Data structures Binomial Heaps - Binomial Trees B0B0 BkBk B k-1.

Data structures Binomial Heaps - Binomial Trees B0B0 BkBk B k-1

Data structures Binomial Heaps - Binomial Trees B0B0 B2B2 B1B1 B3B3

Data structures Binomial Heaps - Binomial Trees B4B4

Data structures Binomial Heaps - Binomial Trees BkBk B k-1 B k-2 B2B2 B1B1 B0B0 

Data structures Binomial Heaps - Binomial Trees Properties of binomial tree B k it has 2 k nodes the height of the tree is k it has exactly C k i nodes at depth i for i = 0, 1, , k C k i = n ! / (i ! (n – i) !) the root has degree k

Data structures Binomial Heaps - Definition Binomial heap H is an ordered list of binomial trees that satisfies the following properties: Each binomial tree is heap-ordered, i.e. for each node its key is greater or equal to the key of its parent There is at most one binomial tree in H whose root has a given degree Binomial trees are contained in the list in the order of increasing degrees

Data structures Binomial Heaps - Example 38 1429 6 27 1117 8 18 1225 110

Data structures Binomial Heaps - Example 38 - 0 14 - 129 - 0 6 - 3 27 - 0 11 - 117 - 0 8 - 2 18 - 0 12 - 125 - 0 1 - 2 10 - 0 KeyDataDegParentLeftNext

Data structures Binomial Heaps - Example [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Binomial Heaps - Min procedure Min(BinomialHeap H): min   P  H.Head while P  0 do if P.Key < min then min  P.Key P  P.Next return min

Data structures Binomial Heaps - Union - Example 17 1044 6 50 4831 2937 318 24 2322 8 55 4532 30 25 712 33 15 41 28 H1H1 H2H2

Data structures Binomial Heaps - Union - Example 17 1044 6 50 4831 2937 318 24 2322 8 55 4532 30 25 712 33 15 41 28 H1H2H1H2

Data structures Binomial Heaps - Union - Example 17 1044 6 50 4831 2937 3 18 24 2322 8 55 4532 30 25 712 33 15 41 28 H1H2H1H2

Data structures Binomial Heaps - Union - Example 17 1044 6 50 4831 2937 3 18 24 2322 8 55 4532 3025 7 12 33 15 41 28 H1H2H1H2

Data structures Binomial Heaps - Union procedure Union(BinomialHeap H1,H2): H  BinomialHeapMakeEmpty() P  H.Head; P1  H1.Head; P2  H2.Head while P1  0 and P2  0 do if P1.Deg < P2.Deg then P.Next  P1; P1  P1.Next else P.Next  P2; P2  P2.Next while P1  0 do P.Next  P1; P1  P1.Next while P2  0 do P.Next  P2; P2  P2.Next

Data structures Binomial Heaps - Union [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Binomial Heaps - Union (continued) P1  H.Head; P2  P1; P3  P1; P4  P1 merge  0 while P1.Next  0 do P4  P3; P3  P2; P2  P1; P1  P1.Next if P1.Deg = P2.Deg then merge  1 if P1.Deg < P2.Deg and merge = 1 then if P2.Key < P3.Key then P2  P3 if P4 = H.Head then H.Head  P3 else P4.Next  P3 P2.Next  P3.Left; P2.Parent  P3 P3.Left  P2; P3.Next  P1; P3.Deg  P3.Deg + 1 merge  0 return H

Data structures Binomial Heaps - Insert procedure Insert(BinomialHeap H, int Key, int Data): H1  BinomialHeapMakeEmpty() E  NewNode() H1.Head  E; E.Next  0 E.Parent  0; E.Left  0 E.Deg  0 E.Key  Key; E.Data  Data; H  Union(H,H1)

Data structures Binomial Heaps - ExtractMin - Example 17 1044 6 50 4831 2918 24 2322 8 55 4532 30 12 33 15 41 28

Data structures Binomial Heaps - ExtractMin - Example 17 1044 50 4831 29 18 24 2322 8 55 4532 30 12 33 15 41 28

Data structures Binomial Heaps - ExtractMin - Example 17 1044 50 4831 29 1824 2322 8 55 4532 301233 15 41 28

Data structures Binomial Heaps - ExtractMin - Example 17 1044 50 4831 291824 2322 8 55 4532 3012 33 15 41 28

Data structures Binomial Heaps - ExtractMin procedure ExtractMin(BinomialHeap H): min   ; M  0; MP  0 P  H.Head; PP  H.Head while P  0 do if P.Key < min then min  P.Key; M  P; MP  PP PP  P; P  P.Next if M = 0 then return fail

Data structures Binomial Heaps - ExtractMin (continued) H1  BinomialHeapMakeEmpty() if MP  H.Head then MP.Next  M.Next else H.Head  M.Next P  M.Left while P  0 do Next  P.Next P.Next  H1.Head; H1.Head  P P.Parent  0 P  Next H  Union(H,H1)

Data structures Binomial Heaps - DecreaseKey - Example 17 1044 50 4831 291824 2322 8 55 4532 3012 33 15 41 28 10

Data structures Binomial Heaps - DecreaseKey - Example 17 1044 10 4831 291824 2322 8 55 4532 3012 33 15 41 28

Data structures Binomial Heaps - DecreaseKey procedure DecreaseKey(BinomialHeap H, Node P, int Key): Parent  P.Parent while Parent  0 and P.Key < Parent.Key do P.Key  Parent.Key P.Data  Parent.Data P  Parent Parent  Parent.Parent

Data structures Binomial Heaps - Delete procedure DeleteKey(BinomialHeap H, Node P): BinomialHeapDecreaseKey(H,P,   ) BinomialHeapExtractMin(H)

