Introduction to Fractals and Fractal Dimension
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Introduction to Fractals and Fractal Dimension. Christos Faloutsos. Intro to Fractals - Outline. Motivation – 3 problems / case studies Definition of fractals and power laws Solutions to posed problems More examples Discussion - putting fractals to work! Conclusions – practitioner’s guide
Introduction to Fractals and Fractal Dimension

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Slide 1

Introduction to Fractalsand Fractal Dimension

Christos Faloutsos

Slide 2

Intro to Fractals - Outline

  • Motivation – 3 problems / case studies

  • Definition of fractals and power laws

  • Solutions to posed problems

  • More examples

  • Discussion - putting fractals to work!

  • Conclusions – practitioner’s guide

  • Appendix: gory details - boxcounting plots

Copyright: C. Faloutsos (2001)

Slide 3

Problem #1: GIS - points

  • Road end-points of Montgomery county:

  • Q : distribution?

    • not uniform

    • not Gaussian

    • ???

    • no rules/pattern at all!?

Copyright: C. Faloutsos (2001)

Slide 4

Problem #2 - spatial d.m.

Galaxies (Sloan Digital Sky Survey w/ B. Nichol)

- ‘spiral’ and ‘elliptical’ galaxies

(stores and households ...)

- patterns?

- attraction/repulsion?

- how many ‘spi’ within r from an ‘ell’?

Copyright: C. Faloutsos (2001)

Slide 5

Poisson

Problem #3: traffic

  • disk trace (from HP - J. Wilkes); Web traffic

#bytes

- how many explosions to expect?

- queue length distr.?

time

Copyright: C. Faloutsos (2001)

Slide 6

Common answer for all three:

  • Fractals / self-similarities / power laws

  • Seminal works from Hilbert, Minkowski, Cantor, Mandelbrot, (Hausdorff, Lyapunov, Ken Wilson, …)

Copyright: C. Faloutsos (2001)

Slide 7

Road map

  • Motivation – 3 problems / case studies

  • Definition of fractals and power laws

  • Solutions to posed problems

  • More examples and tools

  • Discussion - putting fractals to work!

  • Conclusions – practitioner’s guide

  • Appendix: gory details - boxcounting plots

Copyright: C. Faloutsos (2001)

Slide 8

Definitions (cont’d)

  • Simple 2-D figure has finite perimeter and finite area

    • area ≈ perimeter2

  • Do all 2-D figures have finite perimeter, finite area, and area ≈ perimeter2?

Copyright: C. Faloutsos (2001)

Slide 9

What is a fractal?

= self-similar point set, e.g., Sierpinski triangle:

zero area;

infinite length!

...

Copyright: C. Faloutsos (2001)

Slide 10

Definitions (cont’d)

  • Paradox: Infinite perimeter ; Zero area!

  • ‘dimensionality’: between 1 and 2

  • actually: Log(3)/Log(2) = 1.58…

Copyright: C. Faloutsos (2001)

Slide 11

Definitions (cont’d)

  • “self similar”

    • pieces look like whole independent of scale

  • “dimension”

    dim = log(# self-similar pieces) / log(scale)

    • dim(line) = log(2)/log(2) = 1

    • dim(square) = log(4)/log(2) = 2

    • dim(cube) = log(8)/log(2) = 3

Copyright: C. Faloutsos (2001)

Slide 12

Q: fractal dimension of a line?

A: 1 (= log(2)/log(2))

Intrinsic (‘fractal’) dimension

Copyright: C. Faloutsos (2001)

Slide 13

Q: dfn for a given set of points?

x

y

5

1

4

2

3

3

2

4

Intrinsic (‘fractal’) dimension

Copyright: C. Faloutsos (2001)

Slide 14

Q: fractal dimension of a line?

A: nn ( <= r ) ~ r^1

(‘power law’: y=x^a)

Q: fd of a plane?

