No abstract

We give simple algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the out-degree of each vertex is bounded. On one hand, we show how to orient the edges such that the out-degree of each vertex is proportional to the arboricity α of the graph, in a worst-case update time of O(log 2 n log α). On the other hand, motivated by applications in dynamic maximal matching, we obtain a different trade-off, namely the improved worst case update time of O(log n log α) for the problem of maintaining an edge-orientation with at most O(α + log n) out-edges per vertex. Since our algorithms have update times with worst-case guarantees, the number of changes to the solution (i.e. the recourse) is naturally limited.Our algorithms make choices based entirely on local information, which makes them automatically adaptive to the current arboricity of the graph. In other words, they are arboricityoblivious, while they are arboricity-sensitive. This both simplifies and improves upon previous work, by having fewer assumptions or better asymptotic guarantees.As a consequence, one obtains an algorithm with improved efficiency for maintaining a (1+ε) approximation of the maximum subgraph density, and an algorithm for dynamic maximal matching whose worst-case update time is guaranteed to be upper bounded by O(α+log n log α), where α is the arboricity at the time of the update.

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].