Fibonacci. Heaps. Lazy. Binomial. Heaps. Binomial. Heaps. Binary. Heaps. O(1). O(1). O(logn). O(logn). Insert. O(1). O(1). O(1). O(1). Find-min. O(logn). O(logn). In computer science, a binomial heap is a heap similar to a binary heap but also supports quick merging of two heaps. This is achieved by using a special tree. Lazy Binomial Heaps (Today). ○ A powerful building block for designing advanced data structures. ○ Fibonacci Heaps (Wednesday). ○ A heavyweight and.

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Heaos by Hilary Alexander Modified 4 months binomia. Remove the minimum root and meld? Amortized Analysis Let Ti be the number of trees after the ith operation Let r be the rank of the tree containing the min. Hewps link decreases the potential by 1.

This can pay for handling all the trees involved in the link. O log n Decrease-Key, Delete: Define the rank of Bk to be k. Items at the nodes, heap ordered. Doubly link roots and children of every nodes. Parent heapz needed for delete. Produce a Bk from two Bk-1, keep heap order. Basic operation is meld h1,h2: Like addition of binary numbers. B5 B4 B2 B1 h1: Chop off the minimal root.

Meld the subtrees with h. Update minimum pointer if needed. Bubble up, update min ptr if needed All operations take O log n time on the worst case, except find-min h that takes O 1 time. On the worst case increment takes O k. Define a potential of the counter: Concatenate the lists of binomial trees. Update the minimum pointer to be the smaller of the minimums O 1 worst case and amortized.

Chop off the minimum root, add its children to binonial list of trees. Traverse the forest keep linking trees of the same rank, maintain a pointer to the minimum root. Once we encounter a second tree of some rank we link them and keep linking until we do not have two trees of the same rank.

Meld h1,h2Insert x,h — as before Delete x,h: Modify the potential a little: What is the amortized cost of find-min? How many new trees are created by the purging step? One of them has binmial at least ki Therefore in its subtree there are at least 2 ki-1 nodes. Repeat the following step until there is only one tree in the forest: Add the resulting tree to the end of the queue. Denote it also by T Edges with one endpoint in T are binnomial in a heap data structure.

Denoted by h T.

We use binomial queues with lazy meld and deletion. Find e by doing find-min on h T.

The meld implicitly delete edges. We never explicitly delete edges! We can determine whether an edge is deleted or not by two find operations. Then we can determine whether a node is marked deleted in O 1 time, and our analysis is still valid. So, we have at most 2m implicit delete operations that cost O m. The complexity of these find-min operations dominates the complexity of the algorithm. Let pi be the number of deleted edges purged from the heap at the find-min performed by the i-th iteration.

We want to bound the sum of these expressions. Pass 1 is when we remove the original singleton trees from the queue.

## Binomial heap

Pass i is when we remove trees added to the queue at pass i What is the size of a tree removed from the queue at pass j? Prove by induction So how many passes are there? At most log n. My presentations Profile Feedback Log out. Auth with social network: Registration Forgot your password? OK The Intelligent Choice.

### Lazy Binomial Queues

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