A: nn ( <= r ) ~ r^2

Fd = slope of (log(nn) vs log(r) )

Intrinsic (‘fractal’) dimension

Copyright: C. Faloutsos (2001)

Slide 15

Algorithm, to estimate it?

Notice

avg nn(<=r) is exactly

tot#pairs(<=r) / (2*N)

Intrinsic (‘fractal’) dimension

including ‘mirror’ pairs

Copyright: C. Faloutsos (2001)

Slide 16

Dfn of fd:

ONLY for a perfectly self-similar point set:

zero area;

infinite length!

...

=log(n)/log(f) = log(3)/log(2) = 1.58

Copyright: C. Faloutsos (2001)

Slide 17

log(#pairs

within <=r )

1.58

log( r )

Sierpinsky triangle

== ‘correlation integral’

Copyright: C. Faloutsos (2001)

Slide 18

Observations:

  • Euclidean objects have integer fractal dimensions

    • point: 0

    • lines and smooth curves: 1

    • smooth surfaces: 2

  • fractal dimension -> roughness of the periphery

Copyright: C. Faloutsos (2001)

Slide 19

fd = embedding dimension -> uniform pointset

a point set may have several fd, depending on scale

Important properties

Copyright: C. Faloutsos (2001)

Slide 20

Road map

  • Motivation – 3 problems / case studies

  • Definition of fractals and power laws

  • Solutions to posed problems

  • More examples and tools

  • Discussion - putting fractals to work!

  • Conclusions – practitioner’s guide

  • Appendix: gory details - boxcounting plots

Copyright: C. Faloutsos (2001)

Slide 21

Problem #1: GIS points

  • Cross-roads of Montgomery county:

  • any rules?

Copyright: C. Faloutsos (2001)

Slide 22

1.51

Solution #1

A: self-similarity ->

  • <=> fractals

  • <=> scale-free

  • <=> power-laws (y=x^a, F=C*r^(-2))

  • avg#neighbors(<= r ) = r^D

log(#pairs(within <= r))

log( r )

Copyright: C. Faloutsos (2001)

Slide 23

1.51

Solution #1

A: self-similarity

  • avg#neighbors(<= r ) ~ r^(1.51)

log(#pairs(within <= r))

log( r )

Copyright: C. Faloutsos (2001)

Slide 24

Examples:MG county

  • Montgomery County of MD (road end-points)

Copyright: C. Faloutsos (2001)

Slide 25

Examples:LB county

  • Long Beach county of CA (road end-points)

Copyright: C. Faloutsos (2001)

Slide 26

Solution#2: spatial d.m.

Galaxies ( ‘BOPS’ plot - [sigmod2000])

log(#pairs)

log(r)

Copyright: C. Faloutsos (2001)

Slide 27

Solution#2: spatial d.m.

log(#pairs within <=r )

- 1.8 slope

- plateau!

- repulsion!

ell-ell

spi-spi

spi-ell

log(r)

Copyright: C. Faloutsos (2001)

Slide 28

spatial d.m.

log(#pairs within <=r )

- 1.8 slope

- plateau!

- repulsion!

ell-ell

spi-spi

spi-ell

log(r)

Copyright: C. Faloutsos (2001)

Slide 29

r1

r2

r2

r1

spatial d.m.

Heuristic on choosing # of clusters

Copyright: C. Faloutsos (2001)

Slide 30

spatial d.m.

log(#pairs within <=r )

- 1.8 slope

- plateau!

- repulsion!

ell-ell

spi-spi

spi-ell

log(r)

Copyright: C. Faloutsos (2001)

Slide 31

spatial d.m.

log(#pairs within <=r )

  • - 1.8 slope

  • - plateau!

