Very Natural Computing

Piotr Chrząstowski

Mimicking the nature

• Man always tried to learn from nature some fresh ideas.

• The nature rewarded man with many interesting and useful solutions.

• Sometimes it is quite worthy to look around and discover „inventions” ready to use.

Helicopter

Submarine

Polartec

Naps

Planes

How about algorithms?

• Is there any way to use the forces of nature in order to increase our computing abilities?

• Can we learn something just looking around us?

• Does nature compute anything? Or maybe computing is purely human attitude?

Natural Computing

• Genetic and evolutionary algorithms – using natural selection to find better solutions

• Quantum computing – using quantum mechanics to simulate nondeterminism

• Biological computing – DNA plays the role of a processor

• Neural computing – constructing artificial neural networks in order to mimic the learning process of human brain.

Very Natural Computing

• We will use pure forces of nature. No real algorithms will be needed.

• What is needed: a proper experiment setting, and physics will provide us with the solution.

Sorting

• Given n real numbers. List them from the largest to the smallest.

• Solution: Cut the appropriate length sticks and let them freely stand on the table. They are already sorted. What is needed is to take one after one from the tallest to the smallest.

• Gravity sorts in constant time! Only preparation of data and presenting the result takes O(n) time.

Convex hull

• Problem: Find the smallest polygon surrounding given set of points on Euclidean plane

• Solution: Draw the points on the plane and drive nails in perpendicularly one at each of the points. Use rubber stripe to surround the points. The polygon is formed.

• Again, regardless of the number of points given, it takes constant time for a rubber to determine the convex hull.

Jacob Steiner

• Jacob Steiner (1796-1863)

• Swiss mathematician• One of the greatest

geometers in the history of mathematics.

Steiner problem

• Given n points on the Euclidean plane. Span these points with the smallest amount of cable.

• Some extra points may be added, where cable segments meet. They are called Steiner points.

First attempts

Some improvements

Yet not the best...

The solution!

Shortest path joining vertices of a triangle

Discrete Steiner problem

• Find the shortest Steiner tree on a grid

• A lot of research has been done in this srea, but no satisfactory solution has been found.

• This is an NP-complete problem

What is known about Steiner trees?

• Edges are segments,• In each added point (Steiner point) exactly 3 edges

meet at angles 120º• There are at most n-2 Steiner points needed to span

optimally n given points.• Steiner problem is not compositional • The ratio between the total length of the optimal Steiner

tree and the minimum spanning tree (without additional points, easily computable for instance by Kruskal or Prim algorithms) is at least √3/2≈0.87. This result known as Gilbert and Pollak hypothesis from 1968 was proven as late as in 1991 by Zhu and Hwang.

Related 3D problem

• 3D version of Steiner problem: find the minimal surface that connects given set of points.

• Even for such simple shape as 12 edges of cube, the shape is extremely complex. It does not contain any single piece of plane. And in fact no rigorous proof is known that this known shape is minimal.

Cube

Octahedron

Octahedron (2)

Octahedron (3)

The best!

Brain

• Human brain is a very natural computer

• It solves many problems incredibly fast, and we often have no idea, how it does.

• It surprises us

Illusions -

Our brain is sometimes an unpredictable processor