8 Search Results for "Tarjan, Robert E."

Document
Optimal Energetic Paths for Electric Cars

Authors: Dani Dorfman, Haim Kaplan, Robert E. Tarjan, and Uri Zwick

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
A weighted directed graph G = (V,A,c), where A ⊆ V× V and c:A → ℝ, naturally describes a road network in which an electric car, or vehicle (EV), can roam. An arc uv ∈ A models a road segment connecting the two vertices (junctions) u and v. The cost c(uv) of the arc uv is the amount of energy the car needs to travel from u to v. This amount can be positive, zero or negative. We consider both the more realistic scenario where there are no negative cycles in the graph, as well as the more challenging scenario, which can also be motivated, where negative cycles may be present. The electric car has a battery that can store up to B units of energy. The car can traverse an arc uv ∈ A only if it is at u and the charge b in its battery satisfies b ≥ c(uv). If the car traverses the arc uv then it reaches v with a charge of min{b-c(uv),B} in its battery. Arcs with a positive cost deplete the battery while arcs with negative costs may charge the battery, but not above its capacity of B. If the car is at a vertex u and cannot traverse any outgoing arcs of u, then it is stuck and cannot continue traveling. We consider the following natural problem: Given two vertices s,t ∈ V, can the car travel from s to t, starting at s with an initial charge b, where 0 ≤ b ≤ B? If so, what is the maximum charge with which the car can reach t? Equivalently, what is the smallest depletion δ_{B,b}(s,t) such that the car can reach t with a charge of b-δ_{B,b}(s,t) in its battery, and which path should the car follow to achieve this? We also refer to δ_{B,b}(s,t) as the energetic cost of traveling from s to t. We let δ_{B,b}(s,t) = ∞ if the car cannot travel from s to t starting with an initial charge of b. The problem of computing energetic costs is a strict generalization of the standard shortest paths problem. When there are no negative cycles, the single-source version of the problem can be solved using simple adaptations of the classical Bellman-Ford and Dijkstra algorithms. More involved algorithms are required when the graph may contain negative cycles.

Cite as

Dani Dorfman, Haim Kaplan, Robert E. Tarjan, and Uri Zwick. Optimal Energetic Paths for Electric Cars. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 42:1-42:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

@InProceedings{dorfman_et_al:LIPIcs.ESA.2023.42,
author =	{Dorfman, Dani and Kaplan, Haim and Tarjan, Robert E. and Zwick, Uri},
title =	{{Optimal Energetic Paths for Electric Cars}},
booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
pages =	{42:1--42:17},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-295-2},
ISSN =	{1868-8969},
year =	{2023},
volume =	{274},
editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.42},
URN =		{urn:nbn:de:0030-drops-186955},
doi =		{10.4230/LIPIcs.ESA.2023.42},
annote =	{Keywords: Electric cars, Optimal Paths, Battery depletion}
}
Document
Dynamic Planar Embedding Is in DynFO

Authors: Samir Datta, Asif Khan, and Anish Mukherjee

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract
Planar Embedding is a drawing of a graph on the plane such that the edges do not intersect each other except at the vertices. We know that testing the planarity of a graph and computing its embedding (if it exists), can efficiently be computed, both sequentially [John E. Hopcroft and Robert Endre Tarjan, 1974] and in parallel [Vijaya Ramachandran and John H. Reif, 1994], when the entire graph is presented as input. In the dynamic setting, the input graph changes one edge at a time through insertion and deletions and planarity testing/embedding has to be updated after every change. By storing auxilliary information we can improve the complexity of dynamic planarity testing/embedding over the obvious recomputation from scratch. In the sequential dynamic setting, there has been a series of works [David Eppstein et al., 1996; Giuseppe F. Italiano et al., 1993; Jacob Holm et al., 2018; Jacob Holm and Eva Rotenberg, 2020], culminating in the breakthrough result of polylog(n) sequential time (amortized) planarity testing algorithm of Holm and Rotenberg [Jacob Holm and Eva Rotenberg, 2020]. In this paper we study planar embedding through the lens of DynFO, a parallel dynamic complexity class introduced by Patnaik et al [Sushant Patnaik and Neil Immerman, 1997] (also [Guozhu Dong et al., 1995]). We show that it is possible to dynamically maintain whether an edge can be inserted to a planar graph without causing non-planarity in DynFO. We extend this to show how to maintain an embedding of a planar graph under both edge insertions and deletions, while rejecting edge insertions that violate planarity. Our main idea is to maintain embeddings of only the triconnected components and a special two-colouring of separating pairs that enables us to side-step cascading flips when embedding of a biconnected planar graph changes, a major issue for sequential dynamic algorithms [Jacob Holm and Eva Rotenberg, 2020; Jacob Holm and Eva Rotenberg, 2020].