  • repulsion!!

ell-ell

spi-spi

-duplicates

spi-ell

log(r)

Copyright: C. Faloutsos (2001)

Slide 32

#bytes

time

Solution #3: traffic

  • disk traces: self-similar:

Copyright: C. Faloutsos (2001)

Slide 33

20%

80%

Solution #3: traffic

  • disk traces (80-20 ‘law’ = ‘multifractal’)

#bytes

time

Copyright: C. Faloutsos (2001)

Slide 34

Solution#3: traffic

Clarification:

  • fractal: a set of points that is self-similar

  • multifractal: a probability density function that is self-similar

    Many other time-sequences are bursty/clustered: (such as?)

Copyright: C. Faloutsos (2001)

Slide 35

Tape#1

Tape# N

time

Tape accesses

# tapes needed, to retrieve n records?

(# days down, due to failures / hurricanes / communication noise...)

Copyright: C. Faloutsos (2001)

Slide 36

Tape#1

Tape# N

time

Tape accesses

50-50 = Poisson

# tapes retrieved

real

# qual. records

Copyright: C. Faloutsos (2001)

Slide 37

Fast estimation of fd(s):

  • How, for the (correlation) fractal dimension?

  • A: Box-counting plot:

log(sum(pi ^2))

pi

r

log( r )

Copyright: C. Faloutsos (2001)

Slide 38

Definitions

  • pi : the percentage (or count) of points in the i-th cell

  • r: the side of the grid

Copyright: C. Faloutsos (2001)

Slide 39

Fast estimation of fd(s):

  • compute sum(pi^2) for another grid side, r’

log(sum(pi ^2))

pi’

r’

log( r )

Copyright: C. Faloutsos (2001)

Slide 40

Fast estimation of fd(s):

  • etc; if the resulting plot has a linear part, its slope is the correlation fractal dimension D2

log(sum(pi ^2))

log( r )

Copyright: C. Faloutsos (2001)

Slide 41

Hausdorff or box-counting fd:

  • Box counting plot: Log( N ( r ) ) vs Log ( r)

  • r: grid side

  • N (r ): count of non-empty cells

  • (Hausdorff) fractal dimension D0:

Copyright: C. Faloutsos (2001)

Slide 42

Definitions (cont’d)

  • Hausdorff fd:

r

log(#non-empty cells)

D0

log(r)

Copyright: C. Faloutsos (2001)

Slide 43

Observations

  • q=0: Hausdorff fractal dimension

  • q=2: Correlation fractal dimension (identical to the exponent of the number of neighbors vs radius)

  • q=1: Information fractal dimension

Copyright: C. Faloutsos (2001)

Slide 44

Observations, cont’d

  • in general, the Dq’s take similar, but not identical, values.

  • except for perfectly self-similar point-sets, where Dq=Dq’ for any q, q’

Copyright: C. Faloutsos (2001)

Slide 45

Examples:MG county

  • Montgomery County of MD (road end-points)

Copyright: C. Faloutsos (2001)

Slide 46

Examples:LB county

  • Long Beach county of CA (road end-points)

Copyright: C. Faloutsos (2001)

Slide 47

Summary So Far

  • many fractal dimensions, with nearby values

  • can be computed quickly

    (O(N) or O(N log(N))

  • (code: on the web)

Copyright: C. Faloutsos (2001)

Slide 48

Dimensionality Reduction with FD

Problem definition: ‘Feature selection’

  • given N points, with E dimensions

  • keep the k most ‘informative’ dimensions

    [Traina+,SBBD’00]

Copyright: C. Faloutsos (2001)

Slide 49

Dim. reduction - w/ fractals

not informative

Copyright: C. Faloutsos (2001)

Slide 50

Dim. reduction

Problem definition: ‘Feature selection’

  • given N points, with E dimensions

  • keep the k most ‘informative’ dimensions

    Re-phrased: spot and drop attributes with strong (non-)linear correlations

    Q: how do we do that?

