In computer science , a FIBONACCI HEAP is a data structure for priority queue operations, consisting of a collection of heap-ordered trees . It has a better amortized running time than many other priority queue data structures including the binary heap and binomial heap . Michael L. Fredman and Robert E. Tarjan developed Fibonacci heaps in 1984 and published them in a scientific journal in 1987. They named Fibonacci heaps after the Fibonacci numbers , which are used in their running time analysis. For the Fibonacci heap, the find-minimum operation takes constant (O
(1)) amortized time. The insert and decrease key operations also work
in constant amortized time. Deleting an element (most often used in
the special case of deleting the minimum element) works in O(log n)
amortized time, where n is the size of the heap. This means that
starting from an empty data structure, any sequence of a insert and
decrease key operations and b delete operations would take O(a + b log
n) worst case time, where n is the maximum heap size. In a binary or
binomial heap such a sequence of operations would take O((a + b) log
n) time. A
Using Fibonacci heaps for priority queues improves the asymptotic running time of important algorithms, such as Dijkstra\'s algorithm for computing the shortest path between two nodes in a graph, compared to the same algorithm using other slower priority queue data structures. CONTENTS * 1 Structure * 2 Implementation of operations * 3 Proof of degree bounds * 4 Worst case * 5 Summary of running times * 6 Practical considerations * 7 References * 8 External links STRUCTURE Figure 1. Example of a Fibonacci heap. It has three trees of degrees 0, 1 and 3. Three vertices are marked (shown in blue). Therefore, the potential of the heap is 9 (3 trees + 2 × (3 marked-vertices)). A
However, at some point order needs to be introduced to the heap to achieve the desired running time. In particular, degrees of nodes (here degree means the number of children) are kept quite low: every node has degree at most O(log n) and the size of a subtree rooted in a node of degree k is at least Fk+2, where Fk is the kth Fibonacci number . This is achieved by the rule that we can cut at most one child of each non-root node. When a second child is cut, the node itself needs to be cut from its parent and becomes the root of a new tree (see Proof of degree bounds, below). The number of trees is decreased in the operation delete minimum, where trees are linked together. As a result of a relaxed structure, some operations can take a long
time while others are done very quickly. For the amortized running
time analysis we use the potential method , in that we pretend that
very fast operations take a little bit longer than they actually do.
This additional time is then later combined and subtracted from the
actual running time of slow operations. The amount of time saved for
later use is measured at any given moment by a potential function. The
potential of a
where t is the number of trees in the Fibonacci heap, and m is the number of marked nodes. A node is marked if at least one of its children was cut since this node was made a child of another node (all roots are unmarked). The amortized time for an operation is given by the sum of the actual time and c times the difference in potential, where c is a constant (chosen to match the constant factors in the O notation for the actual time). Thus, the root of each tree in a heap has one unit of time stored. This unit of time can be used later to link this tree with another tree at amortized time 0. Also, each marked node has two units of time stored. One can be used to cut the node from its parent. If this happens, the node becomes a root and the second unit of time will remain stored in it as in any other root. IMPLEMENTATION OF OPERATIONS To allow fast deletion and concatenation, the roots of all trees are linked using a circular, doubly linked list . The children of each node are also linked using such a list. For each node, we maintain its number of children and whether the node is marked. Moreover, we maintain a pointer to the root containing the minimum key. Operation FIND MINIMUM is now trivial because we keep the pointer to the node containing it. It does not change the potential of the heap, therefore both actual and amortized cost are constant. As mentioned above, MERGE is implemented simply by concatenating the lists of tree roots of the two heaps. This can be done in constant time and the potential does not change, leading again to constant amortized time. Operation INSERT works by creating a new heap with one element and
doing merge. This takes constant time, and the potential increases by
one, because the number of trees increases. The amortized cost is thus
still constant.
Operation EXTRACT MINIMUM (same as delete minimum) operates in three
phases. First we take the root containing the minimum element and
remove it. Its children will become roots of new trees. If the number
of children was d, it takes time O(d) to process all new roots and the
potential increases by d−1. Therefore, the amortized running time of
this phase is O(d) = O(log n).