Cite as

Samir Datta, Asif Khan, and Anish Mukherjee. Dynamic Planar Embedding Is in DynFO. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 39:1-39:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

@InProceedings{datta_et_al:LIPIcs.MFCS.2023.39,
author =	{Datta, Samir and Khan, Asif and Mukherjee, Anish},
title =	{{Dynamic Planar Embedding Is in DynFO}},
booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
pages =	{39:1--39:15},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-292-1},
ISSN =	{1868-8969},
year =	{2023},
volume =	{272},
editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address =	{Dagstuhl, Germany},
URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.39},
URN =		{urn:nbn:de:0030-drops-185736},
doi =		{10.4230/LIPIcs.MFCS.2023.39},
annote =	{Keywords: Dynamic Complexity, Planar graphs, Planar embedding}
}
Document
Exact and Approximate Range Mode Query Data Structures in Practice

Authors: Meng He and Zhen Liu

Published in: LIPIcs, Volume 265, 21st International Symposium on Experimental Algorithms (SEA 2023)

Abstract
We conduct an experimental study on the range mode problem. In the exact version of the problem, we preprocess an array A, such that given a query range [a, b], the most frequent element in A[a, b] can be found efficiently. For this problem, our most important finding is that the strategy of using succinct data structures to encode more precomputed information not only helped Chan et al. (Linear-space data structures for range mode query in arrays, Theory of Computing Systems, 2013) improve previous results in theory but also helps us achieve the best time/space tradeoff in practice; we even go a step further to replace more components in their solution with succinct data structures and improve the performance further. In the approximate version of this problem, a (1+ε)-approximate range mode query looks for an element whose occurrences in A[a,b] is at least F_{a,b}/(1+ε), where F_{a,b} is the frequency of the mode in A[a,b]. We implement all previous solutions to this problems and find that, even when ε = 1/2, the average approximation ratio of these solutions is close to 1 in practice, and they provide much faster query time than the best exact solution. These solutions achieve different useful time-space tradeoffs, and among them, El-Zein et al. (On Approximate Range Mode and Range Selection, 30th International Symposium on Algorithms and Computation, 2019) provide us with one solution whose space usage is only 35.6% to 93.8% of the cost of storing the input array of 32-bit integers (in most cases, the space cost is closer to the lower end, and the average space cost is 20.2 bits per symbol among all datasets). Its non-succinct version also stands out with query support at least several times faster than other O(n/ε)-word structures while using only slightly more space in practice.

