On finding and updating spanning trees and shortest paths dirtydating

"Is the weight of the edge between x and y larger than the weight of the edge between w and z? The two children of the node correspond to the two possible answers "yes" or "no".In each leaf of the DT, there is a list of edges from G that correspond to an MST.

If it is constrained to bury the cable only along certain paths (e.g.

In a comparison model, in which the only allowed operations on edge weights are pairwise comparisons, Karger, Klein α(m,n)), where α is the classical functional inverse of the Ackermann function.

The function α grows extremely slowly, so that for all practical purposes it may be considered a constant no greater than 4; thus Chazelle's algorithm takes very close to linear time. m/n ≥ log log log n), then a deterministic algorithm by Fredman and Tarjan finds the MST in time O(m). Each phase executes Prim's algorithm many times, each for a limited number of steps. If the number of vertices before a phase is Whether the problem can be solved deterministically for a general graph in linear time by a comparison-based algorithm remains an open question.

along roads), then there would be a graph representing which points are connected by those paths.

Some of those paths might be more expensive, because they are longer, or require the cable to be buried deeper; these paths would be represented by edges with larger weights.

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