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Metric Space `epsilon`-Nets and Their Applications
Ken Clarkson
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Outline
- Metric spaces, `epsilon`-nets, coverings, packings, Delone sets...
- Building nets
- Using nets:
- as roadmaps
- for nearest neighbor searching
- as mesh vertices for approximating curves (and surfaces)
- as triangulation vertices for approximating functions
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Metric Spaces
- A metric space is a collection of points `U` and a distance measure `D` so that:
- `D(x,y)=0 iff x=y`
- `D(x,y)=D(y,x)`
- `D(x,z) <= D(x,y) + D(y,z)`
- Examples:
- Nodes in a graph
- Pixels in an image
- Points on a surface
- Sets
- Functions
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Packings, Coverings, and Nets
- Subsets of the points that are "nicely distributed"
- `N subset U` is a:
- Covering if every point in `U` has some point in `N` nearby
- Packing if no pairs of points in `N` is too close together
- Net or Delone Set if `N` is both a covering and a packing
- An `epsilon`-net has covering and packing distances `epsilon`
- Delone `equiv` Delaunay `equiv` Делоне
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Building Nets: the Greedy Algorithm
- Nets exist: here's a simple greedy algorithm:
- Start with `N` containing any point of `U`
- Repeat until have `k` points:
- Pick the point in `U setminus N` farthest from `N`, add to `N`
random
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Nets and the Greedy Algorithm
- Optimal approximation algorithm, in a sense [Gonzalez]
- Within a factor of two of best possible
- Also called, for example:
- Bawden-Lajiness algorithm, in computational chemistry
- Farthest Point Sampling, in image processing [Eldar et al.]
- Due to Kolmogorov/Tikhonov?
- Typically `|N| = Theta(1//{:epsilon^d:})` as `epsilon -> 0`
- In `d`-space; so
`{:-log |N|:}//{:log epsilon:}`
estimates `d` [Kégl]
- `equiv` box dimension
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Nets as Roadmaps: Motion Planning
- Consider moving a small object (a point) among obstacles
- Want a plan to move from `a` to `b`
- By using more dimensions, can model more complicated motions and objects
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Nets as (Non)-Probabilistic Roadmaps
- Original Approach:
- Pick points in "free" space at random
- Find straight-line paths between nearby points
- Random points are nice, but
- `epsilon`-nets (low dispersion) are nicer
- Grid points seem to work ok, Sobol sets, etc.
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Nets as Roadmaps, Analysis
- Consider case where shortest path from `a` to `b` is a curve
- Suppose `N` is a `delta^2`-net, `delta ≤ 1`, `diam U = 1`
- Suppose every `x,y in N cup \{a,b\}` with `D(x,y)≤2delta` are connected
- Then this scheme has `2 delta ` overhead:
- Split path into segments of length `delta`
- Detour to `N` at each split, adds `≤ 1/delta (2delta^2) = 2delta`
- Using triangle inequality, path along `N` is not longer
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Nets as "Beacons" or "Landmarks"
- In networks: pick a net `N`
- To route from `a->b`:
- Let `hat a` be nearest to `a` in `N`
- Let `hat b` nearest to `b` in `N`
- Walk `a -> hat a -> hat b -> b
- Simpler addressing, simpler network
- Collection of distances to landmarks also used as a "coordinate system"
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Approximate Nearest Neighbor Searching
- Given `(U,D)`, `S subset U`, `diam S =1`
- Build a data structure so that for `q in U`, can find nearest `a in S`
- Divide-and-conquer approach for approximation algorithm:
- Pick `delta ≤ 1`, find `delta^2`-net `N subset S`
- Recursively build data structure for each `S_p := B(p, 3delta) cap S`, `p in N`
- To search: given `q`, find nearest `p in N`; recursively search `S_p`
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Approximate NN Searching, Analysis
- It can be shown that either:
- `D(q,a)>=delta`, and `p` itself within `1+delta` of nearest, or
- Nearest to `q` is in `S_p = B(p, 3delta) cap S`
- At level `k`, balls of size `(3delta)^k`; depth bounded for small "spread"
- Proof:
- Suppose `a` is nearest to `q` in `S`
- Suppose `p_a` is nearest to `a` in `N`
- We have `D(q,p) ≤ D(q,p_a) ≤ D(q,a) + D(a,p_a) ≤ D(q,a) + delta^2`
- If `D(q,a) >= delta`, then `p` is ok; otherwise
`D(a,p)
≤ D(a,q) + D(q,p)
≤ 2D(a,q) + delta^2
≤ 2delta + delta^2 ≤ 3delta`
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Nets as Vertices in Approximations
- Polygonal approximation of smooth curves:
- Pick a distance measure for points on curve
- Pick an `epsilon`-net using that distance
- Connect the dots
- Due to McClure/Vitale, 1975; for `d>1`, by Gruber, Schneider,...
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What Distance Measure?
- Could use arc length, but misses large curvature
- Could use turning angle, but misses scale
- Sum of both works ok
- Best may be `{:sqrt {:kappa(x):}:}` as a weight for distance
- `kappa(x)` is the curvature
Arc Length
Angle Only
Both
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Nets as Vertices of Graded Triangulations
- Given a region and "walls", split into triangles so that:
- Triangles are close to equilateral
- Triangle edges don't cross the walls
- Triangles must be small near small features, get bigger gradually
- Implies a "local feature size" `F(x)` at point `x`
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A Metric Space Scaled to Local Feature Size
- For "nice" functions `F` on `U`, there is a metric $D_F$ so that:
- `D_F(x,y) approx {:D(x,y):}//{:F(x):}` when `D_F` is small
- Greedy algorithm then `approx` Chew/Ruppert algorithms
- Metric is:
`D_F(x,y) := min { 1, {: 2D(x,y):}/{:D(x,y) + F(x) + F(y):} }`
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Building a triangulation
Initial
Edges
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The Steinhaus Transform
- `D_F` is proven to be a metric using the Steinhaus (biotope) transform
- For `(U,D)` and `a in U`,
`hat D(x,y) := {:2D(x,y):} / {:D(x,a) + D(y,a) + D(x,y):}
is also a metric
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Steinhaus Transform for Sets
- Suppose `U` is all subsets of a set, `D(A,B) := |A Delta B|`
- For `a=O/`, biotope transform is
`hat D(A,B) = {:|A Delta B|:} / {:|A uu B|:}`
- Called the Jaccard or Tanimoto or Steinhaus distance, or resemblance
- Used in information retrieval, ecology, comp. chemistry, web page comparison
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Scaling by Distance to Boundary
- For `T subset U`, replacing `D(x,a) + D(y,a)` by `min_{a in T} D(x,a) + D(y,a)` also a metric
- Using a circle as `T`, space is `approx` Poincaire disk model of hyperbolic space:
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`epsilon`-nets vs. RMS
- `D(p, N) := min_{p' in N} D(p,p')`
- While `epsilon`-net `N` minimizes
`max_{p in U} D(p,N)`
- How about minimizing
`sum_{p in U} D(p,N)^2` ?
- In the smooth case, `epsilon`-nets are also pretty good for the latter
- Implies results for optimum vector quantization, quadrature [Gruber]
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Concluding Remarks
- `epsilon`-nets of functions used in learning theory
- Size of net is doubly exponential in domain dimension
- In surface reconstruction: when `T` is the medial axis,
net in scaled distance `D_T` implies provably good results [AB]
- In motion planning, scaling by distance to boundary helps find a short path with good clearance
- Having a "doubling constant" means that all subsets of `U` have small nets
- Thanks!