic Strongly Connected Components in Planar Digraphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 95:1-95:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{karczmarz_et_al:LIPIcs.ICALP.2024.95,
  author =	{Karczmarz, Adam and Smulewicz, Marcin},
  title =	{{Fully Dynamic Strongly Connected Components in Planar Digraphs}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{95:1--95:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.95},
  URN =		{urn:nbn:de:0030-drops-202388},
  doi =		{10.4230/LIPIcs.ICALP.2024.95},
  annote =	{Keywords: dynamic strongly connected components, dynamic strong connectivity, dynamic reachability, planar graphs}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2022.10
Clustering with Faulty Centers

Authors: Kyle Fox, Hongyao Huang, and Benjamin Raichel

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

Abstract
In this paper we introduce and formally study the problem of k-clustering with faulty centers. Specifically, we study the faulty versions of k-center, k-median, and k-means clustering, where centers have some probability of not existing, as opposed to prior work where clients had some probability of not existing. For all three problems we provide fixed parameter tractable algorithms, in the parameters k, d, and ε, that (1+ε)-approximate the minimum expected cost solutions for points in d dimensional Euclidean space. For Faulty k-center we additionally provide a 5-approximation for general metrics. Significantly, all of our algorithms have a small dependence on n. Specifically, our Faulty k-center algorithms have only linear dependence on n, while for our algorithms for Faulty k-median and Faulty k-means the dependence is still only n^(1 + o(1)).
Cite as

Kyle Fox, Hongyao Huang, and Benjamin Raichel. Clustering with Faulty Centers. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fox_et_al:LIPIcs.ISAAC.2022.10,
  author =	{Fox, Kyle and Huang, Hongyao and Raichel, Benjamin},
  title =	{{Clustering with Faulty Centers}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{10:1--10:14},
  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.2022.10},
  URN =		{urn:nbn:de:0030-drops-172950},
  doi =		{10.4230/LIPIcs.ISAAC.2022.10},
  annote =	{Keywords: clustering, approximation, probabilistic input, uncertain input}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2022.13
Computation of Cycle Bases in Surface Embedded Graphs

Authors: Kyle Fox and Thomas Stanley

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

Abstract
We present an O(n³ g²log g + m) + Õ(n^{ω + 1}) time deterministic algorithm to find the minimum cycle basis of a directed graph embedded on an orientable surface of genus g. This result improves upon the previous fastest known running time of O(m³n + m²n² log n) applicable to general directed graphs. While an O(n^ω + 2^{2g}n² + m) time deterministic algorithm was known for undirected graphs, the use of the underlying field ℚ in the directed case (as opposed to ℤ₂ for the undirected case) presents extra challenges. It turns out that some of our new observations are useful for both variants of the problem, so we present an O(n^ω + n² g² log g + m) time deterministic algorithm for undirected graphs as well.
Cite as

Kyle Fox and Thomas Stanley. Computation of Cycle Bases in Surface Embedded Graphs. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fox_et_al:LIPIcs.ISAAC.2022.13,
  author =	{Fox, Kyle and Stanley, Thomas},
  title =	{{Computation of Cycle Bases in Surface Embedded Graphs}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{13:1--13:13},
  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.2022.13},
  URN =		{urn:nbn:de:0030-drops-172982},
  doi =		{10.4230/LIPIcs.ISAAC.2022.13},
  annote =	{Keywords: cycle basis, surface embedded graphs, homology}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2021.72
A Faster Algorithm for Maximum Flow in Directed Planar Graphs with Vertex Capacities

