PRIORITY QUEUES
‣ binary heaps ‣ d-ary heaps ‣ binomial heaps ‣ Fibonacci heaps
Lecture slides by Kevin Wayne Copyright © 2005 Pearson-Addison Wesley http://www.cs.princeton.edu/~wayne/kleinberg-tardos
Last updated on 7/25/17 11:07 AM Priority queue data type
A min-oriented priority queue supports the following core operations: ・MAKE-HEAP(): create an empty heap. ・INSERT(H, x): insert an element x into the heap. ・EXTRACT-MIN(H): remove and return an element with the smallest key. ・DECREASE-KEY(H, x, k): decrease the key of element x to k.
The following operations are also useful: ・IS-EMPTY(H): is the heap empty? ・FIND-MIN(H): return an element with smallest key. ・DELETE(H, x): delete element x from the heap. ・MELD(H1, H2): replace heaps H1 and H2 with their union.
Note. Each element contains a key (duplicate keys are permitted) from a totally-ordered universe.
2 Priority queue applications
Applications. ・A* search. ・Heapsort. ・Online median. ・Huffman encoding. ・Prim’s MST algorithm. ・Discrete event-driven simulation. ・Network bandwidth management. ・Dijkstra’s shortest-paths algorithm. ・…
http://younginc.site11.com/source/5895/fos0092.html 3 PRIORITY QUEUES
‣ binary heaps ‣ d-ary heaps ‣ binomial heaps Algorithms ‣ Fibonacci heaps FOURTH EDITION
ROBERT SEDGEWICK | KEVIN WAYNE
SECTION 2.4 Complete binary tree
Binary tree. Empty or node with links to two disjoint binary trees.
Complete tree. Perfectly balanced, except for bottom level.
complete tree with n = 16 nodes (height = 4)
Property. Height of complete binary tree with n nodes is ⎣log2 n⎦. Pf. Height increases (by 1) only when n is a power of 2. ▪
5 A complete binary tree in nature
Binary heap. Heap-ordered complete binary tree.
Heap-ordered tree. For each child, the key in child ≥ key in parent.
6
parent
10 8
child 12 18 child 11 25
21 17 19
7 Explicit binary heap
Pointer representation. Each node has a pointer to parent and two children. ・Maintain number of elements n. ・Maintain pointer to root node. ・Can find pointer to last node or next node in O(log n) time.
root
6
10 8
12 18 11 25
21 17 19
last next
8 Implicit binary heap
Array representation. Indices start at 1. ・Take nodes in level order. ・Parent of node at k is at ⎣k / 2⎦. ・Children of node at k are at 2k and 2k + 1.
1 6
2 3 10 8
4 5 6 7 12 18 11 25
8 9 10 21 17 19
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
6 10 8 12 18 11 25 21 17 19
9 Binary heap demo
heap ordered
6
10 8
12 18 11 25
21 17 19
10 Binary heap: insert
Insert. Add element in new node at end; repeatedly exchange new element with element in its parent until heap order is restored.
6
10 8
12 18 11 25
add key to heap 21 17 19 7 (violates heap order)
6
7 8
12 10 11 25
21 17 19 18 swim up
11 Binary heap: extract the minimum
Extract min. Exchange element in root node with last node; repeatedly exchange element in root with its smaller child until heap order is restored.
element to 6 remove 7 8
12 10 11 25
exchange 21 17 19 18 with root
sink down
violates 18 7 heap order 7 8 10 8
12 10 11 25 12 18 11 25
remove 21 17 19 6 21 17 19 6 from heap
12 Binary heap: decrease key
Decrease key. Given a handle to node, repeatedly exchange element with its parent until heap order is restored.
decrease key of node x to 11
6
10 8
12 18 11 25
21 17 19
x
13 Binary heap: analysis
Theorem. In an implicit binary heap, any sequence of m INSERT, EXTRACT-MIN, and DECREASE-KEY operations with n INSERT operations takes O(m log n) time. Pf. ・Each heap op touches nodes only on a path from the root to a leaf; the height of the tree is at most log2 n. ・The total cost of expanding and contracting the arrays is O(n). ▪
Theorem. In an explicit binary heap with n nodes, the operations INSERT, DECREASE-KEY, and EXTRACT-MIN take O(log n) time in the worst case.
