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

In a strongly connected graph G = (V,E), a cut arc (also called strong bridge) is an arc e ∈ E whose removal makes the graph no longer strongly connected. Equivalently, there exist u,v ∈ V, such that all u-v walks contain e. Cut arcs are a fundamental graph-theoretic notion, with countless applications, especially in reachability problems.
In this paper we initiate the study of cut paths, as a generalisation of cut arcs, which we naturally define as those paths P for which there exist u,v ∈ V, such that all u-v walks contain P as subwalk. We first prove various properties of cut paths and define their remainder structures, which we use to present a simple O(m)-time verification algorithm for a cut path (|V| = n, |E| = m).
Secondly, we apply cut paths and their remainder structures to improve several reachability problems from bioinformatics, as follows. A walk is called safe if it is a subwalk of every node-covering closed walk of a strongly connected graph. Multi-safety is defined analogously, by considering node-covering sets of closed walks instead. We show that cut paths provide simple O(m)-time algorithms verifying if a walk is safe or multi-safe. For multi-safety, we present the first linear time algorithm, while for safety, we present a simple algorithm where the state-of-the-art employed complex data structures. Finally we show that the simultaneous computation of remainder structures of all subwalks of a cut path can be performed in linear time, since they are related in a structured way. These properties yield an O(mn)-time algorithm outputting all maximal multi-safe walks, improving over the state-of-the-art algorithm running in time O(m²+n³).
The results of this paper only scratch the surface in the study of cut paths, and we believe a rich structure of a graph can be revealed, considering the perspective of a path, instead of just an arc.

Massimo Cairo, Shahbaz Khan, Romeo Rizzi, Sebastian Schmidt, Alexandru I. Tomescu, and Elia C. Zirondelli. Cut Paths and Their Remainder Structure, with Applications. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cairo_et_al:LIPIcs.STACS.2023.17, author = {Cairo, Massimo and Khan, Shahbaz and Rizzi, Romeo and Schmidt, Sebastian and Tomescu, Alexandru I. and Zirondelli, Elia C.}, title = {{Cut Paths and Their Remainder Structure, with Applications}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {17:1--17:17}, 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.2023.17}, URN = {urn:nbn:de:0030-drops-176690}, doi = {10.4230/LIPIcs.STACS.2023.17}, annote = {Keywords: reachability, cut arc, strong bridge, covering walk, safety, persistence, essentiality, genome assembly} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Minimum flow decomposition (MFD) is the NP-hard problem of finding a smallest decomposition of a network flow X on directed graph G into weighted source-to-sink paths whose superposition equals X. We focus on a common formulation of the problem where the path weights must be non-negative integers and also on a new variant where these weights can be negative. We show that, for acyclic graphs, considering the width of the graph (the minimum number of s-t paths needed to cover all of its edges) yields advances in our understanding of its approximability. For the non-negative version, we show that a popular heuristic is a O(log |X|)-approximation (|X| being the total flow of X) on graphs satisfying two properties related to the width (satisfied by e.g., series-parallel graphs), and strengthen its worst-case approximation ratio from Ω(√m) to Ω(m / log m) for sparse graphs, where m is the number of edges in the graph. For the negative version, we give a (⌈log ║X║⌉+1)-approximation (║X║ being the maximum absolute value of X on any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary flows (║X║ ≤ 1) into at most width paths. We also disprove a conjecture about the linear independence of minimum (non-negative) flow decompositions posed by Kloster et al. [ALENEX 2018], but show that its useful implication (polynomial-time assignments of weights to a given set of paths to decompose a flow) holds for the negative version.

