Characterization of minimum cycle basis in weighted partial 2-trees

Narayanaswamy, N. S.; Ramakrishna, G.

doi:10.1016/j.dam.2014.05.008

Cited by 1 publication

(8 citation statements)

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“…It remains a challenging question if our result can be extended to partial k-trees for k > 2. We remark that it was noted in [19] that already for partial 3-trees the set of lex short cycles do not necessarily form a minimum cycle basis. Since in particular the proof of Theorem 20 is based on this, extending our result to partial 3-trees may therefore require substantially new ideas.…”

Section: Discussionmentioning

confidence: 86%

“…For the proof of Theorem 1 it will be crucial that the set of lex short cycles (cf. Section 2.3) in any weighted partial 2-tree forms a minimum cycle basis [19]. As lex short cycles are inherently defined by shortest paths, we will need a data structure that reports the distance between two vertices in constant time (e.g., for checking whether an edge is the shortest path between its two endpoints).…”

Section: Our Resultsmentioning

confidence: 99%

“…Equivalently, a partial 2-tree is outerplanar if and only if it does not contain a K 2,3 -subdivision (as a subgraph). The following statement is taken from [19].…”

Section: Lemmamentioning

confidence: 99%

“…Lemma 6 ( [13,19]). For any edge-weighted simple graph G, there is a set B ⊆ LSC(G) such that B is a minimum cycle basis for G. Additionally, the set of lex short cycles LSC(G) forms a minimum cycle basis if G is a weighted partial 2-tree.…”

Section: Lex Shortest Paths and Lex Short Cyclesmentioning

confidence: 99%

“…Lemma 3 (Lemma 2.4 in [19]). A partial 2-tree G is not outerplanar if and only if G contains a K 2,3subdivision.…”

Section: Tree Decompositions and Partial 2-treesmentioning

confidence: 99%

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Computing Minimum Cycle Bases in Weighted Partial 2-Trees in Linear Time

Doerr¹,

Ramakrishna²,

Schmidt³

2014

JGAA

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We present a linear time algorithm for computing an implicit linear space representation of a minimum cycle basis (MCB) in weighted partial 2-trees; i.e., graphs of treewidth two. The implicit representation can be made explicit in a running time that is proportional to the size of the MCB. For planar graphs, Borradaile, Sankowski, and Wulff-Nilsen [Min st-cut Oracle for Planar Graphs with Near-Linear Preprocessing Time, FOCS 2010] showed how to compute an implicit O(n log n) space representation of an MCB in O(n log 5 n) time. For the special case of partial 2-trees, our algorithm improves this result to linear time and space. Such an improvement was achieved previously only for outerplanar graphs [Liu and Lu: Minimum Cycle Bases of Weighted Outerplanar Graphs, IPL 110:970-974, 2010]. IntroductionA cycle basis of a graph G is a minimum-cardinality set C of cycles in G such that every cycle C in G can be written as the exclusive-or sum of a subset of cycles in C. A minimum cycle basis (MCB) of G is a cycle basis that minimizes the total weight of the cycles in the basis. Minimum cycle bases have numerous applications in the analysis of electrical networks, biochemistry, periodic timetabling, surface reconstruction, and public transportation, and have been intensively studied in the computer science literature. We refer the interested reader to [16] for an exhaustive survey. It is-both from a practical and a theoretical viewpoint-an interesting task to compute minimum cycle bases efficiently.All graphs considered in this work are simple graphs G = (V, E) with a non-negative edge-weight function w : E → R ≥0 . (Computing MCBs for graphs with cycles of negative weight is an NP-hard problem [16]. In all previous work that we are aware of it is therefore assumed that the edge-weights are non-negative.) Throughout this work, m = |E| denotes the size of the edge set and n = |V | the size of the vertex set of G. Previous WorkThe first polynomial-time algorithm for computing MCBs was presented by Horton in 1987 [15]. His algorithm has running time O(m 3 n). This was improved subsequently in a series of papers by different authors, cf. [16] or [18] for surveys of the history. The currently fastest algorithms for general graphs are a deterministic O(m 2 n/ log n) algorithm of Amaldi, Iuliano, and Rizzi [2] and a Monte Carlo based algorithm by Amaldi, Iuliano, Jurkiewicz, Mehlhorn, and Rizzi [1] of running time O(m ω ), where ω is the matrix multiplication constant.The algorithm from [1] is deterministic on planar graphs, and has a running time of O(n 2 ). This improved the previously best known bound by Hartvigsen and Mardon [13], which is of order n 2 log n. The currently best known algorithm on planar graphs is due to Borradaile, Sankowski, and Wulff-Nilsen [7]. It constructs an O(n log n) space implicit representation of an MCB in planar graphs in time O(n log 5 n). 5 ⋆ This is the full version of [11] which has appeared in the

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Section: Discussionmentioning

confidence: 86%

Section: Our Resultsmentioning

confidence: 99%

“…Equivalently, a partial 2-tree is outerplanar if and only if it does not contain a K 2,3 -subdivision (as a subgraph). The following statement is taken from [19].…”

Section: Lemmamentioning

confidence: 99%

Section: Lex Shortest Paths and Lex Short Cyclesmentioning

confidence: 99%

“…Lemma 3 (Lemma 2.4 in [19]). A partial 2-tree G is not outerplanar if and only if G contains a K 2,3subdivision.…”

Section: Tree Decompositions and Partial 2-treesmentioning

confidence: 99%

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