Algorithm 2 Mergable Heap Implemented with Unsorted List

1:procedure MakeHeap()

2:Let $L$ be an empty doubly linked list with a sentinel node, i.e. $L.nil.prev = L.nil.next = L.nil$

3:return $L$

4:end procedure

6:procedure Insert( $L, x$ )

7: $x.prev = L.nil$

8: $x.next = L.nil.next$

9: $L.nil.next.prev = x$

10: $L.nil.next = x$

11:end procedure

12:

13:procedure Minimum( $L$ )

14:if $L.nil.next = L.nil$ then

15:return $L.nil$

16:end if

17: $x = L.nil.next$

18: $p = x$

19:while $p \ne L.nil$ do

20:if $p.key < x.key$ then

21: $x = p$

22:end if

23: $p = p.next$

24:end while

25:return $x$

26:end procedure

27:

28:procedure ExtractMin( $L$ )

29: $x =$ Minimum( $L$ )

30:if $x \ne L.nil$ then

31: $x.prev.next = x.next$

32: $x.next.prev = x.prev$

33:end if

34:return $x$

35:end procedure

36:

37:procedure Union( $L_1, L_2$ )

38: $L_1.nil.prev.next = L_2.nil.next$

39: $L_2.nil.next.prev = L_1.nil.prev$

40: $L_2.nil.prev.next = L_1.nil$

41: $L_1.nil.prev = L_2.nil.prev$

42:return $L_1$

43:end procedure