introduction to fractals and fractal dimension
Download
Skip this Video
Download Presentation
Introduction to Fractals and Fractal Dimension

Loading in 2 Seconds...

play fullscreen
1 / 115

Introduction to Fractals and Fractal Dimension - PowerPoint PPT Presentation


  • 655 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Introduction to Fractals and Fractal Dimension' - zamora


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
intro to fractals outline
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)

problem 1 gis points
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)

problem 2 spatial d m
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)

problem 3 traffic

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)

common answer for all three
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)

road map
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)

definitions cont d
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)

what is a fractal
What is a fractal?

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

zero area;

infinite length!

...

Copyright: C. Faloutsos (2001)

definitions cont d10
Definitions (cont’d)
  • Paradox: Infinite perimeter ; Zero area!
  • ‘dimensionality’: between 1 and 2
  • actually: Log(3)/Log(2) = 1.58…

Copyright: C. Faloutsos (2001)

definitions cont d11
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)

intrinsic fractal dimension
Q: fractal dimension of a line?

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

Intrinsic (‘fractal’) dimension

Copyright: C. Faloutsos (2001)

intrinsic fractal dimension14
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)

intrinsic fractal dimension15
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)

dfn of fd
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)

sierpinsky triangle

log(#pairs

within <=r )

1.58

log( r )

Sierpinsky triangle

== ‘correlation integral’

Copyright: C. Faloutsos (2001)

observations
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)

important properties
fd = embedding dimension -> uniform pointset

a point set may have several fd, depending on scale

Important properties

Copyright: C. Faloutsos (2001)

road map20
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)

problem 1 gis points21
Problem #1: GIS points
  • Cross-roads of Montgomery county:
  • any rules?

Copyright: C. Faloutsos (2001)

solution 1

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)

solution 123

1.51

Solution #1

A: self-similarity

  • avg#neighbors(<= r ) ~ r^(1.51)

log(#pairs(within <= r))

log( r )

Copyright: C. Faloutsos (2001)

examples mg county
Examples:MG county
  • Montgomery County of MD (road end-points)

Copyright: C. Faloutsos (2001)

examples lb county
Examples:LB county
  • Long Beach county of CA (road end-points)

Copyright: C. Faloutsos (2001)

solution 2 spatial d m
Solution#2: spatial d.m.

Galaxies ( ‘BOPS’ plot - [sigmod2000])

log(#pairs)

log(r)

Copyright: C. Faloutsos (2001)

solution 2 spatial d m27
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)

spatial d m
spatial d.m.

log(#pairs within <=r )

- 1.8 slope

- plateau!

- repulsion!

ell-ell

spi-spi

spi-ell

log(r)

Copyright: C. Faloutsos (2001)

spatial d m29

r1

r2

r2

r1

spatial d.m.

Heuristic on choosing # of clusters

Copyright: C. Faloutsos (2001)

spatial d m30
spatial d.m.

log(#pairs within <=r )

- 1.8 slope

- plateau!

- repulsion!

ell-ell

spi-spi

spi-ell

log(r)

Copyright: C. Faloutsos (2001)

spatial d m31
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)

solution 3 traffic

#bytes

time

Solution #3: traffic
  • disk traces: self-similar:

Copyright: C. Faloutsos (2001)

solution 3 traffic33

20%

80%

Solution #3: traffic
  • disk traces (80-20 ‘law’ = ‘multifractal’)

#bytes

time

Copyright: C. Faloutsos (2001)

solution 3 traffic34
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)

tape accesses

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)

tape accesses36

Tape#1

Tape# N

time

Tape accesses

50-50 = Poisson

# tapes retrieved

real

# qual. records

Copyright: C. Faloutsos (2001)

fast estimation of fd s
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)

definitions
Definitions
  • pi : the percentage (or count) of points in the i-th cell
  • r: the side of the grid

Copyright: C. Faloutsos (2001)

fast estimation of fd s39
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)

fast estimation of fd s40
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)

hausdorff or box counting fd
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)

definitions cont d42
Definitions (cont’d)
  • Hausdorff fd:

r

log(#non-empty cells)

D0

log(r)

Copyright: C. Faloutsos (2001)

observations43
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)

observations cont d
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)

examples mg county45
Examples:MG county
  • Montgomery County of MD (road end-points)

Copyright: C. Faloutsos (2001)

examples lb county46
Examples:LB county
  • Long Beach county of CA (road end-points)

