Strict Fibonacci Heaps Gerth Stølting Brodal Aarhus University George Lagogiannis Robert Endre Tarjan Agricultural University of Athens Princeton University & HP 44th Annual ACM Symposium on Theory of Computing, New Yorker Hotel, New York, May 22, 2012

The Problem — Priority Queues Insert(value, key) FindMin DeleteMin / Delete(&value) Meld(Q1,Q2) DecreaseKey(&value, Δ) (C,11) (M,36) (B,14) (X,86) 12 (A,27) (K,54) A 15 (D,24) (Z,29) (W,6) DecreaseKey = Delete+Insert Borůvka = problem to electrify the Czech Republic Jarnik = Prim(1957)s algorithm β(m,n) is for all pratical valures 3-4. Applications 15 1 8 12 9 16 25 11 3 13 22 5 14 10 6 2 17 4 19 t s Insert/DeleteMin Shortest Path Dijkstra (1956) Minimum Spanning Tree Borůvka (1926) Jarník (1930) (n node, m edges) (m+n)∙log n m+n∙log n m∙β(m,n) Fredman, Tarjan 1984 + DecreaseKey 15 1 8 12 9 16 25 11 3 13 22 5 14 10 6 2 17 4 19 MST only Fredman, Tarjan 1984 + DecreaseKey

Amortized complexity (Tarjan 1983) History Binomial queues Run-relaxed heaps Strict Fibonacci heaps Binary heaps Fibonacci heaps Williams 1964 Vuillemin 1978 Fredman Tarjan 1984 Driscoll et al. 1988 Brodal 1995 Brodal 1996 Lagogianis STOC 2012 Insert log n 1 FindMin Delete n Meld - DecreaseKey Run-relaxed heaps : SSSP MST real time applications, parallel algorithms worst-case guarantees important, single source shortest paths Since 90s people working on the problem incl Katajainen, Elm-Asry, Tarjan, Kaplan... trying to simplify solution Arrays Complicated Pointer Based Amortized complexity (Tarjan 1983)

Global partial control Technical History Min 8 7 4 6 86 17 11 16 27 27 12 6 13 24 42 54 36 86 Local control = consequence of wanting Meld to be O(1) Binary heaps Fibonacci heaps Binary heaps 1964 Binomial queues 1978 Fibonacci heaps 1984 Run-relaxed heaps 1988 Brodal 1995 1996 Strict Fibonacci heaps 2012 Heap-order Rigid structure Forest Linking Subtrees cut Cascades  Amortized DecreaseKey Global control Redundant counters Local control Local redundant counters Heap order violations Global partial control Pigenhole principle single tree

Ideas Meld smallest tree red + link 4 12 16 8 5 42 12 6 21 11 7 11 Invariants white nodes share one color record white children left of red root is red color white 15 14 23 15 14 23 color red Meld smallest tree red + link white root 2 Intuition Only maintain structure for white nodes Red non-root nodes never increase degree Delete: O(1) red nodes white 1 Definitions white node rank = #white children DecreaseKey: cut + mark parent (if white) Invariants i’th rightmost white child of a white node: rank + #marks ≥ i - 1 #marks + #white roots = O(log n) Theorem max rank O(log n)

Priority Queue Operations 2 FindMin = return root Insert = make single red node + Meld Delete = DecreaseKey to - + DeleteMin Meld = color smaller tree red + link + O(1) transformations DecreaseKey = cut + link with root + O(1) transformations DeleteMin = cut root + find new root + O(log n) link + O(log n) transformations root degree +1 root degree +1; white root (+1); marks (+1) root degree +O(log n); white root +O(log n) Invariant R = α·log (3/2 · #white nodes + #red nodes) + β degree unmarked white nodes ≤ R degree red and marked white nodes ≤ R – 1

One node mark reduction Two node mark reduction Transformations Root degree reduction Converts two red nodes to white; reduces the root degree by two; creates one new white root White root reduction Two white roots of rank r is replaced by one rank r+1; increases root degree by one x root z 0/0 1/0 y y x z below root r/0 One node mark reduction White node with ≥ 2 marks becomes a white root (unmarked); increases the root degree by one Two node mark reduction Two nodes of equal rank r with 1 mark become unmarked; one parent one more mark below root y x ≥ 2 marks z y x r/1 r / m = rang / #marks

Representation ref-count active ... Q-prev x left-child rank left right parent flag color record Q-next singles color-record root size non-linkable-child Q-head heap record 3 1 4 2 fix-list rank-list left right transformable nodes with marks white roots node marked pointers from unmarked white nodes with rank 1 and with white parent inc dec rank maximum

Thank You Williams Vuillemin Fredman Tarjan Driscoll et al. Binomial queues Binary heaps Fibonacci heaps Run-relaxed heaps Strict Fibonacci heaps Williams 1964 Vuillemin 1978 Fredman Tarjan 1984 Driscoll et al. 1988 Brodal 1995 Brodal 1996 Lagogianis STOC 2012 Insert log n 1 FindMin Delete n Meld - DecreaseKey Arrays Pointer Based Amortized Thank You