Space-efficient Euler partition and bipartite edge coloring

Hagerup, Torben; Kammer, Frank; Laudahn, Moritz

doi:10.1016/j.tcs.2018.01.008

Cited by 9 publications

(6 citation statements)

References 12 publications

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“…and an indexable dictionary over a bitvector S of size + L with [8,25,31] for more information on use-cases and detailed descriptions.…”

Section: Space-efficient Graph Coarsening With Applications To Succin...mentioning

confidence: 99%

Space-Efficient Graph Coarsening with Applications to Succinct Planar Encodings

Kammer¹,

Meintrup²

2022

Preprint

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We present a novel space-efficient graph coarsening technique for n-vertex separable graphs G, in particular for planar graphs, called cloud partition, which partitions the vertices V (G) into disjoint sets C of size O(log n) such that each C induces a connected subgraph of G. Using this partition P we construct a so-called structure-maintaining minor F of G via specific contractions within the disjoint sets such that F has O(n/ log n) vertices. The combination of (F, P) is referred to as a cloud decomposition.We call a graph G = (V, E) separable if it admits to an O(n c )-separator theorem for some constant c < 1 meaning there exists a separator S ⊂ V that partitions V into {A, S, B} such that no vertices of A and B are adjacent in G and neither A nor B contain more than c n vertices for a fixed constant c < 1. Due to the last property such separators are called balanced. This famously includes planar graphs, which admit an O( √ n)-separator theorem. For planar graphs we show that a cloud decomposition can be constructed in O(n) time and using O(n) bits. Given a cloud decomposition (F, P) constructed for a planar graph G we are able to find a balanced separator of G in O(n/ log n) time. Contrary to related publications, we do not make use of an embedding of the input graph. This allows us to construct the succinct encoding scheme for planar graphs due to Blelloch and Farzan (CPM 2010) in O(n) time and O(n) bits improving both runtime and space by a factor of Θ(log n). As an additional application of our cloud decomposition we show that a tree decomposition for planar graphs of width O(n 1/2+ ) for any > 0 can be constructed in O(n) bits and a time linear in the size of the tree decomposition. A similar result by Izumi and Otachi (ICALP 2020) constructs a tree decomposition of width O(k √ n log n) for graphs of treewidth k ≤ √ n in sublinear space and polynomial time. Finally, we generalize our cloud decomposition from planar graphs to arbitrary separable graphs.

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“…and an indexable dictionary over a bitvector S of size + L with [8,25,31] for more information on use-cases and detailed descriptions.…”

Section: Space-efficient Graph Coarsening With Applications To Succin...mentioning

confidence: 99%

Space-Efficient Graph Coarsening with Applications to Succinct Planar Encodings

Kammer¹,

Meintrup²

2022

Preprint

Self Cite

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“…We assume a representation of an undirected input graph G = (V, E) that is practically identical to the one used in [14]: For some known integer n ≥ 1, V = {1, . .…”

Section: Preliminariesmentioning

confidence: 99%

Space-Efficient DFS and Applications to Connectivity Problems: Simpler, Leaner, Faster

Hagerup

2019

Algorithmica

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The problem of space-efficient depth-first search (DFS) is reconsidered. A particularly simple and fast algorithm is presented that, on a directed or undirected input graph G = (V, E) with n vertices and m edges, carries out a DFS in O(n + m) timeA slightly more complicated variant of the algorithm works in the same time with at most n + (4/5)m + O(log n) bits. It is also shown that a DFS can be carried out in a graph with n vertices and m edges in O(n + m log * n) time with O(n) bits or in O(n + m) time with either O(n log log(4 + m/n)) bits or, for arbitrary integer k ≥ 1, O(n log (k) n) bits. These results among them subsume or improve most earlier results on space-efficient DFS. Some of the new time and space bounds are shown to extend to applications of DFS such as the computation of cut vertices, bridges, biconnected components and 2-edge-connected components in undirected graphs.

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“…There are several algorithms for the read-only word RAM, e.g., for sorting [4,23], geometric problems [1,3], or graph algorithms [2,7,10,12,17,18,19]. Unfortunately, most of the algorithms on n-vertex graphs (including depth first search (DFS) and breadth first search (BFS)) have to use roughly Ω(n) bits of working memory in the read-only RAM model since there is a lower bound for the reachability problem, i.e., the problem to find out if two given vertices of a given graph are in the same connected component.…”

Section: Introductionmentioning

confidence: 99%

Linear-Time In-Place DFS and BFS on the Word RAM

Kammer

Sajenko

2019

Lecture Notes in Computer Science

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We present an in-place depth first search (DFS) and an inplace breadth first search (BFS) that runs on a word RAM in linear time such that, if the adjacency arrays of the input graph are given in a sorted order, the input is restored after running the algorithm. To obtain our results we use properties of the representation used to store the given graph and show several linear-time in-place graph transformations from one representation into another.

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Space-efficient Euler partition and bipartite edge coloring

Cited by 9 publications

References 12 publications

Space-Efficient Graph Coarsening with Applications to Succinct Planar Encodings

Space-Efficient Graph Coarsening with Applications to Succinct Planar Encodings

Space-Efficient DFS and Applications to Connectivity Problems: Simpler, Leaner, Faster

Linear-Time In-Place DFS and BFS on the Word RAM

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