Abstract
We investigate a distributed maximal independent set reconfiguration problem, in which there are two MIS for which every node is given its membership status, and the nodes need to communicate with their neighbors to find a reconfiguration schedule from the first MIS to the second. We forbid two neighbors to change their membership status at the same step. We provide efficient solutions when the intermediate sets are only required to be independent and 4-dominating, which is almost always possible. Consequently, our goal is to pin down the tradeoff between the possible length of the schedule and the number of communication rounds. We prove that a constant length schedule can be found in O(MIS+R32) rounds. For bounded degree graphs, this is O(log⁎n) rounds and we show that it is necessary. On the other extreme, we show that with a constant number of rounds we can find a linear length schedule.
Highlights
Consider a distributed setting in which each node of a network receives an input from a higher-level application which tells it whether it is selected or not, such that the set of selected nodes is a maximal independent set (MIS), which we will denote by α
Requiring 3-domination for intermediate steps is costly: 1. There exists a class of inputs G = (V, E) with two MIS α and β such that there is no reconfiguration schedule with 3-dominating intermediate steps
There exists a class of inputs G = (V, E) with two MIS α and β such that any reconfiguration schedule is of length Ω(n) and needs Θ(n) rounds to be found, if intermediate steps must be 3-dominating
Summary
Consider a distributed setting in which each node of a network receives an input from a higher-level application which tells it whether it is selected or not, such that the set of selected nodes is a maximal independent set (MIS), which we will denote by α. 2. There exists a class of inputs G = (V, E) with two MIS α and β such that any reconfiguration schedule is of length Ω(n) and needs Θ(n) rounds to be found, if intermediate steps must be 3-dominating. For any graph G = (V, E) of diameter greater than 3 and any input of two MIS α, β, there exists a reconfiguration schedule of constant length 28, with independent 4-dominating intermediate steps. Such a schedule can be found in O(MIS + R32) rounds, where MIS is the complexity of finding an MIS on a worst-case graph and R32 is the complexity of finding a (3, 2)-ruling set on a worst-case graph. This result implies that our algorithm from Theorem 2, combined with a trivial algorithm that collects the entire graph when the diameter is a small constant, produces an efficient reconfiguration schedule in all cases for which it exists
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