How Bad is the Freedom to Flood-It?

International audienceFixed-Flood-It and Free-Flood-It are combinatorial problems on graphs that generalize a very popular puzzle called Flood-It. Both problems consist of recoloring moves whose goal is to produce a monochromatic ("flooded") graph as quickly as possible. Their difference is that in Free-Flood-It the player has the additional freedom of choosing the vertex to play in each move. In this paper, we investigate how this freedom affects the complexity of the problem. It turns out that the freedom is bad in some sense. We show that some cases trivially solvable for Fixed-Flood-It become intractable for Free-Flood-It. We also show that some tractable cases for Fixed-Flood-It are still tractable for Free-Flood-It but need considerably more involved arguments. We finally present some combinatorial properties connecting or separating the two problems. In particular, we show that the length of an optimal solution for Fixed-Flood-It is always at most twice that of Free-Flood-It, and this is tight

How Bad is the Freedom to Flood-It?

Fixed-Flood-It and Free-Flood-It are combinatorial problems on graphs that generalize a very popular puzzle called Flood-It. Both problems consist of recoloring moves whose goal is to produce a monochromatic ("flooded") graph as quickly as possible. Their difference is that in Free-Flood-It the player has the additional freedom of choosing the vertex to play in each move. In this paper, we investigate how this freedom affects the complexity of the problem. It turns out that the freedom is bad in some sense. We show that some cases trivially solvable for Fixed-Flood-It become intractable for Free-Flood-It. We also show that some tractable cases for Fixed-Flood-It are still tractable for Free-Flood-It but need considerably more involved arguments. We finally present some combinatorial properties connecting or separating the two problems. In particular, we show that the length of an optimal solution for Fixed-Flood-It is always at most twice that of Free-Flood-It, and this is tight

Extremal properties of flood-filling games

The problem of determining the number of "flooding operations" required to make a given coloured graph monochromatic in the one-player combinatorial game Flood-It has been studied extensively from an algorithmic point of view, but basic questions about the maximum number of moves that might be required in the worst case remain unanswered. We begin a systematic investigation of such questions, with the goal of determining, for a given graph, the maximum number of moves that may be required, taken over all possible colourings. We give several upper and lower bounds on this quantity for arbitrary graphs and show that all of the bounds are tight for trees; we also investigate how much the upper bounds can be improved if we restrict our attention to graphs with higher edge-density.Comment: Final version, accepted to DMTC

The complexity of Free-Flood-It on 2xn boards

We consider the complexity of problems related to the combinatorial game Free-Flood-It, in which players aim to make a coloured graph monochromatic with the minimum possible number of flooding operations. Our main result is that computing the length of an optimal sequence is fixed parameter tractable (with the number of colours present as a parameter) when restricted to rectangular 2xn boards. We also show that, when the number of colours is unbounded, the problem remains NP-hard on such boards. This resolves a question of Clifford, Jalsenius, Montanaro and Sach (2010)

EZ-AG: Structure-free data aggregation in MANETs using push-assisted self-repelling random walks

This paper describes EZ-AG, a structure-free protocol for duplicate insensitive data aggregation in MANETs. The key idea in EZ-AG is to introduce a token that performs a self-repelling random walk in the network and aggregates information from nodes when they are visited for the first time. A self-repelling random walk of a token on a graph is one in which at each step, the token moves to a neighbor that has been visited least often. While self-repelling random walks visit all nodes in the network much faster than plain random walks, they tend to slow down when most of the nodes are already visited. In this paper, we show that a single step push phase at each node can significantly speed up the aggregation and eliminate this slow down. By doing so, EZ-AG achieves aggregation in only O(N) time and messages. In terms of overhead, EZ-AG outperforms existing structure-free data aggregation by a factor of at least log(N) and achieves the lower bound for aggregation message overhead. We demonstrate the scalability and robustness of EZ-AG using ns-3 simulations in networks ranging from 100 to 4000 nodes under different mobility models and node speeds. We also describe a hierarchical extension for EZ-AG that can produce multi-resolution aggregates at each node using only O(NlogN) messages, which is a poly-logarithmic factor improvement over existing techniques

Exact Byzantine Consensus on Arbitrary Directed Graphs Under Local Broadcast Model

We consider Byzantine consensus in a synchronous system where nodes are connected by a network modeled as a directed graph, i.e., communication links between neighboring nodes are not necessarily bi-directional. The directed graph model is motivated by wireless networks wherein asymmetric communication links can occur. In the classical point-to-point communication model, a message sent on a communication link is private between the two nodes on the link. This allows a Byzantine faulty node to equivocate, i.e., send inconsistent information to its neighbors. This paper considers the local broadcast model of communication, wherein transmission by a node is received identically by all of its outgoing neighbors, effectively depriving the faulty nodes of the ability to equivocate. Prior work has obtained sufficient and necessary conditions on undirected graphs to be able to achieve Byzantine consensus under the local broadcast model. In this paper, we obtain tight conditions on directed graphs to be able to achieve Byzantine consensus with binary inputs under the local broadcast model. The results obtained in the paper provide insights into the trade-off between directionality of communication links and the ability to achieve consensus

A high resolution coupled hydrologic–hydraulic model (HiResFlood-UCI) for flash flood modeling

HiResFlood-UCI was developed by coupling the NWS's hydrologic model (HL-RDHM) with the hydraulic model (BreZo) for flash flood modeling at decameter resolutions. The coupled model uses HL-RDHM as a rainfall-runoff generator and replaces the routing scheme of HL-RDHM with the 2D hydraulic model (BreZo) in order to predict localized flood depths and velocities. A semi-automated technique of unstructured mesh generation was developed to cluster an adequate density of computational cells along river channels such that numerical errors are negligible compared with other sources of error, while ensuring that computational costs of the hydraulic model are kept to a bare minimum. HiResFlood-UCI was implemented for a watershed (ELDO2) in the DMIP2 experiment domain in Oklahoma. Using synthetic precipitation input, the model was tested for various components including HL-RDHM parameters (a priori versus calibrated), channel and floodplain Manning n values, DEM resolution (10 m versus 30 m) and computation mesh resolution (10 m+ versus 30 m+). Simulations with calibrated versus a priori parameters of HL-RDHM show that HiResFlood-UCI produces reasonable results with the a priori parameters from NWS. Sensitivities to hydraulic model resistance parameters, mesh resolution and DEM resolution are also identified, pointing to the importance of model calibration and validation for accurate prediction of localized flood intensities. HiResFlood-UCI performance was examined using 6 measured precipitation events as model input for model calibration and validation of the streamflow at the outlet. The Nash–Sutcliffe Efficiency (NSE) obtained ranges from 0.588 to 0.905. The model was also validated for the flooded map using USGS observed water level at an interior point. The predicted flood stage error is 0.82 m or less, based on a comparison to measured stage. Validation of stage and discharge predictions builds confidence in model predictions of flood extent and localized velocities, which are fundamental to reliable flash flood warning