Near Optimal Parallel Algorithms for Dynamic DFS in Undirected Graphs
Depth first search (DFS) tree is a fundamental data structure for solving graph problems. The classical algorithm [SiComp74] for building a DFS tree requires \(O(m+n)\) time for a given graph \(G\) having \(n\) vertices and \(m\) edges. Recently, Baswana et al. [SODA16] presented a simple algorithm...
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| Abstract | Depth first search (DFS) tree is a fundamental data structure for solving graph problems. The classical algorithm [SiComp74] for building a DFS tree requires \(O(m+n)\) time for a given graph \(G\) having \(n\) vertices and \(m\) edges. Recently, Baswana et al. [SODA16] presented a simple algorithm for updating DFS tree of an undirected graph after an edge/vertex update in \(\tilde{O}(n)\) time. However, their algorithm is strictly sequential. We present an algorithm achieving similar bounds, that can be adopted easily to the parallel environment. In the parallel model, a DFS tree can be computed from scratch using \(m\) processors in expected \(\tilde{O}(1)\) time [SiComp90] on an EREW PRAM, whereas the best deterministic algorithm takes \(\tilde{O}(\sqrt{n})\) time [SiComp90,JAlg93] on a CRCW PRAM. Our algorithm can be used to develop optimal (upto polylog n factors deterministic algorithms for maintaining fully dynamic DFS and fault tolerant DFS, of an undirected graph. 1- Parallel Fully Dynamic DFS: Given an arbitrary online sequence of vertex/edge updates, we can maintain a DFS tree of an undirected graph in \(\tilde{O}(1)\) time per update using \(m\) processors on an EREW PRAM. 2- Parallel Fault tolerant DFS: An undirected graph can be preprocessed to build a data structure of size O(m) such that for a set of \(k\) updates (where \(k\) is constant) in the graph, the updated DFS tree can be computed in \(\tilde{O}(1)\) time using \(n\) processors on an EREW PRAM. Moreover, our fully dynamic DFS algorithm provides, in a seamless manner, nearly optimal (upto polylog n factors) algorithms for maintaining a DFS tree in semi-streaming model and a restricted distributed model. These are the first parallel, semi-streaming and distributed algorithms for maintaining a DFS tree in the dynamic setting. |
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| AbstractList | Depth first search (DFS) tree is a fundamental data structure for solving graph problems. The classical algorithm [SiComp74] for building a DFS tree requires \(O(m+n)\) time for a given graph \(G\) having \(n\) vertices and \(m\) edges. Recently, Baswana et al. [SODA16] presented a simple algorithm for updating DFS tree of an undirected graph after an edge/vertex update in \(\tilde{O}(n)\) time. However, their algorithm is strictly sequential. We present an algorithm achieving similar bounds, that can be adopted easily to the parallel environment. In the parallel model, a DFS tree can be computed from scratch using \(m\) processors in expected \(\tilde{O}(1)\) time [SiComp90] on an EREW PRAM, whereas the best deterministic algorithm takes \(\tilde{O}(\sqrt{n})\) time [SiComp90,JAlg93] on a CRCW PRAM. Our algorithm can be used to develop optimal (upto polylog n factors deterministic algorithms for maintaining fully dynamic DFS and fault tolerant DFS, of an undirected graph. 1- Parallel Fully Dynamic DFS: Given an arbitrary online sequence of vertex/edge updates, we can maintain a DFS tree of an undirected graph in \(\tilde{O}(1)\) time per update using \(m\) processors on an EREW PRAM. 2- Parallel Fault tolerant DFS: An undirected graph can be preprocessed to build a data structure of size O(m) such that for a set of \(k\) updates (where \(k\) is constant) in the graph, the updated DFS tree can be computed in \(\tilde{O}(1)\) time using \(n\) processors on an EREW PRAM. Moreover, our fully dynamic DFS algorithm provides, in a seamless manner, nearly optimal (upto polylog n factors) algorithms for maintaining a DFS tree in semi-streaming model and a restricted distributed model. These are the first parallel, semi-streaming and distributed algorithms for maintaining a DFS tree in the dynamic setting. |
| Author | Khan, Shahbaz |
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| DOI | 10.48550/arxiv.1705.03637 |
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| Snippet | Depth first search (DFS) tree is a fundamental data structure for solving graph problems. The classical algorithm [SiComp74] for building a DFS tree requires... |
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| SubjectTerms | Algorithms Apexes Computation Data structures Fault tolerance Graph theory Processors |
| Title | Near Optimal Parallel Algorithms for Dynamic DFS in Undirected Graphs |
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