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Binomial Heaps

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Binomial Heaps

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    2. Binomial trees: B0 has a single node Bk: Ex

    3. Lemma1 For the binomial tree Bk, 1. there are 2k nodes, 2. the height if the tree is k, 3. there are exactly nodes at depth i for i= 0, 1, …., k, 4. the root has degree k, which is greater than that of any other node; moreover if the children of the root are numbered from left to right by k-1, k-2, …, 0, child I is the root of a subtree Bi Proof: 1) By induction, 2k-1 + 2k-1 = 2k 2) By induction, 1 + (k-1) = k 3) , by induction

    4. Corollary: The maximum degree of any node in an n-node binomial tree is lg(n) Binomial heaps: H: a set of binomial trees satisfying the following: 1. Each binomial tree in H is heap-ordered: the key of a node is greater than or equal to the key of its parent 2. There is at most one binomial tree in H whose root has a given degree Note: By 2. an n-node binomial heap H consists of at most binomial trees

    7. Operations on binomial heaps Creating a new binomial heap Finding the minimum key Uniting 2 binomial heaps

    8. Binomial-Heap-Merge

    16. Insert a node Extracting the node with minimum key

    19. Decreasing a key Deleting a key

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