IntroducingNatural Algorithms

A passionate affair

Nature and Numbers – Friends for ever!

I started it !

Describing nature with mathematics

The universe is written in the language of mathematics,

and its characters are triangles, circles, and other

geometric figures

Describing nature with computation

The universe is written in the language of Cellular

Automata, and its characters are tiny cells with discrete

states.

How aboutdescribing computation with natural phenomena?

… Let Nature be your teacher.

- Wordsworth

Natural laws can evoke “computational” effects

Soap bubbles minimize the amount of surface area for a given volume

Molecules of a metal arrange themselves in the minimal energystate, when annealed

Meandering rivers: ratio between actual length of rivers from

source to mouth and their direct length as the crow flies = π

Shortest path between cities

AA

B B (Source)(Source)

CC

DD

E E (Destination)(Destination)

ShortestShortestpathpath

As the source, destination nodes in the physical graph-model are stretched apart, the shortest path forms a straight line between them.

Water finds the same level in all limbs due to atmospheric pressure.

l1l2 l3 (l1 + l2 + l3 ) /

3

Average of quantities

Graph Connectivity Problem- Solved with an analog electrical network*

*A. Vergis, K. Steiglitz, B. Dickinson. The complexity of analog computation, Mathematics and Computers in Simulation 28, 1986, 91-113.

Given a graph G = (V, E) and two distinguished vertices s, t in V, is there a path in G from s to t?

s t

Vvolts

iamperes

Location Allocation Problem- Solved using motion of charged particles*

*R. Simha, W. Cai, V. Spitkovsky.Simulated N-body: A new particle physics-based heuristic for a Euclidean location allocation problem, J. Heuristics 7, 1 , 2001, 23-36.

1 3

4

2

5

resident home

police-station

Given a set of facilities, each of which can serve a certain number of nodes on a graph, the objective is to place the facilities on the graph so that the average distance between each node and the facility that serves it is minimized.

17310309456440 possible location allocations exist when 10 police stations have to be allocated in a city with 100 homes.

Location Allocation (cont’d)

boundary of repulsive force

charged particles that repel each other

Travelling Salesman Problem

Given a set of points, either nodes on a graph or cities on a map, the objective is to find the shortest possible tour that visits every single point exactly once and then returns to its starting point.

C

D

G

EFA

B

A B C D E F G A

A C B D E F G A

A C B E D F G A

.

.

.

No. of possible tours = (n-1)! / 2 for a (complete) graph with n vertices.For a 16-city problem, there are 653, 837, 184, 000 possible solutions.

Travelling Salesman Problem- Solved using interaction between charged particles*

a possible tourcities (negatively charged particles)

elastic band

positively charged particle

*Z. Zhang, S. Bai, G. Li. Emergent collective computational abilities in interacting particle systems,Evo Workshops, 1994, 61-72.

Maximum Clique ProblemA clique is a graph such that any two of its vertices are connected by an edge. For a given graph G with n vertices, the MCP asks for the maximum number k, such that G contains a subgraph of k vertices that forms a clique.

1

23

1

2

3

4

5

6

Maximum Clique Problem- Solved using liquid flow*

*D. T. Chiu, E. Pezzoli, H. Wu, A. D. Stroock, and G. M. Whitesides.Using three-dimensional microfluidic networks for solving computationally hard problems, Proceedings of the National Academy of Sciences USA 98, 2961 - 2966 (2001).

Subgraphs (wells)

(1,2)

(1,3)

(2,3)

{1,2}

{1,3}

{2,3}

{1,2,3}

Edges (reservoirs)

{}

{1}

{2}

{3}

1

23

Ring-Sort !

rings whose radii represent positive integers (dropped in any order)

rings fall down in sorted order

source

destination

The Maze Problem- Can it be solved using an electrical circuit?

What is a natural algorithm?(Prose version)

Traditionally, when computists solve problems, they try to achieve the desired end by painstakingly developing a suitable means---an “algorithm”.

On the other hand, when natural computists solve problems, they try to discover a natural (computing) system, one that is bound to produce the desired end (or something “close” to such an end) and whose capacity to produce such an end is innate. (That is, the system’s ability to reach the desired end is not something the computist deliberately assigns to it, but something which the system has been endowed with.)

The means by which natural systems realize an end is something that comes “for free”; the computist need not bother to know the exact means by which the system would achieve the desired end, but simply be aware of the fact that such an end will somehow be achieved.

Natural Algorithm: a “free” means to an anticipated end

What is a natural algorithm?(Poem version)

"What, my dear Sir, is a Natural Algorithm?"So asked Boswell."Bah, that is but a simple idea", said Dr. Johnson.An algorithm is nothing but a means,Not as hard as it seems;One which humans so meticulously design---And all that, my friend,Is for the computer---to achieve an 'end'.

A natural algorithm is also a means,But one that you get “for free”:All you need, my dear Boswell, is to seekFor when you seek, you shall findThat piece of nature's machinery which does what you wantBe it sorting, be it searching or solving SAT!It's right there, neat and clean---The end you seek;Just take a peek.

"But, Sir, by what means does nature reach its end?"Why bother, my dear Boswell,When nature does it well.The means is but free, andFor us (and for nature), it's the end that matters.What matters for starters,Though, is by one means or the other Will it reach its end!