However to complete the extract minimum operation, we need to update the pointer to the root with minimum key. Unfortunately there may be up to n roots we need to check. In the second phase we therefore decrease the number of roots by successively linking together roots of the same degree. When two roots u and v have the same degree, we make one of them a child of the other so that the one with the smaller key remains the root. Its degree will increase by one. This is repeated until every root has a different degree. To find trees of the same degree efficiently we use an array of length O(log n) in which we keep a pointer to one root of each degree. When a second root is found of the same degree, the two are linked and the array is updated. The actual running time is O(log n + m) where m is the number of roots at the beginning of the second phase. At the end we will have at most O(log n) roots (because each has a different degree). Therefore, the difference in the potential function from before this phase to after it is: O(log n) − m, and the amortized running time is then at most O(log n + m) + c(O(log n) − m). With a sufficiently large choice of c, this simplifies to O(log n). In the third phase we check each of the remaining roots and find the
minimum. This takes O(log n) time and the potential does not change.
The overall amortized running time of extract minimum is therefore
O(log n).
Operation DECREASE KEY will take the node, decrease the key and if the heap property becomes violated (the new key is smaller than the key of the parent), the node is cut from its parent. If the parent is not a root, it is marked. If it has been marked already, it is cut as well and its parent is marked. We continue upwards until we reach either the root or an unmarked node. Now we set the minimum pointer to the decreased value if it is the new minimum. In the process we create some number, say k, of new trees. Each of these new trees except possibly the first one was marked originally but as a root it will become unmarked. One node can become marked. Therefore, the number of marked nodes changes by −(k − 1) + 1 = − k + 2. Combining these 2 changes, the potential changes by 2(−k + 2) + k = −k + 4. The actual time to perform the cutting was O(k), therefore (again with a sufficiently large choice of c) the amortized running time is constant. Finally, operation DELETE can be implemented simply by decreasing the key of the element to be deleted to minus infinity, thus turning it into the minimum of the whole heap. Then we call extract minimum to remove it. The amortized running time of this operation is O(log n). PROOF OF DEGREE BOUNDS The amortized performance of a
Consider any node x somewhere in the heap (x need not be the root of one of the main trees). Define SIZE(x) to be the size of the tree rooted at x (the number of descendants of x, including x itself). We prove by induction on the height of x (the length of a longest simple path from x to a descendant leaf), that SIZE(x) ≥ Fd+2, where d is the degree of x. BASE CASE: If x has height 0, then d = 0, and SIZE(x) = 1 = F2. INDUCTIVE CASE: Suppose x has positive height and degree d>0. Let y1, y2, ..., yd be the children of x, indexed in order of the times they were most recently made children of x (y1 being the earliest and yd the latest), and let c1, c2, ..., cd be their respective degrees. We CLAIM that ci ≥ i-2 for each i with 2≤i≤d: Just before yi was made a child of x, y1,...,yi−1 were already children of x, and so x had degree at least i−1 at that time. Since trees are combined only when the degrees of their roots are equal, it must have been that yi also had degree at least i-1 at the time it became a child of x. From that time to the present, yi can only have lost at most one child (as guaranteed by the marking process), and so its current degree ci is at least i−2. This proves the CLAIM. Since the heights of all the yi are strictly less than that of x, we can apply the inductive hypothesis to them to get SIZE(yi) ≥ Fci+2 ≥ F(i−2)+2 = Fi. The nodes x and y1 each contribute at least 1 to SIZE(x), and so we have size ( x ) 2 + i = 2 d size ( y i ) 2 + i = 2 d F i = 1 + i = 0 d F i . {displaystyle {textbf {size}}(x)geq 2+sum _{i=2}^{d}{textbf {size}}(y_{i})geq 2+sum _{i=2}^{d}F_{i}=1+sum _{i=0}^{d}F_{i}.} A routine induction proves that 1 + i = 0 d F i = F d + 2 {displaystyle 1+sum _{i=0}^{d}F_{i}=F_{d+2}} for any d 0 {displaystyle dgeq 0} , which gives the desired lower bound on SIZE(x). WORST CASE Although Fibonacci heaps look very efficient, they have the following
