Fully Dynamic Arboricity Maintenance

Banerjee, Niranka; Raman, Venkatesh; Saurabh, Saket

doi:10.1007/978-3-030-26176-4_1

Cited by 3 publications

(3 citation statements)

References 13 publications

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“…The corollary complements a result by Banerjee et al [2] who presented an algorithm for dynamically maintaining an arboricity decomposition consisting of exactly α forests with O(m • poly(log n)) update time, where m is the number of edges currently in the graph. Thus the result in the corollary obtains an exponentially faster update time by increasing the number of forests by a constant factor.…”

Section: Theoremsupporting

confidence: 78%

Explicit and Implicit Dynamic Coloring of Graphs with Bounded Arboricity

Henzinger¹,

Neumann²,

Wiese³

2020

Preprint

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Graph coloring is a fundamental problem in computer science. We study the fully dynamic version of the problem in which the graph is undergoing edge insertions and deletions and we wish to maintain a vertex-coloring with small update time after each insertion and deletion.We show how to maintain an O(α log n)-coloring with polylogarithmic update time, where n is the number of vertices in the graph and α is the current arboricity of the graph. This improves upon a result by Solomon and Wein (ESA'18) who maintained an O(αmax log 2 n)-coloring, where αmax is the maximum arboricity of the graph over all updates.Furthermore, motivated by a lower bound by Barba et al. (Algorithmica'19), we initiate the study of implicit dynamic colorings. Barba et al. showed that dynamic algorithms with polylogarithmic update time cannot maintain an f (α)-coloring for any function f when the vertex colors are stored explicitly, i.e., for each vertex the color is stored explicitly in the memory. Previously, all dynamic algorithms maintained explicit colorings. Therefore, we propose to study implicit colorings, i.e., the data structure only needs to offer an efficient query procedure to return the color of a vertex (instead of storing its color explicitly). We provide an algorithm which breaks the lower bound and maintains an implicit 2 O(α) -coloring with polylogarithmic update time. In particular, this yields the first dynamic O(1)-coloring for graphs with constant arboricity such as planar graphs or graphs with bounded tree-width, which is impossible using explicit colorings.To obtain our implicit coloring result we show how to dynamically maintain a partition of the graph's edges into O(α) forests with polylogarithmic update time. We believe this data structure is of independent interest and might have more applications in the future.

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Section: Theoremsupporting

confidence: 78%

Explicit and Implicit Dynamic Coloring of Graphs with Bounded Arboricity

Henzinger¹,

Neumann²,

Wiese³

2020

Preprint

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“…(1 + ε)α + 2 O(log 5 (n)α 2 /ε 6 )) adaptive Table 1 Different dynamic algorithms for maintaining out-orientations.…”

Section: Resultsmentioning

confidence: 99%

“…Furthermore, Blumenstock & Fischer provide an algorithm computing a (1 + ε)α + 1 arboricity decomposition in Õ(m log n) time. Bannerjee et al [5] provide an Õ(m) dynamic algorithm maintaining the exact arboricity α of a dynamic graph, and show a lower bound of Ω(log n) for dynamically maintaining arboricity. Henzinger et al [25] provide a dynamic algorithm for maintaining a 2α arboricity decomposition, given access to any black box dynamic α out-degree orientation algorithm (See Table 2).…”

Section: Arboricity Decompositionsmentioning

confidence: 99%

Fully-dynamic $α+ 2$ Arboricity Decomposition and Implicit Colouring

Christiansen¹,

Rotenberg²

2022

Preprint

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In the implicit dynamic colouring problem, the task is to maintain a representation of a proper colouring as a dynamic graph is subject to insertions and deletions of edges, while facilitating interspersed queries to the colours of vertices. The goal is to use few colours, while still efficiently handling edge-updates and responding to colour-queries. For an n-vertex dynamic graph of arboricity α, we present an algorithm that maintains an implicit vertex colouring with 4•2 α colours, in amortised poly-log n update time, and with O(α log n) worst-case query time. The previous best implicit dynamic colouring algorithm uses 2 40α colours, and has a more efficient update time of O(log 3 n) and the same query time of O(α log n) [25].For graphs undergoing arboricity α preserving updates, we give a fully-dynamic α + 2 arboricity decomposition in poly(log n, α) time, which matches the number of forests in the best near-linear static algorithm by Blumenstock and Fischer [12] who obtain α + 2 forests in near-linear time.Our construction goes via dynamic bounded out-degree orientations, where we present a fullydynamic explicit, deterministic, worst-case algorithm for (1+ε)α +2 bounded out-degree orientation with update time O(ε −6 α 2 log 3 n). The state-of-the-art explicit, deterministic, worst-case algorithm for bounded out-degree orientations maintains a β • α + log β n out-orientation in O(β 2 α 2 + βα log β n) time [28].

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