Copyright: C. Faloutsos (2001)

Slide 51

Dim. reduction

A: Hint: correlated attributes do not affect the intrinsic/fractal dimension, e.g., if

y = f(x,z,w)

we can drop y

(hence: ‘partial fd’ (PFD) of a set of attributes = the fd of the dataset, when projected on those attributes)

Copyright: C. Faloutsos (2001)

Slide 52

Dim. reduction - w/ fractals

global FD=1

PFD=1

PFD~0

Copyright: C. Faloutsos (2001)

Slide 53

Dim. reduction - w/ fractals

global FD=1

PFD=1

PFD=1

Copyright: C. Faloutsos (2001)

Slide 54

Dim. reduction - w/ fractals

global FD=1

PFD~1

PFD~1

Copyright: C. Faloutsos (2001)

Slide 55

(problem: given N points in E-d, choose k best dimensions)

Q: Algorithm?

Dim. reduction - w/ fractals

Copyright: C. Faloutsos (2001)

Slide 56

Q: Algorithm?

A: e.g., greedy - forward selection:

keep the attribute with highest partial fd

add the one that causes the highest increase in pfd

etc., until we are within epsilon from the full f.d.

Dim. reduction - w/ fractals

Copyright: C. Faloutsos (2001)

Slide 57

(backward elimination: ~ reverse)

drop the attribute with least impact on the p.f.d.

repeat

until we are epsilon below the full f.d.

Dim. reduction - w/ fractals

Copyright: C. Faloutsos (2001)

Slide 58

Q: what is the smallest # of attributes we should keep?

Dim. reduction - w/ fractals

Copyright: C. Faloutsos (2001)

Slide 59

Q: what is the smallest # of attributes we should keep?

A: we should keep at least as many as the f.d. (and probably, a few more)

Dim. reduction - w/ fractals

Copyright: C. Faloutsos (2001)

Slide 60

Results: E.g., on the ‘currency’ dataset

(daily exchange rates for USD, HKD, BP, FRF, DEM, JPY - i.e., 6-d vectors, one per day - base currency: CAD)

Dim. reduction - w/ fractals

FRF

e.g.:

USD

Copyright: C. Faloutsos (2001)

Slide 61

E.g., on the ‘currency’ dataset

log(#pairs(<=r))

correlation integral

1.98

log(r)

Copyright: C. Faloutsos (2001)

Slide 62

if unif + indep.

E.g., on the ‘currency’ dataset

Copyright: C. Faloutsos (2001)

Slide 63

E.g., on the eigenface dataset

16-d vectors,

one for each

of ~1K faces

4.25

Copyright: C. Faloutsos (2001)

Slide 64

E.g., on the eigenface dataset

Copyright: C. Faloutsos (2001)

Slide 65

global FD=1

PFD~1

PFD~1

Dim. reduction - w/ fractals

  • Conclusion:

  • can do non-linear dim. reduction

Copyright: C. Faloutsos (2001)

Slide 66

More tools

  • Zipf’s law

  • Korcak’s law / “fat fractals”

Copyright: C. Faloutsos (2001)

Slide 67

A famous power law: Zipf’s law

  • Q:vocabulary word frequency in a document - any pattern?

freq.

aaron

zoo

Copyright: C. Faloutsos (2001)

Slide 68

A famous power law: Zipf’s law

log(freq)

“a”

  • Bible - rank vs frequency (log-log)

“the”

log(rank)

Copyright: C. Faloutsos (2001)

Slide 69

A famous power law: Zipf’s law

log(freq)

  • Bible - rank vs frequency (log-log)

  • similarly, in manyother languages; for customers and sales volume; city populations etc etc

log(rank)

Copyright: C. Faloutsos (2001)

Slide 70

A famous power law: Zipf’s law

log(freq)

  • Zipf distr:

    • freq = 1/ rank

  • generalized Zipf:

    • freq = 1 / (rank)^a

log(rank)

Copyright: C. Faloutsos (2001)

Slide 71

Olympic medals (Sidney):

log(#medals)

rank

Copyright: C. Faloutsos (2001)

Slide 72

More power laws: areas – Korcak’s law

Scandinavian lakes

Any pattern?

