Gerth stølting Brodal

1

Closing a Classical Data Structure Problem

Gerth Stølting Brodal

:

Strict Fibonacci Heaps

joint work with

George Lagogiannis Robert Endre Tarjan Athens Princeton & HP

44th Annual ACM Symposium on Theory of Computing, May 2012

Gerth stølting Brodal

2

(J,6)

The Problem

(A,27)

(C,11)(M,36) (B,14)

(X,86) INSERT(value, key) FINDMIN DELETE / DELETEMIN MELD(Q1,Q2) DECREASEKEY(value, Δ)

(K,54)

(D,24)(Z,29)(W,6)

Q1

Q2

12

A 15

15

1

812

9

1625

11 3

1322

115

14

10

6217 4

19

Applications t

s Shortest Path Problem Dijkstra (1956)

Minimum Spanning Tree Borůvka (1926)

Jarník (1930)

(n node, m edges)

15

1

812

9

1625

11 3

1322

115

14

10

6217 4

19

(m+n)∙log n

m+n∙log n

m∙β(m,n)

MST only

INSERT/DELETEMIN

+ DECREASEKEY

+ DECREASEKEY

Fredman, Tarjan 1984

Fredman, Tarjan 1984

Priority queue

Gerth stølting Brodal

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Williams1964

Vuillemin1978

FredmanTarjan1984

Tarjanet al.1988

Brodal1995

Brodal1996

BrodalLagogianis

Tarjan2012

Insert log n log n 1 1 1 1 1 1 1

FindMin 1 1 1 1 1 1 1 1 1

Delete log n log n log n n log n log n log n log n log n

Meld - log n 1 1 1 log n 1 1 1

DecreaseKey log n log n log n n 1 1 log n 1 1

History

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 1990s ... were extremely complicated The data structure is genuinely simple

Binary heaps

Binomial queues

Fibonacc

i heaps

Run-relaxed heaps

Arrays Complicated

Strict

Fibonacc

i

h

eaps

Gerth stølting Brodal

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Binary heaps1964

Binomial queues

1978

Fibonacciheaps1984

Run-relaxed heaps1988

Brodal1995

Brodal1996

Strict Fibonacci heaps2012

Heap-order

Rigid structure

Forest

Linking

Subtrees cut

Cascades Amortized

DECREASEKEY

Global control

Redundant counters

Local control

Redundant counters

Local redundant counters

Heap order violations

Global partial control

Pigenhole principle

Technical History

36

27

13

54

86

6

Binary heaps

7

16

42

1117

86

64

Min

8 243

Bionomial queuesFibonacci heaps

single tree

27

12

34

Gerth stølting Brodal

5

Thank You

don’t give up Peter Gabriel (1985)

((( I worked on the heap problem 1994-2012 )))