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Is it a know problem this problem? (an extension of Nearest-Neighbor Problem)

I met a problem which can be considered as an extension of
Nearest-Neighbor Problem.

Suppose we have n (e.g. n = 10) robots and m (e.g., m = n = 10)
locations to explore in a 3-dimensional space. We want to assign each
robot to explore a possible location, while minimizing the summation of
the distances between each robot-location pair.

If n = 1 or m = 1, it is clear this is a nearest-neighbor problem. What
if both n and m are greater than 1 (n and m are not necessarily equal)?
Is it a known problem which has been addressed by some heuristic
algorithm?

Thanks.

0
jingqiao (1)
10/23/2006 11:53:20 PM
comp.graphics.algorithms 6674 articles. 1 followers. Post Follow

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This sounds more like an optimization problem.  You might ask this question in 
sci.math.num-analysis and sci.op-reesearch.

In article <1161647600.014183.289010@f16g2000cwb.googlegroups.com>, "Joe 
Zhang" <jingqiao@gmail.com> wrote:
>I met a problem which can be considered as an extension of
>Nearest-Neighbor Problem.
>
>Suppose we have n (e.g. n = 10) robots and m (e.g., m = n = 10)
>locations to explore in a 3-dimensional space. We want to assign each
>robot to explore a possible location, while minimizing the summation of
>the distances between each robot-location pair.
>
>If n = 1 or m = 1, it is clear this is a nearest-neighbor problem. What
>if both n and m are greater than 1 (n and m are not necessarily equal)?
>Is it a known problem which has been addressed by some heuristic
>algorithm?
>
>Thanks.
>
0
10/24/2006 12:06:52 PM
"Joe Zhang" <jingqiao@gmail.com> wrote in message 
news:1161647600.014183.289010@f16g2000cwb.googlegroups.com...
>I met a problem which can be considered as an extension of
> Nearest-Neighbor Problem.
>
> Suppose we have n (e.g. n = 10) robots and m (e.g., m = n = 10)
> locations to explore in a 3-dimensional space. We want to assign each
> robot to explore a possible location, while minimizing the summation of
> the distances between each robot-location pair.
>
> If n = 1 or m = 1, it is clear this is a nearest-neighbor problem. What
> if both n and m are greater than 1 (n and m are not necessarily equal)?
> Is it a known problem which has been addressed by some heuristic
> algorithm?

I assume you intend to have at most one robot exploring a
location and at most one location explored by a robot.  Since
the problem is symmetric in robots/location, you may assume
that n <= m.  One way to view this is to have a matrix M of
squared distances between robots and locations.  Each row
corresponds to a robot (use index R); each column corresponds
to a location (use index L).  Your goal is to choose pairs (R,L)
so that the sum of the matrix entries M(R,L) is minimized.  Once
you have chosen a pair (R,L), any other pair (R',L') must satisfy
the conditions R' is different from R and L' is different from L.

This has the flavor of dynamic programming.  If you have
assigned k < n robots to k of the m locations, the assignment
of the remaining n-k robots to the remaining m-k locations is
an optimization problem of the same type.  Perhaps a search
of books/literature on dynamic programming will find you an
algorithm.  It appears that to find the optimal solution, you
must evaluate all squared distances, so the algorithm is at
least order O(n*m).

--
Dave Eberly
http://www.geometrictools.com


0
dNOSPAMeberly (1228)
10/24/2006 1:51:58 PM
Joe Zhang <jingqiao@gmail.com> wrote:
> I met a problem which can be considered as an extension of
> Nearest-Neighbor Problem.

Not really.  It's closer to a variation of a graph theory problem
known as "matching".

> Suppose we have n (e.g. n = 10) robots and m (e.g., m = n = 10)
> locations to explore in a 3-dimensional space. We want to assign each
> robot to explore a possible location, while minimizing the summation of
> the distances between each robot-location pair.

As written, this is a little too vague.  You only really described
what you want for n equal to m.  What's supposed to happen if n is
larger or smaller than m?  Does each robot get to search more than one
location, or will some locations remain unvisited?  Does one location
get multiple visits, or will some robots remain unoccupied?

