Closing a Classical Data Structure Problem Strict Fibonacci

Closing a Classical Data Structure Problem : Strict Fibonacci Heaps Gerth Stølting Brodal joint work with George Lagogiannis Athens Robert Endre Tarjan Princeton & HP 44 th Annual ACM Symposium on Theory of Computing, May 2012 Gerth stølting Brodal

The Problem § § § Q 1 INSERT(value, key) FINDMIN Priority DELETE / DELETEMIN queue MELD(Q 1, Q 2) A 15 DECREASEKEY(value, Δ) Applications § Shortest Path Problem § Dijkstra (1956) 17 2 25 16 6 5 22 11 13 19 INSERT/DELETEMIN Fredman, Tarjan 1984 8 12 § Minimum Spanning Tree § Borůvka (1926) Jarník (1930) (n node, m edges) 17 25 2 16 6 5 10 9 + DECREASEKEY 3 11 14 4 15 1 19 (W, 6) (Z, 29) (D, 24) (m+n)∙log n 1 14 9 4 15 10 s Q 2 3 t 11 (C, 11) (B, 14) (M, 36) (J, 6) (X, 86) 12 (A, 27) (K, 54) m+n∙log n 22 11 13 MST only Fredman, Tarjan 1984 + DECREASEKEY 8 m∙β(m, n) 12 Gerth stølting Brodal 2

History B ry a n i Williams 1964 a he ps es eu u q ial i cc ea h a n ibo m ino Fredman Tarjan Vuillemin 1978 1984 ed x a el ps a he i cc na ibo F ct eaps i r h St -r n Ru F B ps Tarjan et al. Brodal Lagogianis Tarjan 1988 1995 1996 2012 Insert log n 1 1 1 1 Find. Min 1 1 1 1 1 Delete log n n log n log n Meld - log n 1 1 1 Decrease. Key log n n 1 1 log n 1 1 1 log n Arrays Complicated Amortized complexity (Tarjan 1983) CONFERENCE REVIEWS § This paper closes one of the few open problems in the § § area of comparison-based priority queues Other PQs from the 1990 s. . . were extremely complicated The data structure is genuinely simple Gerth stølting Brodal 3

Technical History Min 7 12 17 11 16 27 13 27 54 36 86 4 6 3 42 34 8 24 6 86 Binary heaps Bionomial Fibonacci heaps queues Binomial Binary heaps queues 1964 1978 Heap-order Forest Rigid structure Linking Fibonacci heaps Run-relaxed heaps Brodal Strict Fibonacci heaps 1984 1988 1995 1996 2012 Subtrees cut Global control Local redundant counters Cascades Redundant Amortized DECREASEKEY counters Global partial control Heap order violations Pigenhole principle single tree Gerth stølting Brodal 4

don’t give up Peter Gabriel (1985) ((( I worked on the heap problem 1994 -2012 ))) Thank You Gerth stølting Brodal 5