Question

Consider two Binomial heaps: first H1 comprised of B2,B3 and second H2 comprised of B2,B3. Draw an example of those two heaps and draw the sequence of transformations that are done during the operation union(H1,H2). (I expect 5 stages, but you could have more if you include more details. That is fine.)

Answer 1

Given two Binomial Heaps H1 and H2, first H1 comprised of B2,B3 and second H2 comprised of B2,B3. Union(H1, H2) creates a
(single) Binomial Heap.
Step1:-
Merge the two heaps H1 and H2 in increasing (non decreasing) order of degrees.
You can view these in figure (b)

2)After the merge operation , there must be at most one Binomial Tree of any order.
To achieve this, one has to combine Binomial Trees of same order. Then we have to traverse the list of merged roots,
we keep track of three pointers, prev-y, y and next. There can be following 4 cases when we traverse the list of roots.
Case 1: If Orders of y and next are not same, we simply move ahead.

In following 3 cases orders of y and next are same.
Case 2: If order of next-next is also same, move ahead.
Case 3: If key of y is smaller than or equal to key of next, then make next as a child of y by linking it with y.
Case 4: If key of y is greater, then make y as child of next.

(a) headHl-12 headlH2l18 28) (33 29) (10 44 30 (23) (22) (48) (31) (17 BINOMIAL-HEAP-MERGE 45) (32) (24 50 next-х (b) headH12 18 28) (33 10) (44 41 30) (23 48 31 17 Case 3 next-x (c) headM 12 25 29 (10) (44 41 23) (22) (48) (31 (17 Case 2 45) (32) (24 50 prev-х next-x (d) headlH12 15 18 25 28) (33 29 (10) (44 48) (31) (17 Case 4 45) (32 (24 prev-x next-х (e) headH - 15 28) (33 29 (10 44 30 (23) (22) (48) (31) (17 Case 3 45) 32 (24 50 prev-х next-х (f) headlH12 18 15 (7 3 29 10 44 28) (33) (25 30) (23) (22) (48) (31) (17 Case 41 45) (32) (24 50