Copyright: C. Faloutsos (2001)

summary so far
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)

dimensionality reduction with fd
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)

dim reduction w fractals
Dim. reduction - w/ fractals

not informative

Copyright: C. Faloutsos (2001)

dim reduction
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)

dim reduction51
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)

dim reduction w fractals52
Dim. reduction - w/ fractals

global FD=1

PFD=1

PFD~0

Copyright: C. Faloutsos (2001)

dim reduction w fractals53
Dim. reduction - w/ fractals

global FD=1

PFD=1

PFD=1

Copyright: C. Faloutsos (2001)

dim reduction w fractals54
Dim. reduction - w/ fractals

global FD=1

PFD~1

PFD~1

Copyright: C. Faloutsos (2001)

dim reduction w fractals56
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)

dim reduction w fractals57
(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)

dim reduction w fractals59
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)

dim reduction w fractals60
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)

e g on the currency dataset
E.g., on the ‘currency’ dataset

log(#pairs(<=r))

correlation integral

1.98

log(r)

Copyright: C. Faloutsos (2001)

e g on the eigenface dataset
E.g., on the eigenface dataset

16-d vectors,

one for each

of ~1K faces

4.25

Copyright: C. Faloutsos (2001)

slide64

E.g., on the eigenface dataset

Copyright: C. Faloutsos (2001)

dim reduction w fractals65

global FD=1

PFD~1

PFD~1

Dim. reduction - w/ fractals
  • Conclusion:
  • can do non-linear dim. reduction

Copyright: C. Faloutsos (2001)

more tools
More tools
  • Zipf’s law
  • Korcak’s law / “fat fractals”

Copyright: C. Faloutsos (2001)

a famous power law zipf s law
A famous power law: Zipf’s law
  • Q:vocabulary word frequency in a document - any pattern?

freq.

aaron

zoo

Copyright: C. Faloutsos (2001)

a famous power law zipf s law68
A famous power law: Zipf’s law

log(freq)

“a”

  • Bible - rank vs frequency (log-log)

“the”

log(rank)

Copyright: C. Faloutsos (2001)

a famous power law zipf s law69
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)

a famous power law zipf s law70
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)

olympic medals sidney
Olympic medals (Sidney):

log(#medals)

rank

Copyright: C. Faloutsos (2001)

more power laws areas korcak s law
More power laws: areas – Korcak’s law

Scandinavian lakes

Any pattern?

Copyright: C. Faloutsos (2001)

more power laws areas korcak s law73
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)

more power laws korcak
More power laws: Korcak

Japan islands

Copyright: C. Faloutsos (2001)

more power laws korcak75
More power laws: Korcak

log(count( >= area))

Japan islands;

area vs cumulative count (log-log axes)

log(area)

Copyright: C. Faloutsos (2001)

korcak s law aegean islands
(Korcak’s law: Aegean islands)

Copyright: C. Faloutsos (2001)

korcak s law fat fractals
Korcak’s law & “fat fractals”

How to generate such regions?

Copyright: C. Faloutsos (2001)

korcak s law fat fractals78
Korcak’s law & “fat fractals”

Q: How to generate such regions?

A: recursively, from a single region

Copyright: C. Faloutsos (2001)

conclusions
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)

fractals overall conclusions
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)

nn queries
NN queries
  • Q: What about L1, Linf?
  • A: Same slope, different intercept

log(#neighbors)

log(d)

Copyright: C. Faloutsos (2001)

practitioner s guide

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)

practitioner s guide83

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)

practitioner s guide84

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)

resources
Resources:

Copyright: C. Faloutsos (2001)

books
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)

references
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)

road map88
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)

nn queries89
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)

nn queries90
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)

nn queries91
NN queries
  • Q: What about L1, Linf?

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

D2

L2

r

log(d)

Copyright: C. Faloutsos (2001)

nn queries92
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)

nn queries93

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)

nn queries94
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)

nn queries95
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)

nn queries96
NN queries

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

Copyright: C. Faloutsos (2001)

nn queries97
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)

internet topology

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)

more power laws on the internet

-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)

more power laws internet

-2.2

More power laws - internet
  • pdf of degrees: (slope: 2.2 )

Log(count)

Log(degree)

Copyright: C. Faloutsos (2001)

even more power laws on the internet
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)

more apps brain scans

Log(#octants)

2.63 = fd

octree levels

More apps: Brain scans
  • Oct-trees; brain-scans

Copyright: C. Faloutsos (2001)

more apps medical images
More apps: Medical images

[Burdett et al, SPIE ‘93]:

  • benign tumors: fd ~ 2.37
  • malignant: fd ~ 2.56

Copyright: C. Faloutsos (2001)

more fractals

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)

more power laws
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)

even more power laws
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)

power laws cont ed
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)

resources109
Resources:

Copyright: C. Faloutsos (2001)

books110
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)

references111
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)

references112
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)

references113
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)

references114
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)

references115
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)

ad