Amortized Analysis via Potential Method INSERT: actual (1) amortized (1) O O X EXTRACT-MIN: actual (trees(H)+d(n)) amortized (d(n)) ? O O DECREASE-KEY: actual (# cuts) (marks(H)) amortized (1) ? O O O Φ(H)=trees(H)+2 marks(H) · Lifecycle of a node 7 18 38 18 Consolidate 24 17 23 21 39 41 Loses second child 26 46 30 52 18 18 35 Loses first child 5.2: Fibonacci Heaps (Analysis) T.S. 3 5.2 Fibonacci Heaps (Analysis) Frank Stajano Thomas Sauerwald Lent 2016 Outline Recap of INSERT,EXTRACT-MIN and DECREASE-KEY Glimpse at the Analysis Amortized Analysis Bounding the Maximum Degree 5.2: Fibonacci Heaps (Analysis) T.S. 2 and update min-pointer (if necessary) Create a singleton tree Add to root list 21 Actual Costs: (1) min O 21 Fibonacci Heap: INSERT INSERT 21 17 24 23 7 3 30 26 46 18 52 41 35 39 44 5.2: Fibonacci Heaps T.S. 13 and update min-pointer (if necessary) Add to root list 21 Actual Costs: (1) min O 21 Fibonacci Heap: INSERT INSERT Create a singleton tree 21 17 24 23 7 3 30 26 46 18 52 41 35 39 44 5.2: Fibonacci Heaps T.S. 13 and update min-pointer (if necessary) 21 Actual Costs: (1) min O Fibonacci Heap: INSERT INSERT Create a singleton tree Add to root list 21 17 24 23 7 21 3 30 26 46 18 52 41 35 39 44 5.2: Fibonacci Heaps T.S. 13 and update min-pointer (if necessary) 21 Actual Costs: (1) min O Fibonacci Heap: INSERT INSERT Create a singleton tree Add to root list 21 17 24 23 7 21 3 30 26 46 18 52 41 35 39 44 5.2: Fibonacci Heaps T.S. 13 21 Actual Costs: (1) O Fibonacci Heap: INSERT INSERT Create a singleton tree Add to root list and update min-pointer (if necessary) 21 min 17 24 23 7 21 3 30 26 46 18 52 41 35 39 44 5.2: Fibonacci Heaps T.S. 13 21 Fibonacci Heap: INSERT INSERT Create a singleton tree Add to root list and update min-pointer (if necessary) 21 Actual Costs: (1) min O 17 24 23 7 21 3 30 26 46 18 52 41 35 39 44 5.2: Fibonacci Heaps T.S. 13 + d(n) X X (# children) X X Every root becomes child of d(n) is the maximum degree of a another root at most once! root in any Fibonacci heap of size n (trees(H) ) O 41 44 Delete min Meld childen into root list and unmark them Consolidate so that no roots have the same degree Update minimum degree 0 1 2 3 Actual Costs: min 1 degree=22 0 degree=00 1 0 1 17 18 52 17 24 23 41 39 23 26 46 44 35 Fibonacci Heap: EXTRACT-MIN EXTRACT-MIN min 7 24 23 17 3 30 26 46 18 52 41 35 39 44 5.2: Fibonacci Heaps T.S. 14 + d(n) X (# children) X X Every root becomes child of d(n) is the maximum degree of a another root at most once! root in any Fibonacci heap of size n (trees(H) ) O 41 44 X Meld childen into root list and unmark them Consolidate so that no roots have the same degree Update minimum degree 0 1 2 3 Actual Costs: min 1 degree=22 0 degree=00 1 0 1 17 18 52 17 24 23 41 39 23 26 46 44 35 Fibonacci Heap: EXTRACT-MIN EXTRACT-MIN Delete min min 7 24 23 17 3 30 26 46 18 52 41 35 39 44 5.2: Fibonacci Heaps T.S. 14 + d(n) X (# children) X X Every root becomes child of d(n) is the maximum degree of a another root at most once! root in any Fibonacci heap of size n (trees(H) ) O 41 44 Meld childen into root list and unmark them Consolidate so that no roots have the same degree Update minimum degree 0 1 2 3 Actual Costs: min min 1 degree=22 0 degree=00 1 0 1 17 18 52 17 24 23 41 39 23 26 46 44 35 Fibonacci Heap: EXTRACT-MIN EXTRACT-MIN Delete min X 7 24 23 17 3 30 26 46 18 52 41 35 39 44 5.2: Fibonacci Heaps T.S. 14 + d(n) (# children) X X Every root becomes child of d(n) is the maximum degree of a another root at most once! root in any Fibonacci heap of size n (trees(H) ) O 41 44 X Consolidate so that no roots have the same degree Update minimum degree 0 1 2 3 Actual Costs: min min 1 degree=22 0 degree=00 1 0 1 17 183 52 17 24 23 