the existing operations by increasing the relevant node priority to infinitybubbling-upand using the root delete operationagain with (log ncomplexity overall sothe complexity of delete is always (log nexercisefind pseudocode versions of the mergeinsert and delete algorithms for binomial heapsand see exactly how their time complexities arise fibonacci heaps fibonacci heap is another collection of trees that satisfy the standard priority-ordering property it can be used to implement priority queue in similar way to binary or binomial heapsbut the structure of fibonacci heaps are more flexible and efficientwhich allows them to have better time complexities they are named after the fibonacci numbers that restrict the tree sizes and appear in their time complexity analysis the flexibility and efficiency of fibonacci heaps comes at the cost of more complexitythe trees do not have fixed shapeand in the extreme cases every element in the heap can be in separate tree normallythe roots of all the trees are stored using circular doubly linked listand the children of each node are handled in the same way pointer to the highest priority root node is maintainedmaking it trivial to find the highest priority node in the heap the efficiency is achieved by performing many operations in lazy mannerwith much of the work postponed for later operations to deal with fibonacci heaps can easily be merged with ( complexity by simply concatenating the two lists of root nodesand then insertion can be done by merging the existing heap with new heap consisting only of the new node by inserting items one at timea whole heap can be created from scratch with (ncomplexity obviouslyat some pointorder needs to be introduced into the heap to achieve the overall efficiency this is done by keeping the number of children of all nodes to be at most (log )and the size of subtree rooted in node with children is at least fk+ where fk is the kth fibonacci number the number of trees in the heap is decreased as part of the delete operation that is used to remove the highest priority node and update the pointer to the highest priority root this delete algorithm is quite complex first it removes the highest priority rootleaving its children to become roots of new trees within the heapthe processing of which will be (log nthen the number of trees is reduced by linking together trees that have roots with the same number of childrensimilar to binomial heapuntil every root has different number of childrenleaving at most (log ntrees finally the roots of those trees are checked to reset the pointer to the highest priority it can be shown that all the required processes can be completed with (log naverage time complexity for each nodea record is kept of its number of children and whether it is marked the mark indicates that at least one of its children has been separated since the node was made child of another nodeso all roots are unmarked the mark is used by the algorithm for increasing node prioritywhich is also complexbut can be achieved with ( complexity this gives fibonacci heaps an important advantage over both binary and binomial heaps for which this operation has (log ntime complexity finallyan arbitrary node can be deleted from the heap by increasing its node priority to infinity and applying the delete highest priority algorithmresulting in an overall time complexity of (log