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Introduction Complexity Theory 2002/03. Peter van Emde Boas KNOW YOUR NUMBERS ! The Impact of Complexity Peter van Emde Boas ILLC-FNWI-UvA 2003 See: http://staff.science.uva.nl/~peter/teaching/ct03.html

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Introduction Complexity Theory 2002/03. Peter van Emde Boas Moore’s Law in action © Scientifi American 274, 199601, p45 Computation speed doubles every 1.5 years Corresponding rate of increase of Memory capacity

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Introduction Complexity Theory 2002/03. Peter van Emde Boas 19821985

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Introduction Complexity Theory 2002/03. Peter van Emde Boas 19861988

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Introduction Complexity Theory 2002/03. Peter van Emde Boas 19901992

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Introduction Complexity Theory 2002/03. Peter van Emde Boas 19941996

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Introduction Complexity Theory 2002/03. Peter van Emde Boas 19982000

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Introduction Complexity Theory 2002/03. Peter van Emde Boas 2003

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Introduction Complexity Theory 2002/03. Peter van Emde Boas Is it really so nice ? sum := 0 for i = 1 to N do sum := sum + a[i] ; print ( sum ) Back in 1977 this would work in less than a second for N = 10 000 Can we deal with N = 400 000 000 today within a second ? sum := 0 for i = 1 to N do j:= index[i] ; sum := sum + a[j] ; print ( sum ) And what about this one ?

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Introduction Complexity Theory 2002/03. Peter van Emde Boas Information must be moved! Source: Ph.d. Thesis Stefan Manegold Understanding, Modeling, and Improving Main-Memory Database Performance UvA Dec 16, 2002

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Introduction Complexity Theory 2002/03. Peter van Emde Boas Information must be moved! Source: Ph.d. Thesis Peter A. Boncz Monet; A next-Generation DBMS Kernel for Query-Intensive Applications UvA Mei 31, 2002

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Introduction Complexity Theory 2002/03. Peter van Emde Boas Know your numbers ! # InstructionsTime 1 0000.00001 sec 1 000 0000.01 secInvisible 10 000 0000.1 secPerceptive 100 000 0001 sec 1 000 000 00010 sec 10 000 000 000100 secCoffee 100 000 000 00016.7 minChat 1 000 000 000 0002 hr 47 minWHAARGH! 10 000 000 000 00027 hrthe Limit!! 100 000 000 000 00011.25 daysHopeless 1 000 000 000 000 0004 months 10 000 000 000 000 0003.33 yearsUnrealistic 100 000 000 000 000 00033 years 1 000 000 000 000 000 0003.3 century Processing times on a 100 MIPS system 1 day = 86400 sec1 000 000 sec ≈ 11 days 1 year ≈ 31 536 000 sec1 000 000 000 sec ≈ 35 years

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Introduction Complexity Theory 2002/03. Peter van Emde Boas WHAT IS COMPLEXITY ? Relation between size of the input instance and the time required for solving this instance. Traversing a listLinearO(n) SortingO(n.logn) Multiplication QuadraticO(n 2 ) Matrix multiplicationCubicO(n 3 ) Truth table checkingExponentialO(2 n ) Naive TSPO(n!) What can be done in principle can become undoable in practice Order of growth is more relevant than the implicit constants

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Introduction Complexity Theory 2002/03. Peter van Emde Boas The Order of growth 1020501002001000 n.00001.00002.00005.0001.0002.001 n.logn.00003.00008.00027.0006.0016.01 n 2.0001.0004.0025.01.041 n 3.001.008.1251816 min 2 n.001131 y??!!?*6min Time on a 1 MIPS system in seconds (unless stated otherwise)

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Introduction Complexity Theory 2002/03. Peter van Emde Boas Impact of the faster machine n 1000 times as large n.logn 500 - 1000 times as large n 2 31 times as large n 3 10 times as large 2 n add 10 to input size That’s why Edmonds stated his thesis: Effective ==> Polynomial Time Complexity

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Introduction Complexity Theory 2002/03. Peter van Emde Boas A Fallacy ?? If computation speed doubles every 2 years we can perform (2 N - 1) M operations in 2N years, where M is the number of operations performed in the first 2 years. So why worry about exponential time algorithms ?? What is wrong with this argument ?

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