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Analysis Of Fibonacci Heaps

Analysis Of Fibonacci Heaps. 6. 5. 9. MaxDegree. Let N i = min # of nodes in any min (sub)tree whose root has i children. N 0 = 1. N 1 = 2. b. …. …. c i. c 1. c 2. N i , i > 1. Children of b are labeled in the order in which they became children of b .

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Analysis Of Fibonacci Heaps

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  1. Analysis Of Fibonacci Heaps

  2. 6 5 9 MaxDegree • Let Ni = min # of nodes in any min (sub)tree whose root has i children. • N0 = 1. • N1 = 2.

  3. b … … ci c1 c2 Ni , i > 1 • Children of b are labeled in the order in which they became children of b. • c1became a child before c2 did, and so on. • So, when ck became a child of b, degree(b) >= k –1. • degree(ck) at the time when ckbecame a child of b = degree(b) at the time when ckbecame a child of b >= k – 1.

  4. b … … ci c1 c2 Ni , i > 1 • So, current degree(ck) >= max{0, k – 2}. • So, Ni = N0 + (S0<=q<=i-2Nq ) + 1 =(S0<=q<=i-2Nq ) + 2.

  5. Fibonacci Numbers • F0 = 0. • F1 = 1. • Fi = Fi-1 + Fi-2, i > 1 =(S0<=q<=i-2Fq ) + 1, i > 1. • N0 = 1. • N1 = 2. • Ni =(S0<=q<=i-2Nq ) + 2, i > 1. • Ni =Fi+2~ ((1 + sqrt(5))/2)i , i >= 0.

  6. MaxDegree • MaxDegree <= logfn, where f =(1 + sqrt(5))/2.

  7. Accounting Method • Insert. • Guessed amortized cost = 2. • Use 1 unit to pay for the actual cost of the insert and keep the remaining 1 unit as a credit for a future remove min operation. • Keep this credit with the min tree that is created by the insert operation. • Meld. • Guessed amortized cost = 1. • Use 1 unit to pay for the actual cost of the meld.

  8. 6 1 5 theNode 3 9 7 4 2 8 10 9 5 4 5 9 7 6 6 8 Remove Nonmin Element

  9. 1 5 10 5 3 9 7 9 2 6 8 9 4 5 7 6 6 8 Remove Nonmin Element • Guessed amortized cost = 2logfn + 3. • Use logfn units to pay for setting parent fields to null for subtrees of deleted node. • Use 1 unit to pay for remaining work not related to cascading cut.

  10. 1 5 10 5 3 9 7 9 2 6 8 9 4 5 7 6 6 8 Remove Nonmin Element • Keep logfn units to pay for possible future pairwise combining of the new top-level trees created. • Kept as 1 credit per new top-level tree. • Discard excess credits (if any).

  11. 1 5 10 5 3 9 7 9 2 6 8 9 4 5 7 6 6 8 Remove Nonmin Element • Keep 1 unit to pay for the time when node whose ChildCut field is set to true is cut from its parent, and another 1 unit for the pairwise combining of the cut subtree.

  12. 1 5 10 5 3 9 7 9 2 6 8 9 4 5 7 6 6 8 Remove Nonmin Element • Keep the 2 credits on the node (if any) whose ChildCut field is set to true by the ensuing cascading cut operation. • If there is no such node, discard the credits.

  13. 6 1 5 theNode 3 9 7 4 2 8 10 9 5 4 5 9 7 6 6 8 DecreaseKey(theNode, theAmount)

  14. DecreaseKey(theNode, theAmount) 0 1 6 5 3 10 5 7 9 2 9 8 9 4 5 7 6 6 Guessed amortized cost = 4. 8

  15. 0 1 6 5 3 10 5 7 9 2 9 8 9 4 5 7 6 6 8 DecreaseKey(theNode, theAmount) Use 1 unit to pay for work not related to cascading cut.

  16. 0 1 6 5 3 10 5 7 9 2 9 8 9 4 5 7 6 6 8 DecreaseKey(theNode, theAmount) Keep 1 unit to pay for possible future pairwise combining of the new top-level tree created whose root is theNode. Kept as credit on theNode.

  17. 0 1 6 5 3 10 5 7 9 2 9 8 9 4 5 7 6 6 8 DecreaseKey(theNode, theAmount) Keep 1 unit to pay for the time when node whose ChildCut field is set to true is cut from its parent, and use another 1 unit for the pairwise combining of the cut subtree.

  18. 0 1 6 5 3 10 5 7 9 2 9 8 9 4 5 7 6 6 8 DecreaseKey(theNode, theAmount) • Keep the 2 credits on the node (if any) whose ChildCut field is set to true by the ensuing cascading cut operation. • If there is no such node, discard the credits.

  19. Remove Min • Guessed amortized cost = 3logfn. • Actual cost <= 2logfn – 1 + #MinTrees. • Allocation of amortized cost. • Use 2logfn – 1 to pay part of actual cost. • Keep remaining logfn + 1 as a credit to pay part of the actual cost of a future remove min operation. • Put 1 unit of credit on each of the at most logfn + 1 min trees left behind by the remove min operation. • Discard the remaining credits (if any).

  20. Paying Actual Cost Of A Remove Min • Actual cost <= 2logfn – 1 + #MinTrees • How is it paid for? • 2logfn – 1 ispaid for from the amortized cost of the remove min. • #MinTrees ispaid by the 1 unit credit on each of the min trees in the Fibonacci heap just prior to the remove min operation.

  21. Who Pays For Cascading Cut? • Only nodes with ChildCut = true are cut during a cascading cut. • The actual cost to cut a node is 1. • This cost is paid from the 2 units of credit on the node whose ChildCut field is true. The remaining unit of credit is kept with the min tree that has been cut and now becomes a top-level tree.

  22. Potential Method • P(i) = S[#MinTrees(j) + 2*#NodesWithTrueChildCut(j)] • #MinTrees(j) is #MinTrees for Fibonacci heap j. • When Fibonacci heaps A and B are melded, A and B are no longer included in the sum. • P(0) = 0 • P(i) >= 0 for all i.

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