Copyright: C. Faloutsos (2001)

Slide 73

More power laws: areas – Korcak’s law

log(count( >= area))

Scandinavian lakes area vs complementary cumulative count (log-log axes)

log(area)

Copyright: C. Faloutsos (2001)

Slide 74

More power laws: Korcak

Japan islands

Copyright: C. Faloutsos (2001)

Slide 75

More power laws: Korcak

log(count( >= area))

Japan islands;

area vs cumulative count (log-log axes)

log(area)

Copyright: C. Faloutsos (2001)

Slide 76

(Korcak’s law: Aegean islands)

Copyright: C. Faloutsos (2001)

Slide 77

Korcak’s law & “fat fractals”

How to generate such regions?

Copyright: C. Faloutsos (2001)

Slide 78

Korcak’s law & “fat fractals”

Q: How to generate such regions?

A: recursively, from a single region

Copyright: C. Faloutsos (2001)

Slide 79

Conclusions

  • tool#1: (for points) ‘correlation integral’: (#pairs within <= r) vs (distance r)

  • tool#2: (for categorical values) rank-frequency plot (a’la Zipf)

  • tool#3: (for numerical values) CCDF: Complementary cumulative distr. function (#of elements with value >= a )

Copyright: C. Faloutsos (2001)

Slide 80

Fractals - overall conclusions

  • Real data often disobey textbook assumptions: not Gaussian, not Poisson, not uniform, not independent)

  • self-similar datasets: appear often

  • powerful tools: correlation integral, rank-frequency plot, ...

  • intrinsic/fractal dimension helps:

    • find patterns

    • dimensionality reduction

Copyright: C. Faloutsos (2001)

Slide 81

NN queries

  • Q: What about L1, Linf?

  • A: Same slope, different intercept

log(#neighbors)

log(d)

Copyright: C. Faloutsos (2001)

Slide 82

log(#pairs(within <= r))

log(#pairs)

1.51

2.8

log(hops)

log( r )

Practitioner’s guide:

  • tool#1: #pairs vs distance, for a set of objects, with a distance function (slope = intrinsic dimensionality)

internet

MGcounty

Copyright: C. Faloutsos (2001)

Slide 83

log(degree)

-0.82

log(rank)

Practitioner’s guide:

  • tool#2: rank-frequency plot (for categorical attributes)

Bible

internet domains

log(freq)

log(rank)

Copyright: C. Faloutsos (2001)

Slide 84

log(count( >= area))

log(area)

Practitioner’s guide:

  • tool#3: CCDF, for (skewed) numerical attributes, eg. areas of islands/lakes, UNIX jobs...)

scandinavian lakes

Copyright: C. Faloutsos (2001)

Slide 85

Resources:

  • Software for fractal dimension

    • http://www.cs.cmu.edu/~christos

    • christos@cs.cmu.edu

Copyright: C. Faloutsos (2001)

Slide 86

Books

  • Strongly recommended intro book:

    • Manfred Schroeder Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise W.H. Freeman and Company, 1991

  • Classic book on fractals:

    • B. Mandelbrot Fractal Geometry of Nature, W.H. Freeman, 1977

Copyright: C. Faloutsos (2001)

Slide 87

References

  • Belussi, A. and C. Faloutsos (Sept. 1995). Estimating the Selectivity of Spatial Queries Using the `Correlation' Fractal Dimension. Proc. of VLDB, Zurich, Switzerland.

  • Faloutsos, C. and V. Gaede (Sept. 1996). Analysis of the z-ordering Method Using the Hausdorff Fractal Dimension. VLDB, Bombay, India.

  • Proietti, G. and C. Faloutsos (March 23-26, 1999). I/O complexity for range queries on region data stored using an R-tree. International Conference on Data Engineering (ICDE), Sydney, Australia.

