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Depending on the problem at hand, pre-order, post-order, and especially one of the number of subtrees − 1 in-order operations may be optional. Also, in practice more than one of pre-order, post-order, and in-order operations may be required. For example, when inserting into a ternary tree, a pre-order operation is performed by comparing items.
In pre-order, we always visit the current node; next, we recursively traverse the current node's left subtree, and then we recursively traverse the current node's right subtree. The pre-order traversal is a topologically sorted one, because a parent node is processed before any of its child nodes is done.
An alternative algorithm for topological sorting is based on depth-first search.The algorithm loops through each node of the graph, in an arbitrary order, initiating a depth-first search that terminates when it hits any node that has already been visited since the beginning of the topological sort or the node has no outgoing edges (i.e., a leaf node):
The examples above are problems in which the data-flow value is a set, e.g. the set of reaching definitions (Using a bit for a definition position in the program), or the set of live variables. These sets can be represented efficiently as bit vectors , in which each bit represents set membership of one particular element.
A universal traversal sequence is a sequence of instructions comprising a graph traversal for any regular graph with a set number of vertices and for any starting vertex. A probabilistic proof was used by Aleliunas et al. to show that there exists a universal traversal sequence with number of instructions proportional to O ( n 5 ) for any ...
Animated example of a depth-first search For the following graph: a depth-first search starting at the node A, assuming that the left edges in the shown graph are chosen before right edges, and assuming the search remembers previously visited nodes and will not repeat them (since this is a small graph), will visit the nodes in the following ...
Animated example of a breadth-first search. Black: explored, grey: queued to be explored later on BFS on Maze-solving algorithm Top part of Tic-tac-toe game tree. Breadth-first search (BFS) is an algorithm for searching a tree data structure for a node that satisfies a given property.
An example of a m-ary tree with m=5. In graph theory, an m-ary tree (for nonnegative integers m) (also known as n-ary, k-ary or k-way tree) is an arborescence (or, for some authors, an ordered tree) [1] [2] in which each node has no more than m children. A binary tree is an important case where m = 2; similarly, a ternary tree is one where m = 3.