Fibonacci HeapFibonacci HeapBH&FHBH&FH

Why BH&FH -- Why BH&FH -- 程式目的程式目的MinPQMinPQ

Mergeable MinPQMergeable MinPQ

MaxPQMaxPQ

DEPQDEPQ

Min-Leftist Min-Skew Min-B-Heap

Min-F-Heap

Min Heap

Deap MinMaxSymmetric

Max

Data

Structures

Symmetric

Max

Data

Structures

Inheritance ADT 實作方法

Why BH&FH -- Why BH&FH -- 舉例舉例

Server for one priority shuts downServer for one priority shuts down 。。

Efficient algorithms for Shortest-path prEfficient algorithms for Shortest-path problem and finding minimum spanning troblem and finding minimum spanning treesees 。。 (Improved network optimization)(Improved network optimization)

What’s BH&FH -- DefinitionWhat’s BH&FH -- Definition A binomial heap is a collection of binomial treA binomial heap is a collection of binomial tre

es.es. The The binomial tree binomial tree Bk Bk is an ordered tree defineis an ordered tree define

d recursively. the binomial tree d recursively. the binomial tree BB0 consists of 0 consists of a single node. The binomial tree a single node. The binomial tree Bk Bk consists of consists of two binomial trees two binomial trees BkBk-1 that are -1 that are linked linked togethtogether: the root of one is the leftmost child of the rer: the root of one is the leftmost child of the rootoot

of the other.of the other. Like a binomial heap, a Like a binomial heap, a Fibonacci heap Fibonacci heap is a cois a co

llection of min-heap-ordered trees. The trees illection of min-heap-ordered trees. The trees in a Fibonacci heap are not constrained to be bn a Fibonacci heap are not constrained to be binomial trees. inomial trees.

BH BH 圖解圖解

FHFH 圖解圖解

提供的函式提供的函式 MakeHeap()MakeHeap() Insert(H,x)Insert(H,x) Minimum(H)Minimum(H) Extract-min(H)Extract-min(H) Union(H1,H2)Union(H1,H2) DecreaceKey(H,x,k)DecreaceKey(H,x,k) Delete(H,x)Delete(H,x) 範例網站範例網站

Binomial Heap

Binary Min-Heap

Fibonacci Heap

How Fibonacci InsertHow Fibonacci Insert FIB-HEAP-INSERT(FIB-HEAP-INSERT(HH, , xx)) 1 1 degreedegree[[xx] ← 0] ← 0 2 2 pp[[xx] ← NIL] ← NIL 3 3 childchild[[xx] ← NIL] ← NIL 4 4 leftleft[[xx] ← ] ← xx 5 5 rightright[[xx] ← ] ← xx 6 6 markmark[[xx] ← FALSE] ← FALSE 7 7 concatenate the root list containing concatenate the root list containing x x with root list with root list HH 8 8 if if minmin[[HH] = NIL or ] = NIL or keykey[[xx] < ] < keykey[[minmin[[HH]]]] 9 9 then then minmin[[HH] ← ] ← xx 10 10 nn[[HH] ← ] ← nn[[HH] + 1] + 1

Fibonacci Insert 21Fibonacci Insert 21

How Fibonacci UnionHow Fibonacci Union FIB-HEAP-UNION(FIB-HEAP-UNION(HH1, 1, HH2)2) 1 1 H H ← MAKE-FIB-HEAP()← MAKE-FIB-HEAP() 2 2 minmin[[HH] ← ] ← minmin[[HH1]1] 3 3 concatenate the root list of concatenate the root list of HH2 with the root list of 2 with the root list of HH 4 4 if if ((minmin[[HH1] = NIL) or (1] = NIL) or (minmin[[HH2] ≠ NIL and 2] ≠ NIL and minmin

[[HH2] < 2] < minmin[[HH1])1]) 5 5 then then minmin[[HH] ← ] ← minmin[[HH2]2] 6 6 nn[[HH] ← ] ← nn[[HH1] + 1] + nn[[HH2]2] 7 free the objects 7 free the objects HH1 and 1 and HH22 8 8 return return HH

How Fibonacci Extract-minHow Fibonacci Extract-min FIB-HEAP-EXTRACT-MIN(FIB-HEAP-EXTRACT-MIN(HH)) 1 1 z z ← ← minmin[[HH]] 2 2 if if z z ≠ NIL≠ NIL 3 3 then for then for each child each child x x of of zz 4 4 do do add add x x to the root list of to the root list of HH 5 5 pp[[xx] ← NIL] ← NIL 6 remove 6 remove z z from the root list of from the root list of HH 7 7 if if z z = = rightright[[zz]] 8 8 then then minmin[[HH] ← NIL] ← NIL 9 9 else else minmin[[HH] ← ] ← rightright[[zz]] 10 CONSOLIDATE(10 CONSOLIDATE(HH)) 11 11 nn[[HH] ← ] ← nn[[HH] - 1] - 1 12 12 return return zz