Authors: Julian Enoch, Kyle Fox, Dor Mesica, and Shay Mozes

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Abstract
We give an O(k³ Δ n log n min(k, log² n) log²(nC))-time algorithm for computing maximum integer flows in planar graphs with integer arc and vertex capacities bounded by C, and k sources and sinks. This improves by a factor of max(k²,k log² n) over the fastest algorithm previously known for this problem [Wang, SODA 2019]. The speedup is obtained by two independent ideas. First we replace an iterative procedure of Wang that uses O(k) invocations of an O(k³ n log³ n)-time algorithm for maximum flow algorithm in a planar graph with k apices [Borradaile et al., FOCS 2012, SICOMP 2017], by an alternative procedure that only makes one invocation of the algorithm of Borradaile et al. Second, we show two alternatives for computing flows in the k-apex graphs that arise in our modification of Wang’s procedure faster than the algorithm of Borradaile et al. In doing so, we introduce and analyze a sequential implementation of the parallel highest-distance push-relabel algorithm of Goldberg and Tarjan [JACM 1988].
Cite as

Julian Enoch, Kyle Fox, Dor Mesica, and Shay Mozes. A Faster Algorithm for Maximum Flow in Directed Planar Graphs with Vertex Capacities. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 72:1-72:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{enoch_et_al:LIPIcs.ISAAC.2021.72,
  author =	{Enoch, Julian and Fox, Kyle and Mesica, Dor and Mozes, Shay},
  title =	{{A Faster Algorithm for Maximum Flow in Directed Planar Graphs with Vertex Capacities}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{72:1--72:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.72},
  URN =		{urn:nbn:de:0030-drops-155057},
  doi =		{10.4230/LIPIcs.ISAAC.2021.72},
  annote =	{Keywords: flow, planar graphs, vertex capacities}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.26
Approximating the (Continuous) Fréchet Distance

Authors: Connor Colombe and Kyle Fox

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
We describe the first strongly subquadratic time algorithm with subexponential approximation ratio for approximately computing the Fréchet distance between two polygonal chains. Specifically, let P and Q be two polygonal chains with n vertices in d-dimensional Euclidean space, and let α ∈ [√n, n]. Our algorithm deterministically finds an O(α)-approximate Fréchet correspondence in time O((n³ / α²) log n). In particular, we get an O(n)-approximation in near-linear O(n log n) time, a vast improvement over the previously best know result, a linear time 2^O(n)-approximation. As part of our algorithm, we also describe how to turn any approximate decision procedure for the Fréchet distance into an approximate optimization algorithm whose approximation ratio is the same up to arbitrarily small constant factors. The transformation into an approximate optimization algorithm increases the running time of the decision procedure by only an O(log n) factor.
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Connor Colombe and Kyle Fox. Approximating the (Continuous) Fréchet Distance. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{colombe_et_al:LIPIcs.SoCG.2021.26,
  author =	{Colombe, Connor and Fox, Kyle},
  title =	{{Approximating the (Continuous) Fr\'{e}chet Distance}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{26:1--26:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.26},
  URN =		{urn:nbn:de:0030-drops-138259},
  doi =		{10.4230/LIPIcs.SoCG.2021.26},
  annote =	{Keywords: Fr\'{e}chet distance, approximation algorithm, approximate decision procedure}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.45
A Near-Linear Time Approximation Scheme for Geometric Transportation with Arbitrary Supplies and Spread

Authors: Kyle Fox and Jiashuai Lu

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
The geometric transportation problem takes as input a set of points P in d-dimensional Euclidean space and a supply function μ : P → ℝ. The goal is to find a transportation map, a non-negative assignment τ : P × P → ℝ_{≥ 0} to pairs of points, so the total assignment leaving each point is equal to its supply, i.e., ∑_{r ∈ P} τ(q, r) - ∑_{p ∈ P} τ(p, q) = μ(q) for all points q ∈ P. The goal is to minimize the weighted sum of Euclidean distances for the pairs, ∑_{(p, q) ∈ P × P} τ(p, q) ⋅ ||q - p||₂. We describe the first algorithm for this problem that returns, with high probability, a (1 + ε)-approximation to the optimal transportation map in O(n poly(1 / ε) polylog n) time. In contrast to the previous best algorithms for this problem, our near-linear running time bound is independent of the spread of P and the magnitude of its real-valued supplies.
Cite as