14 Binary heap: find-min
Find the minimum. Return element in the root node.
root
6
7 10
12 8 11 25
21 17 9
15 Binary heap: delete
Delete. Given a handle to a node, exchange element in node with last node; either swim down or sink up the node until heap order is restored.
delete node x or y
6
x 7 10
12 8 11 25
y
21 17 9
last
16 Binary heap: meld
Meld. Given two binary heaps H1 and H2, merge into a single binary heap.
Observation. No easy solution: Ω(n) time apparently required.
H1 H2 7 10
12 8 11 25
21 17 9
17 Binary heap: heapify
Heapify. Given n elements, construct a binary heap containing them. Observation. Can do in O(n log n) time by inserting each element.
Bottom-up method. For i = n to 1, repeatedly exchange the element in node i with its smaller child until subtree rooted at i is heap-ordered.
1 8
2 12 3 9
4 7 5 22 6 3 7 26
8 14 9 11 10 15 11 22
8 12 9 7 22 3 26 14 11 15 22 1 2 3 4 5 6 7 8 9 10 11 18 Binary heap: heapify
Theorem. Given n elements, can construct a binary heap containing those n elements in O(n) time. Pf. ・There are at most ⎡n / 2h+1⎤ nodes of height h. ・The amount of work to sink a node is proportional to its height h. Thus, the total work is bounded by: ・ log n log n 2 2 n/2h+1 h nh/2h logh2=0n logh2=0n 2 2 n/2h+1 h nh/2h n/2 h nh/2 k n h/2h i k 1 h =0 h =0 =2 h=1 2i 2k 2k 1 i=1 =2n n h/2h ▪ 2 h=1 h =1 =2n
Corollary. Given two binary heaps H1 and H2 containing n elements in total, can implement MELD in O(n) time.
19 Priority queues performance cost summary
operation linked list binary heap
MAKE-HEAP O(1) O(1)
ISEMPTY O(1) O(1)
INSERT O(1) O(log n)
EXTRACT-MIN O(n) O(log n)
DECREASE-KEY O(1) O(log n)
DELETE O(1) O(log n)
MELD O(1) O(n)
FIND-MIN O(n) O(1)
20 Priority queues performance cost summary
Q. Reanalyze so that EXTRACT-MIN and DELETE take O(1) amortized time?
operation linked list binary heap binary heap †
MAKE-HEAP O(1) O(1) O(1)
ISEMPTY O(1) O(1) O(1)
INSERT O(1) O(log n) O(log n)
EXTRACT-MIN O(n) O(log n) O(1) †
DECREASE-KEY O(1) O(log n) O(log n)
DELETE O(1) O(log n) O(1) †
MELD O(1) O(n) O(n)
FIND-MIN O(n) O(1) O(1)
† amortized 21 PRIORITY QUEUES
‣ binary heaps ‣ d-ary heaps ‣ binomial heaps ‣ Fibonacci heaps
CHAPTER 6 (2ND EDITION) Priority queues performance cost summary
operation linked list binary heap d-ary heap
MAKE-HEAP O(1) O(1) O(1)
ISEMPTY O(1) O(1) O(1)
INSERT O(1) O(log n) O(logd n)
EXTRACT-MIN O(n) O(log n) O(d logd n)
DECREASE-KEY O(1) O(log n) O(logd n)
DELETE O(1) O(log n) O(d logd n)
MELD O(1) O(n) O(n)
FIND-MIN O(n) O(1) O(1)
Goal. O(log n) INSERT, DECREASE-KEY, EXTRACT-MIN, and MELD.
