1. 5.9 Heaps of optimal complexity
5.10 Double-ended heap structures and multidimensional heaps
5.11 Heap-related structures with constant time updates
AlShahrani Hasan

2. Heaps of optimal complexity
delete_min in O(log n).
insert and other operations in constant time.
Steps towards this direction:
Fibonacci heap: insert, find_min, merge in O (1) amortized time and delete_min in O (log n).
Pairing heap: O(log n)amortized bounds operations , and Ω(log log n) amortized lower bound for decrease_key.
Relaxed heap: O (1) worst-case for insert and decrease_key, O (log n) for find_min and delete_min.

3. The suggested structure: Brodal Heap.
Worst-case guarantee per-operation
The structure is:
The root has rank 0.
Heap-ordered tree. (parent smaller than child )
Each node n has a nonnegative rank as balancing information.
Each node has at most one special lower neighbor which might be of arbitrary rank.
“it is quite complicated and not applicable in practice” . Brodal

4. Brodal heap structure……..
The other normal neighbors are ordered in increasing rank. And have the following properties :
Each rank less than the rank of n occurs at least once and at most three times.
Between two ranks that occur three times there is a rank that occurs only once.
Before the first rank that occurs three times, there is a rank that occurs only once.
For each node the first lower neighbors of each rank that occurs three times are arranged in a linked list, in increasing order.

5. Double-ended heap structures
Two heaps a min-heap and max-heap and insert each element in both linking the two copies by pointers. This means:
Insert: two insert operations.
One delete-min or delete-max to the corresponding deletion heap and arbitrary deletion in the other heap.
Merge: two merge operations supported.

6. Double-ended heap structures ……….

7. Main parts: A min-heap. A max-heap.
Double-ended heap structures ………. Interval heaps (Group elements in pairs )
Main parts:
A min-heap.
A max-heap.
A pairing of the elements of the min-heap and the max-heap.
At most one unmatched element.

8. Operations: Interval heaps………….
Insert : if (there is unmatched element ) {
Insert(x)} // pair x with it .
Else {unmatched element = x}
Find_min: MIN (find-min (min-heap), unmatched element).
Find-max: MAX (find-max (max-heap), unmatched element).
Delete-min: perform  Find_min: MIN (find-min (min-heap), unmatched element)  delete and return the value deleted.
Delete-max: perform  Find_max: MAX (find-max (max-heap), unmatched element)  delete and return the value deleted.
Merge: merge the two min-heaps and merge the two max-heaps. If there are two unmatched elements, one from each of the merged heaps, it matches them and inserts the smaller one in the min-heap and the larger one in the max-heap.

10. Double-ended heap structures …………
Theorem: there is a double-ended heap that supports insert , find- min , find-max, merge in O(1) ; and delete-min , delete-max in O(log n) worst-case .

11. Generalization of the double-ended heap:
d-dimensional min-heap:
A set of objects, each with d key values in a structure that allows inserts and query for and deletion of the object with the minimum ith coordinate.
Double-ended heap is a special case of 2-dimensional heap.
Theorem: there is a d-dimensional min-heap that supports insert, find- min and merge in each coordinate in O(1) ; and delete-min in each coordinate in O(log n) worst-case time .

12. Heap-related structures with constant time updates
Structures that make updates in faster time than O (log n)
1. Doubled stack : A stack to keep track of the minimum value of elements.
Theorem: the doubled stack structure supports push, pop, and find- min in O (1) worst-case time.

13. Heap-related structures with constant time updates………..
2. Minqueue: models sliding window over a sequence of items to keep track of the smallest key in the window.

14. Minqueue……….
enqueue: enqueue the object in the top queue and remove from the rear of the bottom queue all key larger than the key of the new object.
dequeue.
find-min: return the key in the front of the minimum key queue
Theorem: the doubled queue supports enqueue, dequeue and find-min in O (1) amortized time.

15. Thank you