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PRIORITY QUEUES

‣ binary heaps ‣ d-ary heaps ‣ binomial heaps ‣ Fibonacci heaps

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

6 Binary heap

Binary heap. Heap-ordered complete binary tree.

Heap-ordered tree. For each child, the key in child ≥ key in parent.

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

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

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.

10 8

12 18 11 25

add key to heap 21 17 19 7 (violates heap order)

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

10 8

12 18 11 25

21 17 19

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

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

x 7 10

12 8 11 25

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 Algorithms ‣ Fibonacci heaps FOURTH EDITION

ROBERT SEDGEWICK | KEVIN WAYNE

SECTION 2.4 Complete d-ary tree d-ary tree. Empty or node with links to d disjoint d-ary trees.

Complete tree. Perfectly balanced, except for bottom level.

Fact. The height of a complete d-ary tree with n nodes is ≤ ⎡logd n⎤.

23 d-ary heap d-ary heap. Heap-ordered complete d-ary tree. Heap-ordered tree. For each child, the key in child ≥ key in parent.

30 10 20

80 32 34 55 22 34 46 20 40

82 90

24 d-ary heap: insert

Insert. Add node at end; repeatedly exchange element in child with element in parent until heap order is restored.

Running time. Proportional to height = O(logd n).

30 10 20

80 32 34 55 22 34 46 20 40

82 90

25 d-ary heap: extract the minimum

Extract min. Exchange root node with last node; repeatedly exchange element in parent with element in largest child until heap order is restored.

Running time. Proportional to d ⨉ height = O(d logd n).

30 10 20

80 32 34 55 22 34 46 20 40

82 90

26 d-ary heap: decrease key

Decrease key. Given a handle to an element x, repeatedly exchange it with its parent until heap order is restored.

Running time. Proportional to height = O(logd n).

30 10 20

80 32 34 55 22 34 46 20 40

82 90

27 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)

28 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 priority 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

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

33 Binomial heap

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.

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

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

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

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

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

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.