mergeable heap 30 Binomial heaps
A priority queue is a set; each element of such a set has a name, which is used to uniquely identify the element, and a label or priority drawn from a totally ordered set. Elements of the priority queue can be Programming S.L. Graham, R.L. Rivest thought of as awaiting service, where the item with the ~ Techniques Editors smallest label is always to be served next. Ordinary stacks and queues are special cases of priority queues. A Data Structure for A variety of applications directly require using prior- ity queues: job scheduling, discrete simulation languages Manipulating Priority where labels represent the time at which events are to occur, as well as various sorting problems. These are Queues discussed, for example, in [2, 3, 5, 11, 15, 17, 24]. Priority queues also play a central role in several good algorithms, Jean Vuillemin such as optimal code constructions, Chartre's prime Universit6 de Paris-Sud number generator, and Brown's power series multipli- cation (see [16] and [17]); applications have also been found in numerical analysis algorithms [10, 17, 19] and A data structure is described which can be used for in graph algorithms for such problems as finding shortest representing a collection of priority queues. The paths [2, 13] and minimum cost spanning tree [2, 4, 25]. primitive operations are insertion, deletion, union, Typical applications require primitive operations update, and search for an item of earliest priority. among the following five: INSERT, DELETE, MIN, UP- Key Words and Phrases: data structures, DATE, and UNION. The operation INSERT (name, label, implementation of set operations, priority queues, Q) adds an element to queue Q, while DELETE (name) mergeable heaps, binary trees removes the element having that name. Operation MIN CR Categories: 4.34, 5.24, 5.25, 5.32, 8.1 (Q) returns the name of the element in Q having the least label, and UPDATE (name, label) changes the label of the element named. Finally, UNION (Q1, Q2, Q3) merges into I. Introduction Qa all elements of Q1 and Q2; the sets Q1 and Q2 become empty. In what follows, we assume that names are In order to design correct and efficient algorithms for handled in a separate dictionary [2, 17] such as a hash- solving a specific problem, it is often helpful to describe table or a balanced tree. If deletions are restricted to our first approach to a solution in a language close to elements extracted by MIN, such an auxiliary symbol that in which the problem is formulated. One such table is not needed. 1 language is that of set theory, augmented by primitive The heap, a truly elegant data structure discovered set manipulation operations. Once the algorithm is out- by J. W. Williams and R. W. Floyd, handles a sequence lined in terms of these set operations, one can then look of n primitives INSERT, DELETE, and MIN, and runs in for data structures most suitable for representing each of O(nlogn) elementary operations using absolutely mini-31 the sets involved. This choice depends only upon the mal storage [17]. For applications in which UNION is collection of primitive operations required for each set. necessary, more sophisticated data structures have been It is thus important to establish a good catalogue of such devised, such as 2-3 trees [2, 17], leftist trees [5, 17], and data structures. A summary of the state of the art on this binary heaps [9]. question can be found in [2]. In this paper, we add to The data structure we present here handles an arbi- this catalogue a data structure which allows efficient trary sequence of n primitives, each drawn from the five manipulation of priority queues. described above, in O(nlogn) machine operations and O(n) memory cells. It also allows for an efficient treat- ment of a large number of updates, which is crucial in connection with spanning tree algorithms: Our data structure provides an implementation (described in General permission to make fair use in teaching or research of all [25]) of the Cheriton-Tarjan-Yao [3] minimum cost span- or part of this material is granted to individual readers and to nonprofit ning tree algorithm which is much more straightforward libraries acting for them provided that ACM's copyrightnotice is given than the original one. and that reference is made to the publication, to its date of issue, and to the fact that reprinting privileges were granted by permission of the The proposed data structure uses less storage than Association for Computing Machinery. To otherwise reprint a figure, leftist, AVL, or 2-3 trees; in addition, when the primitive table, other substantial excerpt, or the entire work requires specific operations are carefully machine coded from the pro- permission as does republication, or systematic or multiple reproduc- tion. grams given in Section 4, they yield worst case running Author's address: Laboratoire de Recherche en Informatique, times which compare favorably with those of their corn- Brit. 490, Universit6 de Paris-Sud, Centre d'Orsay 91405--Orsay, France. We-g'gsume here that indexing through the symbol table is done © 1978 ACM 0001-0782/78/0400-0309 $00.75 in constant time. 309 Communications April 1978 of Volume 21 the ACM Number 4 Binomial tree
Def. A binomial tree of order k is defined recursively: ・Order 0: single node. ・Order k: one binomial tree of order k – 1 linked to another of order k – 1.
B0 Bk
Bk-1
Bk-1
B0 B1 B2 B3 B4
32 Binomial tree properties
Properties. Given an order k binomial tree Bk, ・Its height is k. ・It has 2k nodes. It has k nodes at depth ・ i i. The degree of its root is k. ・ ! " Deleting its root yields binomial trees . ・ k Bk–1, …, B0
Pf. [by induction on k]
Bk+1
B B1 0 B2
Bk
B4
Def. A binomial heap is a sequence of binomial trees such that: ・Each tree is heap-ordered. ・There is either 0 or 1 binomial tree of order k.
6 3 18
8 29 10 44 37
30 23 22 48 31 17
45 32 24 50
55 B4 B1 B0 34 Binomial heap representation
Binomial trees. Represent trees using left-child, right-sibling pointers.
