Editing Binomial heap
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==Merging== | ==Merging== | ||
− | Let two binomial heaps be denoted <math>A</math> and <math>B</math>, with sizes <math>n</math> and <math>m</math>, respectively. Let <math>A_k</math> denote <math>A</math>'s power-of-two tree of size <math>2^k</math>, if it has one, and <math>B_k</math> denote likewise a power-of-two tree of <math>B</math>. We want to create a new binomial | + | Let two binomial heaps be denoted <math>A</math> and <math>B</math>, with sizes <math>n</math> and <math>m</math>, respectively. Let <math>A_k</math> denote <math>A</math>'s power-of-two tree of size <math>2^k</math>, if it has one, and <math>B_k</math> denote likewise a power-of-two tree of <math>B</math>. We want to create a new binomial tree <math>S</math> that contains all the nodes from either <math>A</math> or <math>B</math> and has size <math>m+n</math>. After this operation, <math>A</math> and <math>B</math> will no longer exist as binomial heaps. |
(In theory, we could create a new binomial heap <math>S</math> that contained all the elements from both <math>A</math> and <math>B</math> ''without'' also destroying <math>A</math> and <math>B</math>. However, this would require copying over all the data from both heaps, which would take linear time (in the sum of their sizes). This would then be no more efficient than simply using the [[binary heap]]s. For this reason, we assume we usually do not want to do this.) | (In theory, we could create a new binomial heap <math>S</math> that contained all the elements from both <math>A</math> and <math>B</math> ''without'' also destroying <math>A</math> and <math>B</math>. However, this would require copying over all the data from both heaps, which would take linear time (in the sum of their sizes). This would then be no more efficient than simply using the [[binary heap]]s. For this reason, we assume we usually do not want to do this.) | ||
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</pre> | </pre> | ||
This procedure performs at most one merge (of a pair of power-of-two trees) at each iteration of the loop, which executes a logarithmic number of times; so it is <math>O(\log n)</math>, overall. | This procedure performs at most one merge (of a pair of power-of-two trees) at each iteration of the loop, which executes a logarithmic number of times; so it is <math>O(\log n)</math>, overall. | ||
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==Insertion== | ==Insertion== | ||
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* A node in a binomial tree can have more than two children. | * A node in a binomial tree can have more than two children. | ||
* Binomial trees are fully max-heap-ordered, rather than simply left-max-heap-ordered like power-of-two trees. | * Binomial trees are fully max-heap-ordered, rather than simply left-max-heap-ordered like power-of-two trees. | ||
− | A binomial tree of height 0 is a single node. A binomial tree of height <math>k > 0</math> consists of a root node with <math>k</math> children. | + | A binomial tree of height 0 is a single node. A binomial tree of height <math>k > 0</math> consists of a root node with <math>k</math> children. The subtree rooted at each child is a tree of a different height. |
When we merge two binomial trees of the same size, we simply make the smaller root the child of the larger root. Observe that this instantly gives a binomial tree of twice the size (whose height is then one greater than the heights of the two original trees). When we split a binomial tree of height <math>k > 0</math>, we simply detach the subtree of height <math>k-1</math>, giving two binomial trees of size <math>k-1</math>. | When we merge two binomial trees of the same size, we simply make the smaller root the child of the larger root. Observe that this instantly gives a binomial tree of twice the size (whose height is then one greater than the heights of the two original trees). When we split a binomial tree of height <math>k > 0</math>, we simply detach the subtree of height <math>k-1</math>, giving two binomial trees of size <math>k-1</math>. | ||