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Orienting Fully Dynamic Graphs with WorstCase Time Bounds
, 2014
"... In edge orientations, the goal is usually to orient (direct) the edges of an undirected network (modeled by a graph) such that all outdegrees are bounded. When the network is fully dynamic, i.e., admits edge insertions and deletions, we wish to maintain such an orientation while keeping a tab on th ..."
Abstract

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In edge orientations, the goal is usually to orient (direct) the edges of an undirected network (modeled by a graph) such that all outdegrees are bounded. When the network is fully dynamic, i.e., admits edge insertions and deletions, we wish to maintain such an orientation while keeping a tab on the update time. Low outdegree orientations turned out to be a surprisingly useful tool for managing networks. Brodal and Fagerberg (1999) initiated the study of the edge orientation problem in terms of the graph’s arboricity, which is very natural in this context. Their solution achieves a constant outdegree and a logarithmic amortized update time for all graphs with constant arboricity, which include all planar and excludedminor graphs. It remained an open question – first proposed by Brodal and Fagerberg, later by Erickson and others – to obtain similar bounds with worstcase update time. We address this 15 year old question by providing a simple algorithm with worstcase bounds that nearly match the previous amortized bounds. Our algorithm is based on a new approach of maintaining a combinatorial invariant, and achieves a logarithmic outdegree with logarithmic worstcase update times. This result has applications to various dynamic network problems such as maintaining a maximal matching, where we obtain logarithmic worstcase update time compared to a similar amortized update time of Neiman and Solomon (2013).