Manuel Cáceres, Massimo Cairo, Andreas Grigorjew, Shahbaz Khan, Brendan Mumey, Romeo Rizzi, Alexandru I. Tomescu, and Lucia Williams. Width Helps and Hinders Splitting Flows. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{caceres_et_al:LIPIcs.ESA.2022.31, author = {C\'{a}ceres, Manuel and Cairo, Massimo and Grigorjew, Andreas and Khan, Shahbaz and Mumey, Brendan and Rizzi, Romeo and Tomescu, Alexandru I. and Williams, Lucia}, title = {{Width Helps and Hinders Splitting Flows}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {31:1--31:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.31}, URN = {urn:nbn:de:0030-drops-169695}, doi = {10.4230/LIPIcs.ESA.2022.31}, annote = {Keywords: Flow decomposition, approximation algorithms, graph width} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Network flow is one of the most studied combinatorial optimization problems having innumerable applications. Any flow on a directed acyclic graph G having n vertices and m edges can be decomposed into a set of O(m) paths. The applications of such a flow decomposition range from network routing to the assembly of biological sequences. However, in some applications, each solution (decomposition) corresponds to some particular data that generated the original flow. Given the possibility of multiple optimal solutions, no optimization criterion ensures the identification of the correct decomposition. Hence, recently flow decomposition was studied [RECOMB22] in the Safe and Complete framework, particularly for RNA Assembly. The proposed solution reported all the safe paths, i.e., the paths which are subpath of every possible solution of flow decomposition.
They presented a characterization of the safe paths, resulting in an O(mn+out_R) time algorithm to compute all safe paths, where out_R is the size of the raw output reporting each safe path explicitly. They also showed that out_R can be Ω(mn²) in the worst case but O(m) in the best case. Hence, they further presented an algorithm to report a concise representation of the output out_C in O(mn+out_C) time, where out_C can be Ω(mn) in the worst case but O(m) in the best case.
In this work, we study how different safe paths interact, resulting in optimal output-sensitive algorithms requiring O(m+out_R) and O(m+out_C) time for computing the existing representations of the safe paths. Our algorithm uses a novel data structure called Path Tries, which may be of independent interest. Further, we propose a new characterization of the safe paths resulting in the optimal representation of safe paths out_O, which can be Ω(mn) in the worst case but requires optimal O(1) space for every safe path reported. We also present a near-optimal algorithm to compute all the safe paths in O(m+out_Olog n) time. The new representation also establishes tighter worst case bounds Θ(mn²) and Θ(mn) bounds for out_R and out_C (along with out_O), respectively.
Overall we further develop the theory of safe and complete solutions for the flow decomposition problem, giving an optimal algorithm for the explicit representation, and a near-optimal algorithm for the optimal representation of the safe paths.

Shahbaz Khan and Alexandru I. Tomescu. Optimizing Safe Flow Decompositions in DAGs. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 72:1-72:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{khan_et_al:LIPIcs.ESA.2022.72, author = {Khan, Shahbaz and Tomescu, Alexandru I.}, title = {{Optimizing Safe Flow Decompositions in DAGs}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {72:1--72:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.72}, URN = {urn:nbn:de:0030-drops-170101}, doi = {10.4230/LIPIcs.ESA.2022.72}, annote = {Keywords: safety, flows, networks, directed acyclic graphs} }

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**Published in:** LIPIcs, Volume 191, 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)

Genome assembly is a fundamental problem in Bioinformatics, where for a given set of overlapping substrings of a genome, the aim is to reconstruct the source genome. The classical approaches to solving this problem use assembly graphs, such as de Bruijn graphs or overlap graphs, which maintain partial information about such overlaps. For genome assembly algorithms, these graphs present a trade-off between overlap information stored and scalability. Thus, Hierarchical Overlap Graph (HOG) was proposed to overcome the limitations of both these approaches.
For a given set P of n strings, the first algorithm to compute HOG was given by Cazaux and Rivals [IPL20] requiring O(||P||+n²) time using superlinear space, where ||P|| is the cumulative sum of the lengths of strings in P. This was improved by Park et al. [SPIRE20] to O(||P||log n) time and O(||P||) space using segment trees, and further to O(||P||(log n)/(log log n)) for the word RAM model. Both these results described an open problem to compute HOG in optimal O(||P||) time and space. In this paper, we achieve the desired optimal bounds by presenting a simple algorithm that does not use any complex data structures. At its core, our solution improves the classical result [IPL92] for a special case of the All Pairs Suffix Prefix (APSP) problem from O(||P||+n²) time to optimal O(||P||) time, which may be of independent interest.

Shahbaz Khan. Optimal Construction of Hierarchical Overlap Graphs. In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, pp. 17:1-17:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{khan:LIPIcs.CPM.2021.17, author = {Khan, Shahbaz}, title = {{Optimal Construction of Hierarchical Overlap Graphs}}, booktitle = {32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)}, pages = {17:1--17:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-186-3}, ISSN = {1868-8969}, year = {2021}, volume = {191}, editor = {Gawrychowski, Pawe{\l} and Starikovskaya, Tatiana}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2021.17}, URN = {urn:nbn:de:0030-drops-139683}, doi = {10.4230/LIPIcs.CPM.2021.17}, annote = {Keywords: Hierarchical Overlap Graphs, String algorithms, Genome assembly} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

In recent years, significant advances have been made in the design and analysis of fully dynamic maximal matching algorithms. However, these theoretical results have received very little attention from the practical perspective. Few of the algorithms are implemented and tested on real datasets, and their practical potential is far from understood. In this paper, we attempt to bridge the gap between theory and practice that is currently observed for the fully dynamic maximal matching problem. We engineer several algorithms and empirically study those algorithms on an extensive set of dynamic instances.