Extract-min 1Extract-min 1

Extract-min 2Extract-min 2

Extract-min 3Extract-min 3

Extract-min FinalExtract-min Final

How Fibonacci ConsolidateHow Fibonacci Consolidate CONSOLIDATE(CONSOLIDATE(HH)) 1 1 for for i i ← 0 ← 0 to to DD((nn[[HH])]) 2 2 do do AA[[ii] ← NIL] ← NIL 3 3 for for each node each node w w in the root list of in the root list of HH 4 4 do do x x ← ← ww 5 5 d d ← ← degreedegree[[xx]] 6 6 while while AA[[dd] ≠ NIL] ≠ NIL 7 7 do do y y ← ← AA[[dd] Another node with the same degree as ▹] Another node with the same degree as ▹ xx.. 8 8 if if keykey[[xx] > ] > keykey[[yy]] 9 9 then then exchange exchange x x ↔ ↔ yy 10 FIB-HEAP-LINK(10 FIB-HEAP-LINK(HH, , yy, , xx)) 11 11 AA[[dd] ← NIL] ← NIL 12 12 d d ← ← d d + 1+ 1 13 13 AA[[dd] ← ] ← xx 14 14 minmin[[HH] ← NIL] ← NIL 15 15 for for i i ← 0 ← 0 to to DD((nn[[HH])]) 16 16 do if do if AA[[ii] ≠ NIL] ≠ NIL 17 17 then then add add AA[[ii] to the root list of ] to the root list of HH 18 18 if if minmin[[HH] = NIL or ] = NIL or keykey[[AA[[ii]] < ]] < keykey[[minmin[[HH]]]] 19 19 then then minmin[[HH] ← ] ← AA[[ii]] FIB-HEAP-LINK(FIB-HEAP-LINK(HH, , yy, , xx)) 1 remove 1 remove y y from the root list of from the root list of HH 2 make 2 make y y a child of a child of xx, incrementing , incrementing degreedegree[[xx]] 3 3 markmark[[yy] ← FALSE] ← FALSE

How Fibonacci Decrease-KeyHow Fibonacci Decrease-Key FIB-HEAP-DECREASE-KEY(FIB-HEAP-DECREASE-KEY(HH, , xx, , kk)) 1 1 if if k k > > keykey[[xx]] 2 2 then error then error "new key is greater than current key""new key is greater than current key" 3 3 keykey[[xx] ← ] ← kk 4 4 y y ← ← pp[[xx]] 5 5 if if y y ≠ NIL and ≠ NIL and keykey[[xx] < ] < keykey[[yy]] 6 6 then then CUT(CUT(HH, , xx, , yy)) 7 CASCADING-CUT(7 CASCADING-CUT(HH, , yy)) 8 8 if if keykey[[xx] < ] < keykey[[minmin[[HH]]]] 9 9 then then minmin[[HH] ← ] ← xx CUT(CUT(HH, , xx, , yy)) 1 remove 1 remove x x from the child list of from the child list of yy, decrementing , decrementing degreedegree[[yy]] 2 add 2 add x x to the root list of to the root list of HH 3 3 pp[[xx] ← NIL] ← NIL 4 4 markmark[[xx] ← FALSE] ← FALSE CASCADING-CUT(CASCADING-CUT(HH, , yy)) 1 1 z z ← ← pp[[yy]] 2 2 if if z z ≠ NIL≠ NIL 3 3 then if then if markmark[[yy] = FALSE] = FALSE 4 4 then then markmark[[yy] ← TRUE] ← TRUE 5 5 else else CUT(CUT(HH, , yy, , zz)) 6 CASCADING-CUT(6 CASCADING-CUT(HH, , zz))

Fibonacci Decrease-KeyFibonacci Decrease-Key

How Fibonacci DeleteHow Fibonacci Delete

FIB-HEAP-DELETE(FIB-HEAP-DELETE(HH, , xx)) 1 FIB-HEAP-DECREASE-KEY(1 FIB-HEAP-DECREASE-KEY(HH, , xx, -∞), -∞) 2 FIB-HEAP-EXTRACT-MIN(2 FIB-HEAP-EXTRACT-MIN(HH))

How Binomial DeleteHow Binomial Delete

BINOMIAL-HEAP-DELETE(BINOMIAL-HEAP-DELETE(HH, , xx)) 1 BINOMIAL-HEAP-DECREASE-KEY(1 BINOMIAL-HEAP-DECREASE-KEY(HH, , xx, -∞), -∞) 2 BINOMIAL-HEAP-EXTRACT-MIN(2 BINOMIAL-HEAP-EXTRACT-MIN(HH))

Note -- Time ComplexityNote -- Time Complexity

Note -- DifferenceNote -- Difference

1 Delete1 Delete DecreaseKeyDecreaseKey 2 Binomial2 Binomial ：： OrderedOrdered FibonacciFibonacci ：： UnorderedUnordered

Note -- ApplicationNote -- Application

Shortest-pathShortest-path

Combine PQCombine PQ

Fibonacci Heap Fibonacci Heap

習題習題

1.1. 請建立一個空的請建立一個空的 Fibonacci HeapFibonacci Heap ，並，並 insertinsert 以以下數值為以以下數值為 keykey 的節點的節點 7070 、、 8080 、、 9090 、、 3030 、、 4040 、、 5050 、、 2020 、、 6060 (hint: insert(hint: insert 至哪個位置並無差別至哪個位置並無差別 ))2.extractMin2.extractMin3.3. 將將 8080 的的 keykey ，， Decrease KeyDecrease Key 為為 10104.4. 給一給一 Fibonacci HeapFibonacci Heap 如圖，請將兩者如圖，請將兩者 UnionUnion