Finding equitable convex partitions of points and applications
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Finding Equitable Convex Partitions of Points and Applications Benjamin Armbruster, John Gunnar Carlsson, Yinyu Ye Research supported by Boeing and NSF; we also like to thank Arroyo, Ge, Mattikalli, Mitchell, and So for providing us valuable references and comments. Problem Statement

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Finding equitable convex partitions of points and applications l.jpg

Finding Equitable Convex Partitions of Points and Applications

Benjamin Armbruster, John Gunnar Carlsson, Yinyu Ye

Research supported by Boeing and NSF; we also like to thank Arroyo, Ge, Mattikalli, Mitchell, and So for providing us valuable references and comments.


Problem statement l.jpg
Problem Statement Applications

n points are scattered inside a convex polygon P (in 2D) with m vertices. Does there exist a partition of P into n sub-regions satisfying the following:

  • Each sub-region is convex

  • Each sub-region contains one point

  • All sub-regions have equal area


Related problem voronoi diagram l.jpg
Related Problem: Voronoi Diagram Applications

In the Voronoi Diagram, we satisfy the first two properties (each sub-region is convex and contains one point), but the sub-regions have different areas.


Our result l.jpg
Our Result Applications

Not only such an equitable partition always exists, but also we can find it exactly in running time O(Nn log N), where N = m + n.


Usa example l.jpg
USA Example Applications


Usa example7 l.jpg
USA Example Applications


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USA Example Applications


Motivation client server network l.jpg
Motivation: Client/Server Network Applications

This problem has applications in heuristic methods for what we call the Broadcast Network class of problems, in which we connect a set of clients to a set of servers, using a fixed underlying network topology.

Example: Multi-Depot Vehicle Routing Problem (MDVRP).

Definition: A set of vehicles located at depots in the plane must visit a set of customers such that the maximum TSP cost is minimized (min-max MDVRP).


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Minimum Spanning Forest Applications

Definition: Find the spanning forest of a node set with fixed roots for which the maximum tree length is minimized.


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Why Equal Area? Applications

  • A well-known combinatorial result, says that the length of an optimal TSP tour in a service region with uniformly distributed points depends only on the area of the region, asymptotically speaking.

  • Moreover, the locations of clients are changing

  • The same can be said of an MST.


Why equal area12 l.jpg
Why Equal Area? Applications

Using the result, we obtain an asymptotically optimal solution for min-max MDVRP with the following algorithm:

  • Create an equal-area partition containing one depot in each sub-region.

  • Solve a TSP problem in each subregion, visiting all clients plus the depot.

    Asymptotically speaking, the load on each vehicle will be equal

    A similar result holds for the MSF variant


Why convexity l.jpg
Why Convexity? Applications

  • Ensures that any route between two points is self-contained in the sub-region

  • Substructures have no overlap

  • Client can be reached by straight line from the sever


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Why Fast Algorithm? Applications

  • Servers may not be stationary either

  • Region of interested is changing and reshaping

    Thus, need to do repartition in real time


Previous work ham sandwich theorem l.jpg
Previous Work: Ham Sandwich Theorem Applications

“The volumes of any n solids of dimension n can always be simultaneously bisected by an (n – 1) dimensional hyperplane” [Steinhaus 1938]

Corollary of Borsuk-Ulam theorem (1933): “any continuous function from an n-sphere into Rn maps some pair of antipodal points to the same point”


Previous work discrete set partition l.jpg
Previous Work: Discrete Set Partition Applications

[Bespamyatnikh, Kirkpatrick, Snoeyink 2000] and [Ito, Uehara, Yokoyama 1998] address a similar problem:

“Given gn red points and gm blue points in the plane in general position, find a subdivision of the plane into g disjoint convex polygons, each of which contains n red points and m blue points.”

We can find an approximate solution to our original problem by filling our original polygon uniformly with (for example) red points. This would give a proof of existence but a poor approximation algorithm.