Copyright: C. Faloutsos (2001)

Slide 88

Road map

  • Motivation – 3 problems / case studies

  • Definition of fractals and power laws

  • Solutions to posed problems

  • More tools and examples

  • Discussion - putting fractals to work!

  • Conclusions – practitioner’s guide

  • Appendix: gory details - boxcounting plots

Copyright: C. Faloutsos (2001)

Slide 89

NN queries

  • Q: in NN queries, what is the effect of the shape of the query region? [Belussi+95]

L2

Linf

r

L1

Copyright: C. Faloutsos (2001)

Slide 90

NN queries

  • Q: in NN queries, what is the effect of the shape of the query region?

  • that is, for L2, and self-similar data:

log(#pairs-within(<=d))

D2

L2

r

log(d)

Copyright: C. Faloutsos (2001)

Slide 91

NN queries

  • Q: What about L1, Linf?

log(#pairs-within(<=d))

D2

L2

r

log(d)

Copyright: C. Faloutsos (2001)

Slide 92

NN queries

  • Q: What about L1, Linf?

  • A: Same slope, different intercept

log(#pairs-within(<=d))

D2

L2

r

log(d)

Copyright: C. Faloutsos (2001)

Slide 93

L2

r

NN queries

  • Q: what about the intercept? Ie., what can we say about N2 and Ninf

Ninf neighbors

N2 neighbors

Linf

r

volume: V2

volume: Vinf

Copyright: C. Faloutsos (2001)

Slide 94

NN queries

  • Consider sphere with volume Vinf and r’ radius

Ninf neighbors

N2 neighbors

r’

Linf

L2

r

r

volume: V2

volume: Vinf

Copyright: C. Faloutsos (2001)

Slide 95

NN queries

  • Consider sphere with volume Vinf and r’ radius

  • (r/r’)^E = V2 / Vinf

  • (r/r’)^D2 = N2 / N2’

  • N2’ = Ninf (since shape does not matter)

  • and finally:

Copyright: C. Faloutsos (2001)

Slide 96

NN queries

( N2 / Ninf ) ^ 1/D2 = (V2 / Vinf) ^ 1/E

Copyright: C. Faloutsos (2001)

Slide 97

NN queries

Conclusions: for self-similar datasets

  • Avg # neighbors: grows like (distance)^D2 , regardless of query shape (circle, diamond, square, e.t.c. )

Copyright: C. Faloutsos (2001)

Slide 98

2.8

Internet topology

  • Internet routers: how many neighbors within h hops?

log(#pairs)

Reachability function: number of neighbors within r hops, vs r (log-log).

Mbone routers, 1995

log(hops)

Copyright: C. Faloutsos (2001)

Slide 99

-0.82

More power laws on the Internet

log(degree)

log(rank)

degree vs rank, for Internet domains (log-log) [sigcomm99]

Copyright: C. Faloutsos (2001)

Slide 100

-2.2

More power laws - internet

  • pdf of degrees: (slope: 2.2 )

Log(count)

Log(degree)

Copyright: C. Faloutsos (2001)

Slide 101

Even more power laws on the Internet

log( i-th eigenvalue)

0.47

log(i)

Scree plot for Internet domains (log-log) [sigcomm99]

Copyright: C. Faloutsos (2001)

Slide 102

Log(#octants)

2.63 = fd

octree levels

More apps: Brain scans

  • Oct-trees; brain-scans

Copyright: C. Faloutsos (2001)

Slide 103

More apps: Medical images

[Burdett et al, SPIE ‘93]:

  • benign tumors: fd ~ 2.37

  • malignant: fd ~ 2.56

Copyright: C. Faloutsos (2001)

Slide 104

1 year

2 years

More fractals:

  • cardiovascular system: 3 (!)

  • stock prices (LYCOS) - random walks: 1.5

  • Coastlines: 1.2-1.58 (Norway!)

