Parameterized dynamic data structure for Split Completion
(2402.08816)Abstract
We design a randomized data structure that, for a fully dynamic graph $G$ updated by edge insertions and deletions and integers $k, d$ fixed upon initialization, maintains the answer to the Split Completion problem: whether one can add $k$ edges to $G$ to obtain a split graph. The data structure can be initialized on an edgeless $n$-vertex graph in time $n \cdot (k d \cdot \log n){\mathcal{O}(1)}$, and the amortized time complexity of an update is $5k \cdot (k d \cdot \log n){\mathcal{O}(1)}$. The answer provided by the data structure is correct with probability $1-\mathcal{O}(n{-d})$.
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