Cite as

Meng He and Zhen Liu. Exact and Approximate Range Mode Query Data Structures in Practice. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 19:1-19:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

@InProceedings{he_et_al:LIPIcs.SEA.2023.19,
author =	{He, Meng and Liu, Zhen},
title =	{{Exact and Approximate Range Mode Query Data Structures in Practice}},
booktitle =	{21st International Symposium on Experimental Algorithms (SEA 2023)},
pages =	{19:1--19:22},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-279-2},
ISSN =	{1868-8969},
year =	{2023},
volume =	{265},
editor =	{Georgiadis, Loukas},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address =	{Dagstuhl, Germany},
URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.19},
URN =		{urn:nbn:de:0030-drops-183693},
doi =		{10.4230/LIPIcs.SEA.2023.19},
annote =	{Keywords: range mode query, exact range mode query, approximate range mode query}
}
Document
Hierarchical Relative Lempel-Ziv Compression

Authors: Philip Bille, Inge Li Gørtz, Simon J. Puglisi, and Simon R. Tarnow

Published in: LIPIcs, Volume 265, 21st International Symposium on Experimental Algorithms (SEA 2023)

Abstract
Relative Lempel-Ziv (RLZ) parsing is a dictionary compression method in which a string S is compressed relative to a second string R (called the reference) by parsing S into a sequence of substrings that occur in R. RLZ is particularly effective at compressing sets of strings that have a high degree of similarity to the reference string, such as a set of genomes of individuals from the same species. With the now cheap cost of DNA sequencing, such datasets have become extremely abundant and are rapidly growing. In this paper, instead of using a single reference string for the entire collection, we investigate the use of different reference strings for subsets of the collection, with the aim of improving compression. In particular, we propose a new compression scheme hierarchical relative Lempel-Ziv (HRLZ) which form a rooted tree (or hierarchy) on the strings and then compress each string using RLZ with parent as reference, storing only the root of the tree in plain text. To decompress, we traverse the tree in BFS order starting at the root, decompressing children with respect to their parent. We show that this approach leads to a twofold improvement in compression on bacterial genome datasets, with negligible effect on decompression time compared to the standard single reference approach. We show that an effective hierarchy for a given set of strings can be constructed by computing the optimal arborescence of a completed weighted digraph of the strings, with weights as the number of phrases in the RLZ parsing of the source and destination vertices. We further show that instead of computing the complete graph, a sparse graph derived using locality-sensitive hashing can significantly reduce the cost of computing a good hierarchy, without adversely effecting compression performance.

Cite as

Philip Bille, Inge Li Gørtz, Simon J. Puglisi, and Simon R. Tarnow. Hierarchical Relative Lempel-Ziv Compression. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 18:1-18:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

@InProceedings{bille_et_al:LIPIcs.SEA.2023.18,
author =	{Bille, Philip and G{\o}rtz, Inge Li and Puglisi, Simon J. and Tarnow, Simon R.},
title =	{{Hierarchical Relative Lempel-Ziv Compression}},
booktitle =	{21st International Symposium on Experimental Algorithms (SEA 2023)},
pages =	{18:1--18:16},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-279-2},
ISSN =	{1868-8969},
year =	{2023},
volume =	{265},
editor =	{Georgiadis, Loukas},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address =	{Dagstuhl, Germany},
URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.18},
URN =		{urn:nbn:de:0030-drops-183680},
doi =		{10.4230/LIPIcs.SEA.2023.18},
annote =	{Keywords: Relative compression, Lempel-Ziv compression, RLZ, LZ77, string collections, compressed representation, data structures, efficient algorithms}
}
Document
Invited Talk
Amortised Analysis of Dynamic Data Structures (Invited Talk)