two drawbacks (as mentioned in the paper "The Pairing Heap: A new form
of Self Adjusting Heap"): "They are complicated when it comes to
coding them. Also they are not as efficient in practice when compared
with the theoretically less efficient forms of heaps, since in their
simplest version they require storage and manipulation of four
pointers per node, compared to the two or three pointers per node
needed for other structures ". These other structures are referred to
Although the total running time of a sequence of operations starting
with an empty structure is bounded by the bounds given above, some
(very few) operations in the sequence can take very long to complete
(in particular delete and delete minimum have linear running time in
the worst case). For this reason Fibonacci heaps and other amortized
data structures may not be appropriate for real-time systems . It is
possible to create a data structure which has the same worst-case
performance as the
SUMMARY OF RUNNING TIMES In the following time complexities O(f) is an asymptotic upper bound and Θ(f) is an asymptotically tight bound (see Big O notation ). Function names assume a min-heap. OPERATION BINARY BINOMIAL FIBONACCI PAIRING BRODAL RANK-PAIRING STRICT FIBONACCI find-min Θ(1) Θ(log n) Θ(1) Θ(1) Θ(1) Θ(1) Θ(1) delete-min Θ(log n) Θ(log n) O(log n) O(log n) O(log n) O(log n) O(log n) insert O(log n) Θ(1) Θ(1) Θ(1) Θ(1) Θ(1) Θ(1) decrease-key Θ(log n) Θ(log n) Θ(1) o(log n) Θ(1) Θ(1) Θ(1) merge Θ(n) O(log n) Θ(1) Θ(1) Θ(1) Θ(1) Θ(1) * ^ Brodal and Okasaki later describe a persistent variant with the same bounds except for decrease-key, which is not supported. Heaps with n elements can be constructed bottom-up in O(n). * ^ A B C D E F G Amortized time. * ^ Lower bound of ( log log n ) , {displaystyle Omega (log log n),} upper bound of O ( 2 2 log log n ) . {displaystyle O(2^{2{sqrt {log log n}}}).} * ^ n is the size of the larger heap. PRACTICAL CONSIDERATIONS THIS SECTION NEEDS EXPANSION. You can help by adding to it . (February 2015) Fibonacci heaps have a reputation for being slow in practice due to large memory consumption per node and high constant factors on all operations. Recent experimental results suggest that Fibonacci heaps are more efficient in practice than most of its later derivatives, including quake heaps, violation heaps, strict Fibonacci heaps, rank pairing heaps, but less efficient than either pairing heaps or array-based heaps. REFERENCES * ^ Cormen, Thomas H. ; Leiserson, Charles E. ; Rivest, Ronald L. ;
Stein, Clifford (2001) . "Chapter 20: Fibonacci Heaps". Introduction
to Algorithms (2nd ed.). MIT Press and McGraw-Hill. pp. 476–497.
ISBN 0-262-03293-7 . Third edition p. 518.
* ^ A B C Fredman, Michael Lawrence ; Tarjan, Robert E. (July
1987). "Fibonacci heaps and their uses in improved network
optimization algorithms" (PDF). Journal of the Association for
Computing Machinery . 34 (3): 596–615. doi :10.1145/28869.28874 .
* ^ Fredman, Michael L. ; Sedgewick, Robert ; Sleator, Daniel D. ;
Tarjan, Robert E. (1986). "The pairing heap: a new form of
self-adjusting heap" (PDF). Algorithmica. 1 (1): 111–129. doi
:10.1007/BF01840439 .
* ^ Gerth Stølting Brodal (1996), "Worst-Case Efficient Priority
Queues", Proc. 7th ACM-SIAM Symposium on Discrete Algorithms, Society
for Industrial and Applied Mathematics : 52–58, CiteSeerX
10.1.1.43.8133 , ISBN 0-89871-366-8 , doi :10.1145/313852.313883
* ^ Brodal, G. S. L.; Lagogiannis, G.; Tarjan, R. E. (2012). Strict
Fibonacci heaps (PDF). Proceedings of the 44th symposium on Theory of
Computing - STOC '12. p. 1177. ISBN 978-1-4503-1245-5 . doi
:10.1145/2213977.2214082 .
* ^ A B C D Cormen, Thomas H. ; Leiserson, Charles E. ; Rivest,
Ronald L. (1990).
EXTERNAL LINKS * Java applet simulation of a Fibonacci heap * MATLAB implementation of |