Kyle Fox and Jiashuai Lu. A Near-Linear Time Approximation Scheme for Geometric Transportation with Arbitrary Supplies and Spread. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 45:1-45:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fox_et_al:LIPIcs.SoCG.2020.45,
  author =	{Fox, Kyle and Lu, Jiashuai},
  title =	{{A Near-Linear Time Approximation Scheme for Geometric Transportation with Arbitrary Supplies and Spread}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{45:1--45:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.45},
  URN =		{urn:nbn:de:0030-drops-122034},
  doi =		{10.4230/LIPIcs.SoCG.2020.45},
  annote =	{Keywords: Transportation map, earth mover’s distance, shape matching, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2019.23
Approximating the Geometric Edit Distance

Authors: Kyle Fox and Xinyi Li

Published in: LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Abstract
Edit distance is a measurement of similarity between two sequences such as strings, point sequences, or polygonal curves. Many matching problems from a variety of areas, such as signal analysis, bioinformatics, etc., need to be solved in a geometric space. Therefore, the geometric edit distance (GED) has been studied. In this paper, we describe the first strictly sublinear approximate near-linear time algorithm for computing the GED of two point sequences in constant dimensional Euclidean space. Specifically, we present a randomized O(n log^2n) time O(sqrt n)-approximation algorithm. Then, we generalize our result to give a randomized alpha-approximation algorithm for any alpha in [1, sqrt n], running in time O~(n^2/alpha^2). Both algorithms are Monte Carlo and return approximately optimal solutions with high probability.
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Kyle Fox and Xinyi Li. Approximating the Geometric Edit Distance. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fox_et_al:LIPIcs.ISAAC.2019.23,
  author =	{Fox, Kyle and Li, Xinyi},
  title =	{{Approximating the Geometric Edit Distance}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{23:1--23:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Lu, Pinyan and Zhang, Guochuan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.23},
  URN =		{urn:nbn:de:0030-drops-115195},
  doi =		{10.4230/LIPIcs.ISAAC.2019.23},
  annote =	{Keywords: Geometric edit distance, Approximation, Randomized algorithms}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.83
FPT-Algorithms for Computing Gromov-Hausdorff and Interleaving Distances Between Trees

Authors: Elena Farahbakhsh Touli and Yusu Wang

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
The Gromov-Hausdorff distance is a natural way to measure the distortion between two metric spaces. However, there has been only limited algorithmic development to compute or approximate this distance. We focus on computing the Gromov-Hausdorff distance between two metric trees. Roughly speaking, a metric tree is a metric space that can be realized by the shortest path metric on a tree. Any finite tree with positive edge weight can be viewed as a metric tree where the weight is treated as edge length and the metric is the induced shortest path metric in the tree. Previously, Agarwal et al. showed that even for trees with unit edge length, it is NP-hard to approximate the Gromov-Hausdorff distance between them within a factor of 3. In this paper, we present a fixed-parameter tractable (FPT) algorithm that can approximate the Gromov-Hausdorff distance between two general metric trees within a multiplicative factor of 14. Interestingly, the development of our algorithm is made possible by a connection between the Gromov-Hausdorff distance for metric trees and the interleaving distance for the so-called merge trees. The merge trees arise in practice naturally as a simple yet meaningful topological summary (it is a variant of the Reeb graphs and contour trees), and are of independent interest. It turns out that an exact or approximation algorithm for the interleaving distance leads to an approximation algorithm for the Gromov-Hausdorff distance. One of the key contributions of our work is that we re-define the interleaving distance in a way that makes it easier to develop dynamic programming approaches to compute it. We then present a fixed-parameter tractable algorithm to compute the interleaving distance between two merge trees exactly, which ultimately leads to an FPT-algorithm to approximate the Gromov-Hausdorff distance between two metric trees. This exact FPT-algorithm to compute the interleaving distance between merge trees is of interest itself, as it is known that it is NP-hard to approximate it within a factor of 3, and previously the best known algorithm has an approximation factor of O(sqrt{n}) even for trees with unit edge length.
Cite as