Tool intermediate value theorem l.jpg
Tool: Intermediate Value Theorem Applications

“If we can find a half-space satisfying property X that cuts off too much area (too many points) and another half-space satisfying property X that cuts off too little area (too few points), then there exists a half-space satisfying property X that cuts off the correct area (correct number of points)”

e.g. Property X: “half-space must cut off point p”


The algorithm divide and conquer l.jpg
The Algorithm: Divide-and-Conquer Applications

Uses a “divide-and-conquer” approach, dividing the initial region into smaller regions

At each iteration, we have, for all subregions Ri, Rj


Divide and conquer l.jpg
Divide-and-Conquer Applications

Uses a “divide-and-conquer” approach, dividing the initial region into smaller regions

At each iteration, we have, for all subregions Ri, Rj


Divide and conquer20 l.jpg
Divide-and-Conquer Applications

Uses a “divide-and-conquer” approach, dividing the initial region into smaller regions

At each iteration, we have, for all subregions Ri, Rj


Divide and conquer21 l.jpg
Divide-and-Conquer Applications

Uses a “divide-and-conquer” approach, dividing the initial region into smaller regions

At each iteration, we have, for all subregions Ri, Rj


Divide and conquer22 l.jpg
Divide-and-Conquer Applications

Uses a “divide-and-conquer” approach, dividing the initial region into smaller regions

At each iteration, we have, for all subregions Ri, Rj


Divide and conquer23 l.jpg
Divide-and-Conquer Applications

Uses a “divide-and-conquer” approach, dividing the initial region into smaller regions

At each iteration, we have, for all subregions Ri, Rj


Definition convex equitable 2 and 3 partitions l.jpg
Definition: Convex Equitable 2- and 3-Partitions Applications

Claim: A convex equitable 2- or 3-partition always exists

We then perform this recursively


Helper lemma 1 ham sandwich l.jpg
Helper Lemma 1: Ham Sandwich Applications

If n is even, we can construct a Ham Sandwich Cut, i.e. a 2-partition that cuts the point set and the polygon in half: [n/2,n/2] 2-partition

Odd extension: If n = 2q + 1 and R contains q points, then if R is too small we can construct a [q,q+1] 2-partition


Helper lemma 2 one point cut l.jpg
Helper Lemma 2: One-point cut Applications

If we can cut off exactly one point with a region that is too small, then we can construct an equitable 2-partition.

Two cases:


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Helper Lemma 2: One-point cut, case 1 Applications

If we can cut off exactly one point with a region that is too small, then we can construct an equitable [1,n-1] 2-partition.


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Helper Lemma 2: One-cut, case 2 Applications

If we can cut off exactly one point with a region that is too small, then we can construct an equitable 2-partition.


Region partition algorithm l.jpg
Region Partition Algorithm Applications

  • If n even, compute a ham-sandwich cut by Helper Lemma 1

  • If n odd,

    • try to use Helper Lemma 1 for a ham-sandwich cut;

    • If this fails, try to use Helper Lemma 2 for a 2-partition;

    • If all these fail, build a 3-partition.






Case 1 l.jpg
Case 1 Applications


Case 2 l.jpg
Case 2 Applications


Case 3 l.jpg
Case 3 Applications


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Torture tests Applications

(MATLAB examples)


Extensions l.jpg
Extensions Applications

  • Nonuniform density μ

    • Polyhedra (in 3-D)

    • Non-convex regions


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Future Work Applications

  • 3-dimensional partitioning

    • Theorem says that three sets (e.g. one polyhedron, two point sets) can be simultaneously partitioned in a polygon

  • Diameter-constrained bicriteria partition

    • Avoid skinny subregions


Final note region based previously adapted 58 vehicle tours total 5580 miles l.jpg
Final Note: Region Based (previously adapted) Applications 58 vehicle-tours, total 5580 miles


Final note equitable area based 32 vehicle tours total 4345 miles l.jpg
Final Note: Equitable-”Area” Based Applications 32 vehicle-tours, total 4345 miles


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