Monika Henzinger, Shahbaz Khan, Richard Paul, and Christian Schulz. Dynamic Matching Algorithms in Practice. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 58:1-58:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{henzinger_et_al:LIPIcs.ESA.2020.58, author = {Henzinger, Monika and Khan, Shahbaz and Paul, Richard and Schulz, Christian}, title = {{Dynamic Matching Algorithms in Practice}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {58:1--58:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.58}, URN = {urn:nbn:de:0030-drops-129243}, doi = {10.4230/LIPIcs.ESA.2020.58}, annote = {Keywords: Matching, Dynamic Matching, Blossom Algorithm} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Depth first search (DFS) tree is a fundamental data structure for solving various graph problems. The classical algorithm for building a DFS tree requires O(m+n) time for a given undirected graph G having n vertices and m edges. In the streaming model, an algorithm is allowed several passes (preferably single) over the input graph having a restriction on the size of local space used.
Now, a DFS tree of a graph can be trivially computed using a single pass if O(m) space is allowed. In the semi-streaming model allowing O(n) space, it can be computed in O(n) passes over the input stream, where each pass adds one vertex to the DFS tree. However, it remains an open problem to compute a DFS tree using o(n) passes using o(m) space even in any relaxed streaming environment.
We present the first semi-streaming algorithms that compute a DFS tree of an undirected graph in o(n) passes using o(m) space. We first describe an extremely simple algorithm that requires at most ceil[n/k] passes to compute a DFS tree using O(nk) space, where k is any positive integer. For example using k=sqrt{n}, we can compute a DFS tree in sqrt{n} passes using O(n sqrt{n}) space. We then improve this algorithm by using more involved techniques to reduce the number of passes to ceil[h/k] under similar space constraints, where h is the height of the computed DFS tree. In particular, this algorithm improves the bounds for the case where the computed DFS tree is shallow (having o(n) height). Moreover, this algorithm is presented in form of a framework that allows the flexibility of using any algorithm to maintain a DFS tree of a stored sparser subgraph as a black box, which may be of an independent interest. Both these algorithms essentially demonstrate the existence of a trade-off between the space and number of passes required for computing a DFS tree. Furthermore, we evaluate these algorithms experimentally which reveals their exceptional performance in practice. For both random and real graphs, they require merely a few passes even when allowed just O(n) space.

Shahbaz Khan and Shashank K. Mehta. Depth First Search in the Semi-streaming Model. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 42:1-42:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{khan_et_al:LIPIcs.STACS.2019.42, author = {Khan, Shahbaz and K. Mehta, Shashank}, title = {{Depth First Search in the Semi-streaming Model}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {42:1--42:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.42}, URN = {urn:nbn:de:0030-drops-102818}, doi = {10.4230/LIPIcs.STACS.2019.42}, annote = {Keywords: Depth First Search, DFS, Semi-Streaming, Streaming, Algorithm} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Let G=(V,E) be a graph with n vertices and m edges, with a designated set of sigma sources S subseteq V. The fault tolerant subgraph for any graph problem maintains a sparse subgraph H=(V,E') of G with E' subseteq E, such that for any set F of k failures, the solution for the graph problem on G\F is maintained in its subgraph H\F. We address the problem of maintaining a fault tolerant subgraph for computing Breath First Search tree (BFS) of the graph from a single source s in V (referred as k FT-BFS) or multiple sources S subseteq V (referred as k FT-MBFS). We simply refer to them as FT-BFS (or FT-MBFS) for k=1, and dual FT-BFS (or dual FT-MBFS) for k=2.
The problem of k FT-BFS was first studied by Parter and Peleg [ESA13]. They designed an algorithm to compute FT-BFS subgraph of size O(n^{3/2}). Further, they showed how their algorithm can be easily extended to FT-MBFS requiring O(sigma^{1/2}n^{3/2}) space. They also presented matching lower bounds for these results. The result was later extended to solve dual FT-BFS by Parter [PODC15] requiring (n^{5/3}) space, again with matching lower bounds. However, their result was limited to only edge failures in undirected graphs and involved very complex analysis. Moreover, their solution doesn't seems to be directly extendible for dual FT-MBFS problem.
We present a similar algorithm to solve dual FT-BFS problem with a much simpler analysis. Moreover, our algorithm also works for vertex failures and directed graphs, and can be easily extended to handle dual FT-MBFS problem, matching the lower bound of O(sigma^{1/3}n^{5/3}) space described by Parter [PODC15]. The key difference in our approach is a much simpler classification of path interactions which formed the basis of the analysis by Parter [PODC15].

Manoj Gupta and Shahbaz Khan. Multiple Source Dual Fault Tolerant BFS Trees. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 127:1-127:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gupta_et_al:LIPIcs.ICALP.2017.127, author = {Gupta, Manoj and Khan, Shahbaz}, title = {{Multiple Source Dual Fault Tolerant BFS Trees}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {127:1--127:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.127}, URN = {urn:nbn:de:0030-drops-74184}, doi = {10.4230/LIPIcs.ICALP.2017.127}, annote = {Keywords: BFS, fault-tolerant, graph, algorithms, data-structures} }

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