Roots of trees. Connect with singly-linked list, with degrees decreasing from left to right.
root
6 3 18 6 3 18
parent 37 29 10 44 37 29
right
left
48 31 17 48 10
50 50 31 17 44
binomial heap leftist power-of-2 heap representation 35 Binomial heap properties
Properties. Given a binomial heap with n nodes: The node containing the min element is a root of or . ・ B0, B1, …, Bk It contains the binomial tree iff , where is binary ・ Bi bi = 1 bk⋅ b2 b1 b0 representation of n. It has binomial trees. ・ ≤ ⎣log2 n⎦ + 1 Its height . ・ ≤ ⎣log2 n⎦
6 3 18
8 29 10 44 37 n = 19 # trees = 3 30 23 22 48 31 17 height = 4 binary = 10011
45 32 24 50
55 B4 B1 B0 36 Binomial heap: meld
Meld operation. Given two binomial heaps H1 and H2, (destructively) replace with a binomial heap H that is the union of the two.
Warmup. Easy if H1 and H2 are both binomial trees of order k. ・Connect roots of H1 and H2. ・Choose node with smaller key to be root of H.
6
8 29 10 44
30 23 22 48 31 17
45 32 24 50
55 H1 H2
37 6 3 18
8 29 10 44 37
30 23 22 48 31 17 15 7 12 45 32 24 50 28 33 25 + 55 41 12
18
6 3 18
8 29 10 44 37
30 23 22 48 31 17 15 7 12 45 32 24 50 28 33 25 + 55 41 3 12
7 37 18
25
6 3 18
8 29 10 44 37
30 23 22 48 31 17 15 7 12 45 32 24 50 28 33 25 + 55 41
12
18 3 3 12
15 7 37 7 37 18
28 33 25 25
41 6 3 18
8 29 10 44 37
30 23 22 48 31 17 15 7 12 45 32 24 50 28 33 25 + 55 41
12
18 3 3 12
15 7 37 7 37 18
28 33 25 25
41 6 3 18
8 29 10 44 37
30 23 22 48 31 17 15 7 12 45 32 24 50 28 33 25 + 55 41
3 12
15 7 37 18
28 33 25
41 3 3 12
15 7 37 7 37 18
28 33 25 25
41 6 3 18
8 29 10 44 37
30 23 22 48 31 17 15 7 12 45 32 24 50 28 33 25 + 55 41
6 3 12
8 29 10 44 15 7 37 18
30 23 22 48 31 17 28 33 25
45 32 24 50 41
55 6 3 18
8 29 10 44 37
30 23 22 48 31 17 15 7 12 45 32 24 50 28 33 25 + 55 41
1 1 1
1 0 0 1 1
19 + 7 = 26 + 0 0 1 1 1
1 1 0 1 0 Binomial heap: meld
Meld operation. Given two binomial heaps H1 and H2, (destructively) replace with a binomial heap H that is the union of the two.
Solution. Analogous to binary addition.
Running time. O(log n).
Pf. Proportional to number of trees in root lists ≤ 2 ( ⎣log2 n⎦ + 1). ▪
1 1 1
1 0 0 1 1
19 + 7 = 26 + 0 0 1 1 1
1 1 0 1 0
45 Binomial heap: extract the minimum
Extract-min. Delete the node with minimum key in binomial heap H. ・Find root x with min key in root list of H, and delete.
3 6 18
8 29 10 44 37
30 23 22 48 31 17 H
45 32 24 50
55 46 Binomial heap: extract the minimum
Extract-min. Delete the node with minimum key in binomial heap H. ・Find root x with min key in root list of H, and delete. ・Hʹ ← broken binomial trees. ・H ← MELD(Hʹ, H).
Running time. O(log n).
6 18
8 29 10 44 37
30 23 22 48 31 17 H
45 32 24 50 H′
55 47 Binomial heap: decrease key
Decrease key. Given a handle to an element x in H, decrease its key to k. Suppose is in binomial tree . ・ x Bk ・Repeatedly exchange x with its parent until heap order is restored.
Running time. O(log n).
3 6 18
8 29 10 44 37
30 23 22 48 31 17 H
x 32 24 50
55 48 Binomial heap: delete
Delete. Given a handle to an element x in a binomial heap, delete it. ・DECREASE-KEY(H, x, -∞). ・DELETE-MIN(H).
