Fractal Dimension and Lower Bounds for Geometric Problems - - PowerPoint PPT Presentation

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Fractal Dimension and Lower Bounds for Geometric Problems - - PowerPoint PPT Presentation

Oct 21, 2023 606 likes •1.06k views

Fractal Dimension and Lower Bounds for Geometric Problems Anastasios Sidiropoulos (University of Illinois at Chicago) Kritika Singhal (The Ohio State University) Vijay Sridhar (The Ohio State University) 1 / 18 The curse of dimensionality

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Fractal Dimension and Lower Bounds for Geometric Problems

Anastasios Sidiropoulos (University of Illinois at Chicago) Kritika Singhal (The Ohio State University) Vijay Sridhar (The Ohio State University)

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The curse of dimensionality

  • Computational complexity of geometric problems increases

with the dimension of the input point set.

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The curse of dimensionality

  • Computational complexity of geometric problems increases

with the dimension of the input point set.

  • Many types of dimension, e.g. Euclidean dimension, doubling

dimension etc.

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Fractals

  • How does fractal dimension affect algorithmic complexity?

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Fractals

  • How does fractal dimension affect algorithmic complexity?
  • Fractals are ubiquitous in nature.

Figure: Fractal arrangement of atoms in Cu46Zr54 (left), and lightning bolts (right).

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Fractal Dimension

Several notions of fractal dimension:

  • Hausdorff dimension
  • Box-counting dimension
  • Information dimension
  • ...

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Fractal dimension and volume

  • Fractal dimension δ if scaling by a factor of r > 0 increases

the total ”volume” by a factor of rδ.

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Fractal dimension and volume

  • Fractal dimension δ if scaling by a factor of r > 0 increases

the total ”volume” by a factor of rδ.

  • Sierpi´

nski carpet has fractal dimension log3 8, since scaling by a factor of 3 increases the volume by a factor of 8.

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Fractal dimension of discrete sets

  • Most definitions of fractal dimension are meaningless for

countable sets.

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Fractal dimension of discrete sets

  • Most definitions of fractal dimension are meaningless for

countable sets.

  • E.g. the Hausdorff dimension of any countable set is zero.

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A definition of fractal dimension for discrete sets

Given a pointset X ⊂ Rd, the fractal dimension of X, denoted by dimf(X), is the infimum over all δ > 0 such that for all x ∈ Rd, for all ǫ > 0, r ≥ 2ǫ and for all ǫ-nets N of X, we have |N ∩ ball(x, r)| = O((r/ǫ)δ) ǫ-net - Maximal N ⊆ X such that for all x, x′ ∈ N, x = x′, dX(x, x′) > ǫ.

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Examples

  • For all X ⊆ Rd, dimf(X) ≤ d.

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Examples

  • For all X ⊆ Rd, dimf(X) ≤ d.
  • dimf({1, . . . , n1/d}d) = d.

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Examples

  • For all X ⊆ Rd, dimf(X) ≤ d.
  • dimf({1, . . . , n1/d}d) = d.
  • Fractal dimension of discrete Sierpi´

nski carpet is log3 8 (left below), and of discrete Cantor crossbar is log3 6 (right below).

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k-Independent Set of Unit Balls

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k-Independent Set of Unit Balls

  • No f (k)no(k1−1/d) algorithm exists, assuming the Exponential

Time Hypothesis (ETH) [Marx, Sidiropoulos’14].

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k-Independent Set of Unit Balls

  • No f (k)no(k1−1/d) algorithm exists, assuming the Exponential

Time Hypothesis (ETH) [Marx, Sidiropoulos’14].

  • If the set of centers has fractal dimension δ > 1, solvable in

time nO(k1−1/δ log n) [Sidiropoulos, Sridhar’17].

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k-Independent Set of Unit Balls

  • No f (k)no(k1−1/d) algorithm exists, assuming the Exponential

Time Hypothesis (ETH) [Marx, Sidiropoulos’14].

  • If the set of centers has fractal dimension δ > 1, solvable in

time nO(k1−1/δ log n) [Sidiropoulos, Sridhar’17].

  • If the set of centers has fractal dimension δ > 1, assuming

ETH, no f (k)no(k1−1/(δ−ǫ)) algorithm exists, for all ǫ > 0 [Sidiropoulos, S., Sridhar’18].

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Euclidean Traveling Salesperson Problem

Problem - Given a set of n points in Rd, find a closed tour of shortest length that visits all the points.

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Euclidean Traveling Salesperson Problem

Problem - Given a set of n points in Rd, find a closed tour of shortest length that visits all the points.

  • Assuming ETH, no 2O(n1−1/d−ǫ) algorithm exists, for all ǫ > 0

[Marx, Sidiropoulos’14].

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Euclidean Traveling Salesperson Problem

Problem - Given a set of n points in Rd, find a closed tour of shortest length that visits all the points.

  • Assuming ETH, no 2O(n1−1/d−ǫ) algorithm exists, for all ǫ > 0

[Marx, Sidiropoulos’14].

  • For pointsets of fractal dimension δ > 1, solvable in time

2O(n1−1/δ log n) [Sidiropoulos, Sridhar’17].

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Euclidean Traveling Salesperson Problem

Problem - Given a set of n points in Rd, find a closed tour of shortest length that visits all the points.

  • Assuming ETH, no 2O(n1−1/d−ǫ) algorithm exists, for all ǫ > 0

[Marx, Sidiropoulos’14].