Data structures Binomial Heaps - Summary Binomial HeapsHeaps Min  (log n) or  (1)  (1) ExtractMin  (log n) DecreaseKey  (log n) Union  (log n) (n)(n) Insert  (log n) Delete  (log n) MakeEmpty  (1) IsEmpty  (1)

Data structures Fibonacci Heaps [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Fibonacci Heaps - Example 39 1852 3237 41 3830 17 35 2646 24 min(H)

Data structures Fibonacci Heaps - Example 39 1852 3237 41 3830 17 35 2646 24 min(H) KeyDataDegParentLeftNextPrevious

Data structures Fibonacci Heaps - Example [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Fibonacci Heaps - Degree and Marked nodes A degree D(x) of a node x in Fibonacci heap is defined as a number of children of node x (unlike as it is for binomial heaps, D(x) may not be equal with height of the tree rooted at x). A node x in Fibonacci Heap is marked if and only if it has lost a child since the last time x was made the child of another node (or since the last time x was made the root)

Data structures Fibonacci Heaps - Min [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Fibonacci Heaps - Insert [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Fibonacci Heaps - Insert - Example 39 1852 3237 41 3830 17 35 2646 24 min(H) 1

Data structures Fibonacci Heaps - Union [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Fibonacci Heaps - ExtractMin [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Fibonacci Heaps - ExtractMin - Example 39 1852 3237 41 3830 17 35 2646 24 min(H) 21

Data structures Fibonacci Heaps - ExtractMin - Example 39 1852237 41 38 30 17 35 2646 24 min(H) 21

Data structures Fibonacci Heaps - ExtractMin - Example 39 1852237 41 38 30 17 35 2646 2421 01234

Data structures Fibonacci Heaps - ExtractMin - Example 39 1852 23 7 41 38 30 17 35 2646 2421 01234

Data structures Fibonacci Heaps - ExtractMin - Example 39 1852 23 7 41 38 30 17 35 2646 24 21 01234

Data structures Fibonacci Heaps - ExtractMin - Example 39 18 52 23 7 41 38 30 17 35 2646 2421 01234

Data structures Fibonacci Heaps - ExtractMin - Example 39 18 52 23 7 41 38 30 17 35 2646 2421 min(H)

Data structures Fibonacci Heaps - DecreaseKey [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Fibonacci Heaps - DecreaseKey - Example 1 39 18 52 23 7 41 38 30 17 35 2646 2421 min(H) 15

Data structures Fibonacci Heaps - DecreaseKey - Example 1 39 18 52 23 7 41 38 30 17 35 26 15 2421 min(H)

Data structures Fibonacci Heaps - DecreaseKey - Example 2 39 18 52 23 7 41 38 30 17 35 26 15 2421 min(H) 5

Data structures Fibonacci Heaps - DecreaseKey - Example 2 39 18 52 23 7 41 38 30 17 26 15 2421 min(H) 5

Data structures Fibonacci Heaps - DecreaseKey - Example 2 39 18 52 23 7 41 38 30 17 2615 2421 min(H) 5

Data structures Fibonacci Heaps - DecreaseKey - Example 2 39 18 52 23 7 41 38 30 17 261524 21 min(H) 5

Data structures Fibonacci Heaps - Delete [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Fibonacci Heaps - Complexity Heap potential P(H) = t(H) + 2m(H), where t(H) - number of nodes in root level, m(H) - number of marked nodes. Amortised complexity T a (H) = T(H) +  P(H)

Data structures Fibonacci Heaps - Complexity Insert Min [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Fibonacci Heaps - Complexity Union [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Fibonacci Heaps - Complexity ExtractMin [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Fibonacci Heaps - Complexity DecreaseKey Delete - just sum of the costs for DecreaseKey and ExtractMin [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Fibonacci Heaps - Degree and size D(n) - maximal possible node degree in Fibonacci heap with n elements Theorem D(n) =  (log n)

Data structures Fibonacci Heaps - Degree and size Observation If only operations Min,Union, Insert, ExtractMin, MakeEmpty and IsEmpty are used, then the resulting Fibonacci heap is just a collection of unordered Binomial trees, thus D(n) =  (log n).

Data structures Fibonacci Heaps - Degree and size Lemma Let x be a node in a Fibonacci heap with degree(x) = k. Let y 1,..., y k be the children of x in the order in which they were linked to x, from the earliest to the latest. Then degree(y 1 )  0 and degree(y i )  i – 2 for i = 2, 3,..., k. [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Fibonacci Heaps - Degree and size Fibonacci numbers F 0 = 0F 1 = 1F k = F k – 1 + F k – 2 Lemma For all k  0: F k + 2 = 1 +  i = 0...k F i

Data structures Fibonacci Heaps - Degree and size Lemma Let x be a node in a Fibonacci heap and let degree(x) = k. Then size(x)  F k + 2   k, where  k = (1 +  5)/2. [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Fibonacci Heaps - Degree and size [Adapted from T.Cormen, C.Leiserson, R. Rivest]

Data structures Fibonacci Heaps - Degree and size Corollary The maximum degree D(n) of any node in an n-node Fibonacci heap is O(log n).

Data structures Fibonacci Heaps - Summary Fibonacci Heaps Binomial HeapsHeaps Min  (1)  (log n) or  (1)  (1) ExtractMin  (log n) DecreaseKey  (1)  (log n) Union  (1)  (log n) (n)(n) Insert  (1)  (log n) Delete  (log n) MakeEmpty  (1) IsEmpty  (1)

Data structures Binomial Heaps - Binomial Trees B0B0 BkBk B k-1.

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