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20-0: Binomial Trees
- B 0 is a tree containing a single node
- To build Bk:
- • Start withAdd B Bk− 1 k− 1 as left subtree 20-1: B 0 Binomial Trees B 1 B 2 B 3 B (^4)
20-2: B 0 Binomial Trees B 1 B 2 B 3 B (^4)
20-3: Binomial Trees
- Equivalent defintion
- • BB (^0) k is a binomial heap with a single nodeis a binomial heap with k children:
- B 0... Bk− 1 20-4: Binomial Trees
B 0 B 1 B 2 B 3 B (^4)
20-5: B 0 Binomial Trees B 1 B 2 B 3 B (^4)
20-6: Binomial Trees
- Properties of binomial trees Bk
- • ContainsHas height 2 k k^ nodes
- Contains^ (^ ki^ ) nodes at depth i for i = 0... k 20-7: Binomial Heaps
- A Binomial Heap is:
- Set of binomial trees, each of which has the heap property • Each node in every tree is <= all of its children
- All trees in the set have a different root degree • Can’t have two B 3 ’s, for instance
20-8: Binomial Heaps
- How long does it take? 20-12: Binomial Heaps
- How can we find the minimum element in a binomial heap?
- Look at the root of each tree in the list, find smallest value
- How long does it take?
- • Heap hasRepresent n n elements as a binary number
- • BNumber of trees in heapk is in heap iff kth binary digit of ∈ O(lg n) n is 1
20-13: Binomial Heaps
- Merging Heaps H 1 and H 2
- • Merge root lists ofWhat property of binomial heaps may be broken? H 1 and H 2
- How do we fix it? 20-14: Binomial Heaps
- Merging Heaps H 1 and H 2
- Merge root lists of • Could now have two trees with same degree H 1 and H 2
- Go through list from smallest degree to largest degree • If two trees have same degree, combine them into one tree of larger degree
- If three trees have same degree (how can this happen?) leave one, combine other two into tree oflarger degree 20-15: Binomial Heaps 10 5 22 7 25
20-16: Binomial Heaps
20-17: Binomial Heaps 10 5 22 7 25
12 13 15 17
8 9 15 20
11
3 14 6 30
20-18: Binomial Heaps 10 5 22 7 25
12 13 15 17
8 9 15 20
11
3 14 6 30
20-19: Binomial Heaps
20-23: Binomial Heaps
- Removing minimum element 10 5 22 7 25
12 13 15 17
8 9 15 20
11
6 14 30
20-24: Binomial Heaps
- Removing minimum element 10 5
22 7
25
12
13 15
17
8
9 15
20
11
6 14
30
20-25: Binomial Heaps
- Removing minimum element 10 5
22 7
25
12
13 15
17
8
9 15
20
11
6
14
30
20-26: Binomial Heaps
10
5
22 7
25
12
13 15
17
8
9 15
11 20
6
14
30 20-27: Binomial Heaps
- Exactly like standard, binary heaps
- Time: O(lg n) 20-32: Binomial Heaps
- How could we delete an arbitrary element (assuming we had a pointer to this element)?
Deletethis key
20-33: Binomial Heaps
- How could we delete an arbitrary element (assuming we had a pointer to this element)?
- • Decrease key toRemove smallest, Time −∞, Time O(lg O n(lg) n)