Authors: Eva Rotenberg

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

Abstract
In dynamic data structures, one is interested in efficiently facilitating queries to a data set, while being able to efficiently perform updates as the data set undergoes changes. Often, relaxing the efficiency measure to the amortised setting allows for simpler algorithms. A well-known example of a data structure with amortised guarantees is the splay tree by Sleator and Tarjan [Daniel D. Sleator and Robert E. Tarjan, 1985]. Similarly, in data structures for dynamic graphs, one is interested in efficiently maintaining some information about the graph, or facilitating queries, as the graph undergoes changes in the form of insertion and deletion of edges. Examples of such information include connectivity, planarity, and approximate sparsity of the graph: is the graph presently connected? Is it planar? Has its arboricity grossly exceeded some specified number α̃? The related queries could be: is a connected to b? Are the edges uv and uw consecutive in the ordering around u in its current planar embedding? Or, report the O(α) out-edges of vertex x. In this talk, we will see Brodal and Fagerberg’s amortised algorithm for orienting sparse graphs (i.e. of arboricity ≤ α), so that each vertex has O(α) out-edges [Gerth Stølting Brodal and Rolf Fagerberg, 1999]. The algorithm itself is extremely simple, and uses an elegant amortised argument in its analysis. Then, we will visit the problem of dynamic planarity testing: is the graph presently planar? Here, we will see an elegant amortised reduction to the seemingly easier problem, where planarity-violating edges may be detected and rejected [Eppstein et al., 1996]. We will see a sketch of how the current state-of-the-art algorithm for efficient planarity testing [Jacob Holm and Eva Rotenberg, 2020] uses ideas similar to those in [Gerth Stølting Brodal and Rolf Fagerberg, 1999] to analyse the behaviour of a greedy algorithm via a possibly inefficient algorithm with provably low recourse [Jacob Holm and Eva Rotenberg, 2020]. If time permits, we will touch upon a recent simple amortised data structure for maintaining information in dynamic forests [Jacob Holm et al., 2023], which builds on ideas from splay trees. The talk concludes with some open questions in the area.

Cite as

Eva Rotenberg. Amortised Analysis of Dynamic Data Structures (Invited Talk). In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 2:1-2:2, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

@InProceedings{rotenberg:LIPIcs.STACS.2023.2,
author =	{Rotenberg, Eva},
title =	{{Amortised Analysis of Dynamic Data Structures}},
booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
pages =	{2:1--2:2},
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.2023.2},
URN =		{urn:nbn:de:0030-drops-176547},
doi =		{10.4230/LIPIcs.STACS.2023.2},
annote =	{Keywords: Amortised analysis, splaying, dynamic graphs, planarity testing}
}
Document
Track A: Algorithms, Complexity and Games
Analysis of Smooth Heaps and Slim Heaps

Authors: Maria Hartmann, László Kozma, Corwin Sinnamon, and Robert E. Tarjan

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
The smooth heap is a recently introduced self-adjusting heap [Kozma, Saranurak, 2018] similar to the pairing heap [Fredman, Sedgewick, Sleator, Tarjan, 1986]. The smooth heap was obtained as a heap-counterpart of Greedy BST, a binary search tree updating strategy conjectured to be instance-optimal [Lucas, 1988], [Munro, 2000]. Several adaptive properties of smooth heaps follow from this connection; moreover, the smooth heap itself has been conjectured to be instance-optimal within a certain class of heaps. Nevertheless, no general analysis of smooth heaps has existed until now, the only previous analysis showing that, when used in sorting mode (n insertions followed by n delete-min operations), smooth heaps sort n numbers in O(nlg n) time. In this paper we describe a simpler variant of the smooth heap we call the slim heap. We give a new, self-contained analysis of smooth heaps and slim heaps in unrestricted operation, obtaining amortized bounds that match the best bounds known for self-adjusting heaps. Previous experimental work has found the pairing heap to dominate other data structures in this class in various settings. Our tests show that smooth heaps and slim heaps are competitive with pairing heaps, outperforming them in some cases, while being comparably easy to implement.

Cite as

Maria Hartmann, László Kozma, Corwin Sinnamon, and Robert E. Tarjan. Analysis of Smooth Heaps and Slim Heaps. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 79:1-79:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

@InProceedings{hartmann_et_al:LIPIcs.ICALP.2021.79,
author =	{Hartmann, Maria and Kozma, L\'{a}szl\'{o} and Sinnamon, Corwin and Tarjan, Robert E.},
title =	{{Analysis of Smooth Heaps and Slim Heaps}},
booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages =	{79:1--79:20},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-195-5},
ISSN =	{1868-8969},
year =	{2021},
volume =	{198},
editor =	{Bansal, Nikhil and Merelli, Emanuela 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.ICALP.2021.79},
URN =		{urn:nbn:de:0030-drops-141482},
doi =		{10.4230/LIPIcs.ICALP.2021.79},
annote =	{Keywords: data structure, heap, priority queue, amortized analysis, self-adjusting}
}
Document
Simple Concurrent Labeling Algorithms for Connected Components