41 39 23 26 46 44 35 Fibonacci Heap: EXTRACT-MIN EXTRACT-MIN Delete min X Meld childen into root list and unmark them 7 24 23 17 30 26 46 18 52 41 35 39 44 5.2: Fibonacci Heaps T.S. 14 + d(n) (# children) X X Every root becomes child of d(n) is the maximum degree of a another root at most once! root in any Fibonacci heap of size n (trees(H) ) O 41 44 X Consolidate so that no roots have the same degree Update minimum degree 0 1 2 3 Actual Costs: min min 1 degree=22 0 degree=00 1 0 1 17 183 52 17 24 23 41 39 23 26 46 44 35 Fibonacci Heap: EXTRACT-MIN EXTRACT-MIN Delete min X Meld childen into root list and unmark them 7 24 23 17 30 26 46 18 52 41 35 39 44 5.2: Fibonacci Heaps T.S. 14 + d(n) (# children) X X Every root becomes child of d(n) is the maximum degree of a another root at most once! root in any Fibonacci heap of size n (trees(H) ) O 41 44 X Consolidate so that no roots have the same degree Update minimum degree 0 1 2 3 Actual Costs: min min 1 degree=22 0 degree=00 1 0 1 17 183 52 17 24 23 41 39 23 26 46 44 35 Fibonacci Heap: EXTRACT-MIN EXTRACT-MIN Delete min X Meld childen into root list and unmark them 7 24 23 17 30 26 46 18 52 41 35 39 44 5.2: Fibonacci Heaps T.S. 14 + d(n) (# children) X X Every root becomes child of d(n) is the maximum degree of a another root at most once! root in any Fibonacci heap of size n (trees(H) ) O Consolidate so that no roots have the same degree Update minimum degree 0 1 2 3 Actual Costs: min min 1 degree=22 0 degree=00 1 0 1 17 3 17 24 23 41 18 52 41 23 26 46 44 39 44 35 Fibonacci Heap: EXTRACT-MIN EXTRACT-MIN Delete min X Meld childen into root list and unmark them X 7 24 23 17 18 52 41 30 26 46 39 44 35 5.2: Fibonacci Heaps T.S. 14 + d(n) X Every root becomes child of d(n) is the maximum degree of a another root at most once! root in any Fibonacci heap of size n (trees(H) ) O (# children) X Update minimum degree 0 1 2 3 Actual Costs: min min 1 degree=22 0 degree=00 1 0 1 17 3 17 24 23 41 18 52 41 23 26 46 44 39 44 35 Fibonacci Heap: EXTRACT-MIN EXTRACT-MIN Delete min X Meld childen into root list and unmark them X Consolidate so that no roots have the same degree 7 24 23 17 18 52 41 30 26 46 39 44 35 5.2: Fibonacci Heaps T.S. 14 + d(n) X Every root becomes child of d(n) is the maximum degree of a another root at most once! root in any Fibonacci heap of size n (trees(H) ) O X Update minimum degree 0 1 2 3 Actual Costs: min min 1 degree=22 0 degree=00 1 0 1 17 3 17 24 23 41 18 52 41 23 26 46 44 39 44 35 Fibonacci Heap: EXTRACT-MIN EXTRACT-MIN Delete min X Meld childen into root list and unmark them X Consolidate so that no roots have the same degree (# children) 7 24 23 17 18 52 41 30 26 46 39 44 35 5.2: Fibonacci Heaps T.S. 14 + d(n) X Every root becomes child of d(n) is the maximum degree of a another root at most once! root in any Fibonacci heap of size n (trees(H) ) O X Update minimum degree 0 1 2 3 Actual Costs: min min 1 2 0 degree=00 1 0 1 17 3 17 24 23 41 18 52 41 23 26 46 44 39 44 35 Fibonacci Heap: EXTRACT-MIN EXTRACT-MIN Delete min X Meld childen into root list and unmark them X Consolidate so that no roots have the same degree (# children) degree=2 7 24 23 17 18 52 41 30 26 46 39 44 35 5.2: Fibonacci Heaps T.S. 14 + d(n) X Every root becomes child of d(n) is the maximum degree of a another root at most once! root in any Fibonacci heap of size n (trees(H) ) O X Update minimum degree 0 1 2 3 Actual Costs: min min 1 degree=22 0 degree=00 1 0 1 17 3 17 24 23 41 18 52 41 23 26 46 44 39 44 35 Fibonacci Heap: EXTRACT-MIN EXTRACT-MIN Delete min X Meld childen into root list and unmark them X Consolidate so that no roots have the same degree (# children) 7 24 23 17 18 52 41 30 26 46 39 44 35 5.2: Fibonacci Heaps T.S.