Running time. O(log n).
3 6 18
8 29 10 44 37
30 23 22 48 31 17 H
45 32 24 50
55 49 Binomial heap: insert
Insert. Given a binomial heap H, insert an element x. ・Hʹ ← MAKE-HEAP( ). ・Hʹ ← INSERT(Hʹ, x). ・H ← MELD(Hʹ, H).
Running time. O(log n).
3 6 18 x
H′ 8 29 10 44 37
30 23 22 48 31 17 H
45 32 24 50
55 50 Binomial heap: sequence of insertions
Insert. How much work to insert a new node x ? ・If n = ...... 0, then only 1 credit. 3 6 x ・If n = ...... 01, then only 2 credits. ・If n = ...... 011, then only 3 credits. 29 10 44 37 If , then only credits. ・ n = ...... 0111 4 48 31 17
50
Observation. Inserting one element can take Ω(log n) time.
if n = 11...111
Theorem. Starting from an empty binomial heap, a sequence of n consecutive INSERT operations takes O(n) time.
Pf. (n / 2) (1) + (n / 4)(2) + (n / 8)(3) + … ≤ 2 n. ▪ k i k 1 =2 2i 2k 2k 1 i=1 2 51 Binomial heap: amortized analysis
Theorem. In a binomial heap, the amortized cost of INSERT is O(1) and the worst-case cost of EXTRACT-MIN and DECREASE-KEY is O(log n).
Pf. Define potential function Φ(Hi) = trees(Hi) = # trees in binomial heap Hi. ・Φ(H0) = 0. ・Φ(Hi) ≥ 0 for each binomial heap Hi.
Case 1. [INSERT] ・Actual cost ci = number of trees merged + 1. ・∆Φ = Φ(Hi) – Φ(Hi–1) = 1 – number of trees merged. ・Amortized cost = ĉi = ci + Φ(Hi) – Φ(Hi–1) = 2.
52 Binomial heap: amortized analysis
Theorem. In a binomial heap, the amortized cost of INSERT is O(1) and the worst-case cost of EXTRACT-MIN and DECREASE-KEY is O(log n).
Pf. Define potential function Φ(Hi) = trees(Hi) = # trees in binomial heap Hi. ・Φ(H0) = 0. ・Φ(Hi) ≥ 0 for each binomial heap Hi.
Case 2. [ DECREASE-KEY ] ・Actual cost ci = O(log n). ・∆Φ = Φ(Hi) – Φ(Hi–1) = 0. ・Amortized cost = ĉi = ci = O(log n).
53 Binomial heap: amortized analysis
Theorem. In a binomial heap, the amortized cost of INSERT is O(1) and the worst-case cost of EXTRACT-MIN and DECREASE-KEY is O(log n).
Pf. Define potential function Φ(Hi) = trees(Hi) = # trees in binomial heap Hi. ・Φ(H0) = 0. ・Φ(Hi) ≥ 0 for each binomial heap Hi.
Case 3. [ EXTRACT-MIN or DELETE ] ・Actual cost ci = O(log n).
・∆Φ = Φ(Hi) – Φ(Hi–1) ≤ Φ(Hi) ≤ ⎣log2 n⎦. ・Amortized cost = ĉi = ci + Φ(Hi) – Φ(Hi–1) = O(log n). ▪
54 Priority queues performance cost summary
operation linked list binary heap binomial heap binomial heap
MAKE-HEAP O(1) O(1) O(1) O(1)
ISEMPTY O(1) O(1) O(1) O(1)
INSERT O(1) O(log n) O(log n) O(1) †
EXTRACT-MIN O(n) O(log n) O(log n) O(log n)
DECREASE-KEY O(1) O(log n) O(log n) O(log n)
DELETE O(1) O(log n) O(log n) O(log n) homework MELD O(1) O(n) O(log n) O(1) †
FIND-MIN O(n) O(1) O(log n) O(1)
† amortized
Hopeless challenge. O(1) INSERT, DECREASE-KEY and EXTRACT-MIN. Why? Challenge. O(1) INSERT and DECREASE-KEY, O(log n) EXTRACT-MIN.