Copyright: C. Faloutsos (2001)

Slide 105

Copyright: C. Faloutsos (2001)

Slide 106

More power laws

  • duration of UNIX jobs [Harchol-Balter]

  • Energy of earthquakes (Gutenberg-Richter law) [simscience.org]

log(freq)

amplitude

day

magnitude

Copyright: C. Faloutsos (2001)

Slide 107

Even more power laws:

  • publication counts (Lotka’s law)

  • Distribution of UNIX file sizes

  • Income distribution (Pareto’s law)

  • web hit counts [Huberman]

Copyright: C. Faloutsos (2001)

Slide 108

Power laws, cont’ed

  • In- and out-degree distribution of web sites [Barabasi], [IBM-CLEVER]

  • length of file transfers [Bestavros+]

  • Click-stream data (w/ A. Montgomery (CMU-GSIA) + MediaMetrix)

Copyright: C. Faloutsos (2001)

Slide 109

Resources:

  • Software for fractal dimension

    • http://www.cs.cmu.edu/~christos

    • christos@cs.cmu.edu

Copyright: C. Faloutsos (2001)

Slide 110

Books

  • Strongly recommended intro book:

    • Manfred Schroeder Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise W.H. Freeman and Company, 1991

  • Classic book on fractals:

    • B. Mandelbrot Fractal Geometry of Nature, W.H. Freeman, 1977

Copyright: C. Faloutsos (2001)

Slide 111

References

  • [ieeeTN94] W. E. Leland, M.S. Taqqu, W. Willinger, D.V. Wilson, On the Self-Similar Nature of Ethernet Traffic, IEEE Transactions on Networking, 2, 1, pp 1-15, Feb. 1994.

  • [pods94] Christos Faloutsos and Ibrahim Kamel, Beyond Uniformity and Independence: Analysis of R-trees Using the Concept of Fractal Dimension, PODS, Minneapolis, MN, May 24-26, 1994, pp. 4-13

Copyright: C. Faloutsos (2001)

Slide 112

References

  • [vldb95] Alberto Belussi and Christos Faloutsos, Estimating the Selectivity of Spatial Queries Using the `Correlation' Fractal Dimension Proc. of VLDB, p. 299-310, 1995

  • [vldb96] Christos Faloutsos, Yossi Matias and Avi Silberschatz, Modeling Skewed Distributions Using Multifractals and the `80-20 Law’ Conf. on Very Large Data Bases (VLDB), Bombay, India, Sept. 1996.

Copyright: C. Faloutsos (2001)

Slide 113

References

  • [vldb96] Christos Faloutsos and Volker Gaede Analysis of the Z-Ordering Method Using the Hausdorff Fractal Dimension VLD, Bombay, India, Sept. 1996

  • [sigcomm99] Michalis Faloutsos, Petros Faloutsos and Christos Faloutsos, What does the Internet look like? Empirical Laws of the Internet Topology, SIGCOMM 1999

Copyright: C. Faloutsos (2001)

Slide 114

References

  • [icde99] Guido Proietti and Christos Faloutsos, I/O complexity for range queries on region data stored using an R-tree International Conference on Data Engineering (ICDE), Sydney, Australia, March 23-26, 1999

  • [sigmod2000] Christos Faloutsos, Bernhard Seeger, Agma J. M. Traina and Caetano Traina Jr., Spatial Join Selectivity Using Power Laws, SIGMOD 2000

Copyright: C. Faloutsos (2001)

Slide 115

References

  • [PODS94] Faloutsos, C. and I. Kamel (May 24-26, 1994). Beyond Uniformity and Independence: Analysis of R-trees Using the Concept of Fractal Dimension. Proc. ACM SIGACT-SIGMOD-SIGART PODS, Minneapolis, MN.

  • [Traina+, SBBD’00] Traina, C., A. Traina, et al. (2000). Fast feature selection using the fractal dimension. XV Brazilian Symposium on Databases (SBBD), Paraiba, Brazil.

Copyright: C. Faloutsos (2001)


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