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A tree whose root node has two subtrees, both of which are full binary trees. A perfect binary tree is a binary tree in which all interior nodes have two children and all leaves have the same depth or same level (the level of a node defined as the number of edges or links from the root node to a node). [18] A perfect binary tree is a full ...
In graph theory, a tree decomposition is a mapping of a graph into a tree that can be used to define the treewidth of the graph and speed up solving certain computational problems on the graph. Tree decompositions are also called junction trees , clique trees , or join trees .
A more general problem is to count spanning trees in an undirected graph, which is addressed by the matrix tree theorem. (Cayley's formula is the special case of spanning trees in a complete graph.) The similar problem of counting all the subtrees regardless of size is #P-complete in the general case (Jerrum (1994)).
For a large class of problems, there is a linear time algorithm to solve a problem from the class if a tree-decomposition with constant bounded treewidth is provided. Specifically, Courcelle's theorem [ 19 ] states that if a graph problem can be expressed in the logic of graphs using monadic second order logic , then it can be solved in linear ...
This is an unbalanced assignment problem. One way to solve it is to invent a fourth dummy task, perhaps called "sitting still doing nothing", with a cost of 0 for the taxi assigned to it. This reduces the problem to a balanced assignment problem, which can then be solved in the usual way and still give the best solution to the problem.
The rate of tree performance degradation is dependent upon the spatial distribution of tree points being added, and the number of points added in relation to the tree size. If a tree becomes too unbalanced, it may need to be re-balanced to restore the performance of queries that rely on the tree balancing, such as nearest neighbour searching.
Finding the frustration index is an NP-hard problem. One can see the NP-hard complexity by observing that the frustration index of an all-negative signed graph is the same as the maximum cut problem in graph theory, which is NP-hard. The frustration index is important in a model of spin glasses, the mixed Ising model. In this model, the signed ...
In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Steiner tree problems may be formulated in a number of settings, they all require an optimal interconnect for a given set of objects and a predefined ...