Elena Farahbakhsh Touli and Yusu Wang. FPT-Algorithms for Computing Gromov-Hausdorff and Interleaving Distances Between Trees. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 83:1-83:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{farahbakhshtouli_et_al:LIPIcs.ESA.2019.83,
  author =	{Farahbakhsh Touli, Elena and Wang, Yusu},
  title =	{{FPT-Algorithms for Computing Gromov-Hausdorff and Interleaving Distances Between Trees}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{83:1--83:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.83},
  URN =		{urn:nbn:de:0030-drops-112048},
  doi =		{10.4230/LIPIcs.ESA.2019.83},
  annote =	{Keywords: Gromov-Hausdorff distance, Interleaving distance, Merge trees}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2017.8
Maintaining Reeb Graphs of Triangulated 2-Manifolds

Authors: Pankaj K. Agarwal, Kyle Fox, and Abhinandan Nath

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

Abstract
Let M be a triangulated, orientable 2-manifold of genus g without boundary, and let h be a height function over M that is linear within each triangle. We present a kinetic data structure (KDS) for maintaining the Reeb graph R of h as the heights of M's vertices vary continuously with time. Assuming the heights of two vertices of M become equal only O(1) times, the KDS processes O((k + g) n \polylog n) events; n is the number of vertices in M, and k is the number of external events which change the combinatorial structure of R. Each event is processed in O(\log^2 n) time, and the total size of our KDS is O(gn). The KDS can be extended to maintain an augmented Reeb graph as well.
Cite as

Pankaj K. Agarwal, Kyle Fox, and Abhinandan Nath. Maintaining Reeb Graphs of Triangulated 2-Manifolds. 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. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{agarwal_et_al:LIPIcs.FSTTCS.2017.8,
  author =	{Agarwal, Pankaj K. and Fox, Kyle and Nath, Abhinandan},
  title =	{{Maintaining Reeb Graphs of Triangulated 2-Manifolds}},
  booktitle =	{37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)},
  pages =	{8:1--8:14},
  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.2017.8},
  URN =		{urn:nbn:de:0030-drops-84043},
  doi =		{10.4230/LIPIcs.FSTTCS.2017.8},
  annote =	{Keywords: Reeb graphs, 2-manifolds, topological graph theory}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.7
Faster Algorithms for the Geometric Transportation Problem

Authors: Pankaj K. Agarwal, Kyle Fox, Debmalya Panigrahi, Kasturi R. Varadarajan, and Allen Xiao

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract
Let R, B be a set of n points in R^d, for constant d, where the points of R have integer supplies, points of B have integer demands, and the sum of supply is equal to the sum of demand. Let d(.,.) be a suitable distance function such as the L_p distance. The transportation problem asks to find a map tau : R x B --> N such that sum_{b in B}tau(r,b) = supply(r), sum_{r in R}tau(r,b) = demand(b), and sum_{r in R, b in B} tau(r,b) d(r,b) is minimized. We present three new results for the transportation problem when d(.,.) is any L_p metric: * For any constant epsilon > 0, an O(n^{1+epsilon}) expected time randomized algorithm that returns a transportation map with expected cost O(log^2(1/epsilon)) times the optimal cost. * For any epsilon > 0, a (1+epsilon)-approximation in O(n^{3/2}epsilon^{-d}polylog(U)polylog(n)) time, where U is the maximum supply or demand of any point. * An exact strongly polynomial O(n^2 polylog n) time algorithm, for d = 2.
Cite as

Pankaj K. Agarwal, Kyle Fox, Debmalya Panigrahi, Kasturi R. Varadarajan, and Allen Xiao. Faster Algorithms for the Geometric Transportation Problem. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2017.7,
  author =	{Agarwal, Pankaj K. and Fox, Kyle and Panigrahi, Debmalya and Varadarajan, Kasturi R. and Xiao, Allen},
  title =	{{Faster Algorithms for the Geometric Transportation Problem}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{7:1--7:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.7},
  URN =		{urn:nbn:de:0030-drops-72344},
  doi =		{10.4230/LIPIcs.SoCG.2017.7},
  annote =	{Keywords: transportation map, earth mover's distance, shape matching, approximation algorithms}
}
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