head [ (step head [hc/ head [ (step head [ (step figure (contdbinomial heap fibonacci heaps in the last sectionwe have seen that binomial heaps support operations such as insertextractminimumdecrease-valuedeleteand union in (log nworst-case time in this sectionwe will discuss fibonacci heaps which support the same operations but have the advantage that operations that do not involve deleting an element run in ( amortized time sotheoreticallyfibonacci heaps are especially desirable when the number of extract-minimum and delete operations is small relative to the number of other operations performed this situation arises in many applicationswhere algorithms for graph problems may call the decrease-value once per edge howeverthe programming complexity of fibonacci heaps makes them less desirable to use fibonacci heap is collection of trees it is loosely based on binomial heaps if neither the decrease-value nor the delete operation is performedeach tree in the heap is like binomial tree fibonacci heaps differ from binomial heaps as they have more relaxed structureallowing improved asymptotic time bounds structure of fibonacci heaps although fibonacci heap is collection of heap-ordered treesthe trees in fibonacci heap are not constrained to be binomial trees that iswhile the trees in binomial heap are orderedthose within fibonacci heaps are rooted but unordered look at the fibonacci heap given in fig the figure shows that each node in the fibonacci heap contains the following pointersa pointer to its parentand pointer to any one of its children note that the children of each node are linked together in min[ha circular doubly linked list which is known as the child list of that node each child in child list contains pointers to its left and right siblings if node is the only child of its parentthen left[xright[xx (refer fig circular doubly linked lists provide an added advantage as they allow node to be removed in ( time alsogiven two circular doubly linked liststhe lists can be concatenated figure fibonacci heap to form one list in ( time