-- 
Hans-Bernhard Broeker (broeker@physik.rwth-aachen.de)
Even if all the snow were burnt, ashes would remain.
0
broeker (2903)
10/24/2006 6:58:04 PM
Joe Zhang wrote:
> I met a problem which can be considered as an extension of
> Nearest-Neighbor Problem.
> 
> Suppose we have n (e.g. n = 10) robots and m (e.g., m = n = 10)
> locations to explore in a 3-dimensional space. We want to assign each
> robot to explore a possible location, while minimizing the summation of
> the distances between each robot-location pair.
> 
> If n = 1 or m = 1, it is clear this is a nearest-neighbor problem. What
> if both n and m are greater than 1 (n and m are not necessarily equal)?
> Is it a known problem which has been addressed by some heuristic
> algorithm?

It is a known problem, but is is not "an extension of Nearest-Neighbor Problem". Not only you failed to described your problem fully (suprisingly as most persons in this newsgroup do), but also seem not to understand it really.

Even if you have 1 robot, it is is not a "nearest-neighbor problem". In this subcase (1 robot, m nodes) it is a classic "traveling salesman problem" with full graph, nodes or vertices are your locations, and there are ((m^2)-m)/2 of edges. Even for the one robot the problem is hard (the computational cost of finding otimal solution grows exponentially to the number of vertices) and you rather won't be able find optimal solution for larger numbers.

The nearest neighbour approach here is merely a very simple approximation algorithm, which is known as "greedy algorithm" or "Johnson's algorithm". I think in worse case it should give you solution something about log(o) times worse than optimal, where o is the cost of optimal solution.

Now if you have n robot and use your nearest neighbour approach and you want to minimize the sum of distances then algorithm is the same, except you choose the best node-robot (smallest distance) pair. But if you insist on sending all robots at once, you don't really want to minmize the sum of distances only, but also the time in which robots end all tasks (ie visit all locations), right?. So you have polyoptimalization problem which does not have optimal solution as such. You will have to define some ralation between time and distance, so you have some order relation defined between pairs of possible solutions (time,distance).

I would start with simple greedy algorithm:

while(there_are_location_with_work_to_do)
{
	if(is_at_least_one_robot_available)
	{
		choose best robot-location pair; //(closest distance)
		send robot to the location;
	}
	else
		wait_for_robot_to_report_it's_available;
}

If that's not enough and the number of location is small (really) you can try to solve multiple traveling salesman problem.

-- 
Grzegorz Wr�bel
http://www.4neurons.com/
677265676F727940346E6575726F6E732E636F6D
0
null30 (56)
10/25/2006 1:35:15 AM
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A team of astronomers using ground-based telescopes in Hawaii, California, and Arizona recently discovered a planetary system orbiting a nearby ...

Robotically discovering Earth's nearest neighbors: 54 light-years away
Astronomers discovered a planetary system orbiting a nearby star that is only 54 light-years away. All three planets orbit their star at a distance ...

Robotically discovering Earth's nearest neighbors
A team of astronomers using ground-based telescopes in Hawaii, California, and Arizona recently discovered a planetary system orbiting a nearby ...

NASA's WISE Mission Shuffles Sun's Nearest Stellar Neighbors
For decades, the ten closest stars to our own Sun made up a pretty static list. Ideas about where to send the first interstellar probes tended ...

Take a wild trip to our nearest stellar neighbor in Proxima
Set centuries from now on a planet orbiting Proxima Centauri, Stephen Baxter's Proxima is an epic hard scifi story with no shortage of futuristic ...

The Nearest Alien Neighbors: Habitable Planet 'Incredibly Close' On Earth Discovered
A possibly inhabitable planet near the planet earth had been discovered by an Australian team of scientists.

Take a wild trip to our nearest stellar neighbor in Proxima
Set centuries from now on a planet orbiting Proxima Centauri, Stephen Baxter's Proxima is an epic hard scifi story with no shortage of futuristic ...

Resources last updated: 3/8/2016 3:14:21 AM