  • For pointsets of fractal dimension δ > 1, solvable in time

2O(n1−1/δ log n) [Sidiropoulos, Sridhar’17].

  • For pointsets of fractal dimension δ > 1, assuming ETH, no

2O(n1−1/(δ−ǫ)) algorithm exists, for all ǫ > 0 [Sidiropoulos, S., Sridhar’18].

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Central Idea

Construct fractal pointsets such that any O(1)-spanner has large treewidth. Large treewidth implies presence of a large grid minor. Large grid minor embeds a large hard instance in the input. Running time lower bounds for geometric problems.

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Lower bound on treewidth of spanners

For a metric space (X, ρ), a c-spanner is a graph G = (X, E) such that for all x, x′ ∈ X, ρ(x, x′) ≤ dG(x, x′) ≤ c · ρ(x, x′).

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Lower bound on treewidth of spanners

For a metric space (X, ρ), a c-spanner is a graph G = (X, E) such that for all x, x′ ∈ X, ρ(x, x′) ≤ dG(x, x′) ≤ c · ρ(x, x′).

  • For every ǫ > 0, pointsets of fractal dimension δ > 1 admit

(1 + ǫ)-spanner with treewidth O(n1−1/δ log n) [Sidiropoulos, Sridhar’17].

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Lower bound on treewidth of spanners

For a metric space (X, ρ), a c-spanner is a graph G = (X, E) such that for all x, x′ ∈ X, ρ(x, x′) ≤ dG(x, x′) ≤ c · ρ(x, x′).

  • For every ǫ > 0, pointsets of fractal dimension δ > 1 admit

(1 + ǫ)-spanner with treewidth O(n1−1/δ log n) [Sidiropoulos, Sridhar’17].

  • For every ǫ > 0 and δ > 1, there exists X ⊂ Rd with |X| = n

such that dimf(X) ≤ δ, and any c-spanner of X has treewidth Ω

  • n1−1/(δ−ǫ)

cd−1

  • [Sidiropoulos, S., Sridhar’18].

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First attempt: Sierpi´ nski carpet

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First attempt: Sierpi´ nski carpet

  • Discrete Sierpi´

nski carpet (ǫ-net).

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First attempt: Sierpi´ nski carpet

  • Discrete Sierpi´

nski carpet (ǫ-net).

  • δ = log3 8.

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First attempt: Sierpi´ nski carpet

  • Discrete Sierpi´

nski carpet (ǫ-net).

  • δ = log3 8.
  • treewidth(G) = O(n1−1/δ−0.01).

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First attempt: Sierpi´ nski carpet

  • Discrete Sierpi´

nski carpet (ǫ-net).

  • δ = log3 8.
  • treewidth(G) = O(n1−1/δ−0.01).

→

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First attempt: Sierpi´ nski carpet

  • Discrete Sierpi´

nski carpet (ǫ-net).

  • δ = log3 8.
  • treewidth(G) = O(n1−1/δ−0.01).

→

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A treewidth extremal fractal: Cantor crossbar

The Cantor crossbar in R2: = R1(CS × [0, 1]) ∪ R2(CS × [0, 1]) The Cantor set (CS):

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A treewidth extremal fractal: Cantor crossbar

The Cantor dust (CDd):

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A treewidth extremal fractal: Cantor crossbar

The Cantor dust (CDd): The Cantor crossbar in Rd: R1(CDd−1 × [0, 1]) ∪ . . . ∪ Rd(CDd−1 × [0, 1])

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The Cantor crossbar

  • We observe that the point set X obtained has a fixed fractal

dimension δ, for each fixed d ≥ 2.

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The Cantor crossbar

  • We observe that the point set X obtained has a fixed fractal

dimension δ, for each fixed d ≥ 2.

  • We can generalize the above construction so that δ takes any

desired value in the range (1, d).

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The Cantor crossbar

  • We observe that the point set X obtained has a fixed fractal

dimension δ, for each fixed d ≥ 2.

  • We can generalize the above construction so that δ takes any

desired value in the range (1, d).

  • In the definition of Cantor dust, we start with a Cantor set of

smaller dimension. This can be done by removing the central interval of length α ∈ (0, 1), instead of 1

3, and recursing on

the remaining two intervals of length 1−α

2 .

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From treewidth to running time lower bounds

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From treewidth to running time lower bounds

Efficient NP-Hardness reductions, using gadgets, as in the NP- Hardness proof of TSP in R2 by Papadimitriou.

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From treewidth to running time lower bounds

Efficient NP-Hardness reductions, using gadgets, as in the NP- Hardness proof of TSP in R2 by Papadimitriou. Arrange gadgets along a Cantor crossbar.

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Conclusions

  • Running time lower bounds for the following problems on

fractal pointsets:

  • k-independent set of unit balls.
  • Euclidean TSP

The lower bounds obtained nearly match the existing upper bounds, assuming ETH.

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Conclusions

  • Running time lower bounds for the following problems on

fractal pointsets:

  • k-independent set of unit balls.
  • Euclidean TSP

The lower bounds obtained nearly match the existing upper bounds, assuming ETH.

  • Connection between treewidth of spanners and computational
  • complexity. This might have a generalization.

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Conclusions

  • Running time lower bounds for the following problems on

fractal pointsets:

  • k-independent set of unit balls.
  • Euclidean TSP

The lower bounds obtained nearly match the existing upper bounds, assuming ETH.

  • Connection between treewidth of spanners and computational
  • complexity. This might have a generalization.

Thank You.

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