Authors: Sixue Liu and Robert E. Tarjan

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

Abstract
We present new concurrent labeling algorithms for finding connected components, and we study their theoretical efficiency. Even though many such algorithms have been proposed and many experiments with them have been done, our algorithms are simpler. We obtain an O(lg n) step bound for two of our algorithms using a novel multi-round analysis. We conjecture that our other algorithms also take O(lg n) steps but are only able to prove an O(lg^2 n) bound. We also point out some gaps in previous analyses of similar algorithms. Our results show that even a basic problem like connected components still has secrets to reveal.

Cite as

Sixue Liu and Robert E. Tarjan. Simple Concurrent Labeling Algorithms for Connected Components. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 3:1-3:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

@InProceedings{liu_et_al:OASIcs.SOSA.2019.3,
author =	{Liu, Sixue and Tarjan, Robert E.},
title =	{{Simple Concurrent Labeling Algorithms for Connected Components}},
booktitle =	{2nd Symposium on Simplicity in Algorithms (SOSA 2019)},
pages =	{3:1--3:20},
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.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.3},
URN =		{urn:nbn:de:0030-drops-100292},
doi =		{10.4230/OASIcs.SOSA.2019.3},
annote =	{Keywords: Connected Components, Concurrent Algorithms}
}
Document
Minimum Cost Flows in Graphs with Unit Capacities

Authors: Andrew V. Goldberg, Haim Kaplan, Sagi Hed, and Robert E. Tarjan

Published in: LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

Abstract
We consider the minimum cost flow problem on graphs with unit capacities and its special cases. In previous studies, special purpose algorithms exploiting the fact that capacities are one have been developed. In contrast, for maximum flow with unit capacities, the best bounds are proven for slight modifications of classical blocking flow and push-relabel algorithms. In this paper we show that the classical cost scaling algorithms of Goldberg and Tarjan (for general integer capacities) applied to a problem with unit capacities achieve or improve the best known bounds. For weighted bipartite matching we establish a bound of O(\sqrt{rm}\log C) on a slight variation of this algorithm. Here r is the size of the smaller side of the bipartite graph, m is the number of edges, and C is the largest absolute value of an arc-cost. This simplifies a result of [Duan et al. 2011] and improves the bound, answering an open question of [Tarjan and Ramshaw 2012]. For graphs with unit vertex capacities we establish a novel O(\sqrt{n}m\log(nC)) bound. We also give the first cycle canceling algorithm for minimum cost flow with unit capacities. The algorithm naturally generalizes the single source shortest path algorithm of [Goldberg 1995].

Cite as

Andrew V. Goldberg, Haim Kaplan, Sagi Hed, and Robert E. Tarjan. Minimum Cost Flows in Graphs with Unit Capacities. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 406-419, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

@InProceedings{goldberg_et_al:LIPIcs.STACS.2015.406,
author =	{Goldberg, Andrew V. and Kaplan, Haim and Hed, Sagi and Tarjan, Robert E.},
title =	{{Minimum Cost Flows in Graphs with Unit Capacities}},
booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
pages =	{406--419},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-78-1},
ISSN =	{1868-8969},
year =	{2015},
volume =	{30},
editor =	{Mayr, Ernst W. and Ollinger, Nicolas},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address =	{Dagstuhl, Germany},
URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.406},
URN =		{urn:nbn:de:0030-drops-49304},
doi =		{10.4230/LIPIcs.STACS.2015.406},
annote =	{Keywords: minimum cost flow, bipartite matching, unit capacity, cost scaling}
}
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