55 FIBONACCI HEAPS
‣ preliminaries ‣ insert ‣ extract the minimum ‣ decrease key ‣ bounding the rank ‣ meld and delete
Lecture slides by Kevin Wayne http://www.cs.princeton.edu/~wayne/kleinberg-tardos
Last updated on 7/25/17 11:07 AM Priority queues performance cost summary
Fibonacci heap operation linked list binary heap binomial heap †
MAKE-HEAP O(1) O(1) O(1) O(1)
IS-EMPTY O(1) O(1) O(1) O(1)
INSERT O(1) O(log n) O(log n) O(1)
EXTRACT-MIN O(n) O(log n) O(log n) O(log n)
DECREASE-KEY O(1) O(log n) O(log n) O(1)
DELETE O(1) O(log n) O(log n) O(log n)
MELD O(1) O(n) O(log n) O(1)
FIND-MIN O(n) O(1) O(log n) O(1)
† amortized
Ahead. O(1) INSERT and DECREASE-KEY, O(log n) EXTRACT-MIN.
2 Fibonacci heaps
Theorem. [Fredman-Tarjan 1986] Starting from an empty Fibonacci heap, any sequence of m INSERT, EXTRACT-MIN, and DECREASE-KEY operations involving n INSERT operations takes O(m + n log n) time.
this statement is a bit weaker than the actual theorem
Fibonacci Heaps and Their Uses in Improved Network Optimization Algorithms
MICHAEL L. FREDMAN
University of California, San Diego, L.a Jolla, California AND
ROBERT ENDRE TARJAN
AT&T Bell Laboratories, Murray HilI, New Jersey
Abstract. In this paper we develop a new data structure for implementing heaps (priority queues). Our structure, Fibonacci heaps (abbreviated F-heaps), extends the binomial queues proposed by Vuillemin and studied further by Brown. F-heaps support arbitrary deletion from an n-item heap in qlogn) amortized time and all other standard heap operations in o( 1) amortized time. Using F-heaps we are able to obtain improved running times for several network optimization algorithms. In particular, we obtain the following worst-case bounds, where n is the number of vertices and m the number of edges in the problem graph: ( 1) O(n log n + m) for the single-source shortest path problem with nonnegative edge lengths, improved from O(mlogfmh+2)n); (2) O(n*log n + nm) for the all-pairs shortest path problem, improved from O(nm lo&,,,+2,n); (3) O(n*logn + nm) for the assignment problem (weighted bipartite matching), improved from O(nmlog0dn+2)n); (4) O(mj3(m, n)) for the minimum spanning tree problem, improved from O(mloglo&,,.+2,n), where j3(m, n) = min {i 1log% 5 m/n). Note that B(m, n) 5 log*n if m 2 n. Of these results, the improved bound for minimum spanning trees is the most striking, although all the results give asymptotic improvements for graphs of appropriate densities.
Categories and Subject Descriptors: E.l [Data]: Data Structures--trees; graphs; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems-computations on discrete structures; sorting and searching; G.2.2 [Discrete Mathematics]: Graph Theory-graph algo- 3 rithms; network problems; trees General Terms: Algorithms, Theory Additional Key Words and Phrases: Heap, matching, minimum spanning tree, priority queue, shortest Pati
’ A preliminary version of this paper appeared in the Proceedings of the 25th IEEE Symposium on the Foundations of Computer Science (Singer Island, Fla., Oct. 24-26). IEEE, New York, 1984, pp. 338- 346, 0 IEEE. Any portion of this paper that appeared in the preliminary version is reprinted with permission. This research was supported in part by the National Science Foundation under grant MCS 82-0403 1. Authors’ addresses: M. L. Fredman, Electrical Engineering and Computer Science Department, Uni- versity of California, San Diego, La Jolla, CA 92093; R. E. Tarjan, AT&T Bell Laboratories, Murray Hill, NJ 07974. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 0 1987 ACM 0004-541 l/87/0700-0596 $1.50
Journal ofthe Association for Computing Machinery, Vol. 34, No. 3, July 1987, Pages 596-615. Fibonacci heaps
Theorem. [Fredman–Tarjan 1986] Starting from an empty Fibonacci heap, any sequence of m INSERT, EXTRACT-MIN, and DECREASE-KEY operations involving n INSERT operations takes O(m + n log n) time.
History. ・Ingenious data structure and application of amortized analysis. ・Original motivation: improve Dijkstra’s shortest path algorithm from O(m log n) to O(m + n log n). ・Also improved best-known bounds for all-pairs shortest paths, assignment problem, minimum spanning trees.
4