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| A '''kinetic minimum spanning tree''' is a [[kinetic data structure]] that maintains the [[minimum spanning tree]] (MST) of a graph whose edge weights are changing as a continuous function of time.
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| == General case ==
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| The most efficient known data structure for the general case uses a [[kinetic sorted list]] to store the edge weights, and a standard [[minimum spanning tree#Algorithms|MST algorithm]] to compute the MST given the sorted edge weights. This data structure must process <math>O(n^2)</math> events, developing a more [[kinetic data structure#Performance|efficient]] data structure remains an open problem.<ref name="lecture"/>
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| == H-minor-free graphs ==
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| Agarwal ''et al.'' developed a data structure that maintains the MST for a graph belonging to a [[Minor-closed_graph_family#Minor-closed_graph_families|minor closed family]]. It uses the idea of a "swap", calculating the amount by which the weight of the MST would increase if some edge in the tree {{math|<var>e</var>}} was replaced by an edge {{math|<var>f</var>}} outside the tree such that the circle induced by {{math|<var>f</var>}} in the tree contains {{math|<var>e</var>}}. Maintaining the tree is then equivalent to finding and swapping the next pair for which this quantity becomes negative. This data structure considers the [[Duality (projective geometry)|dual]] view of the graph, and then [[divide and conquer paradigm|divides]] based on Frederickson's restricted partitions <ref name="frederickson"/> to make this efficient. It results in a total run time <math>O(pn^{\frac{1}{2}}\log^{\frac{3}{2}}n)</math> if <math>p</math> insertions or deletions are made, or <math>O(n^{\frac{19}{12}}\log^{\frac{3}{2}}n)</math> if only weight changes are allowed. These deterministic bounds are slightly improved if randomization is allowed.
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| == References ==
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| {{Reflist|refs=
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| <ref name="lecture">
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| {{cite video | url=http://courses.csail.mit.edu/6.851/spring12/lectures/L04.html | title=MIT 6.851 Advanced Data Structures, lecture video | people=Demaine, Erik D.}}
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| </ref>
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| <ref name="frederickson">
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| {{cite journal | title=Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees | author=Frederickson, G. N. |journal=SIAM Journal of Computing | year=1997 | pages=484–538}}
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| </ref>
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| }}
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| ==Further reading==
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| {{cite conference | url=http://www.ics.uci.edu/~eppstein/pubs/AgaEppGui-FOCS-98.pdf | title=Parametric and Kinetic Minimum Spanning Trees | accessdate=May 19, 2012 | author=Agarwal, Pankaj; Eppstein, David; Guibas, Leonidas J.; Henzinger, Monika R. | year=1998 | conference=FOCS}}
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| <!--- Categories --->
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| [[Category:Articles created via the Article Wizard]]
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| [[Category:Kinetic data structures]]
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| [[Category:Spanning tree]]
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