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OK, so you have some historic data in the form of [say] an array of integers. This, for example, could represent free-space on a server HDD over a two-year period, with each array element representing a daily sample.
The data (free-space in this example) has a downward trend, but also has periodic positive spikes where files have been removed/compressed, Etc.
How would you go about identifying the overall trend for the two-year period, i.e.: iron out the peaks and troughs in the data?
Now, I did A-level statistics and then a stats module in my degree, but I've slept over 7,000 times since then, and well, it's leaked out of my brain.
I'm not after a bit of code as such, more of a description of how you'd approach this problem...
Thanks in advance!
You'll get many different answers, and the one you choose really depends on more specific requirements you may have. Examples:
Low-pass filter, or any other spectral analysis technique, and use the low frequencies to determine trend.
Linear regression (time/value) to find "r" (the correlation between time and the value).
Moving average of last "n" samples. If "n" is large enough this is my favorite as many times this is sufficient, and is very easy to code. It's a sort of approximation to #1 above.
I'm sure they'll be others.
If I was doing this to produce a line through points for me to look at, I would probably use a some variant of Loess, described at http://en.wikipedia.org/wiki/Local_regression, http://stat.ethz.ch/R-manual and /R-patched/library/stats/html/loess.html. Basically, you find the smoothed value at any particular point by doing a weighted regression on the data points near that point, with the nearest points given the most weight.
Background: I want to create a weather service, and since most available APIs limit the number of daily calls, I want to divide the planet in a thousand or so areas.
Obviously, internet users are not uniformly distributed, so the sampling should be finer around densely populated regions.
How should I go about implementing this?
Where can I find data regarding geographical internet user density?
The algorithm will probably be something similar to k-means. However, implementing it on a sphere with oceans may be a bit tricky. Any insight?
Finally, maybe there is a way I can avoid doing all of this?
Very similar to k-mean is the centroidal Voronoi diagram (it is the continuous version of k-means). However, this would produce a uniform tesselation of your sphere that does not account for user density as you wish.
So a similar solution is the same technique but used with a Power Diagram : a Power Diagram is a Voronoi Diagram that accounts for a density (by assigning a weight to each Voronoi seed). Such diagram can be computed using an embedding in a 3D space (instead of 2D) that consists of the first two (x,y) coordinates plus a third one which is the square root of [any large positive constant minus the weight for the given point].
Using that, you can obtain a tesselation of your domain accounting for a user density.
You don't care about internet user density in general. You care about the density of users using your service - and you don't care where those users are, you care where they ask about. So once your site has been going for more than a day you can use the locations people ask about the previous day to work out what the areas should be for the next day.
Dynamic programming on a tree is easy. What I would do for an algorithm is to build a tree of successively more finely divided cells. More cells mean a smaller error, because people get predictions for points closer to them, and you can work out the error, or at least the relative error between more cells and fewer cells. Starting from the bottom up work out the smallest possible total error contributed by each subtree, allowing it to be divided in up to 1,2,3,..N. ways. You can work out the best possible division and smallest possible error for each k=1..N for a node by looking at the smallest possible error you have already calculated for each of its descendants, and working out how best to share out the available k divisions between them.
I would try to avoid doing this by thinking of a different idea. Depending on the way you look at life, there are at least two disadvantages of this:
1) You don't seem to be adding anything to the party. It looks like you are interposing yourself between organizations that actually make weather forecasts and their clients. Organizations lose direct contact with their clients, which might for instance lose them advertising revenue. Customers get a poorer weather forecast.
2) Most sites have legal terms of service, which must clients can ignore without worrying. My guess is that you would be breaking those terms of service, and if your service gets popular enough to be noticed they will be enforced against you.
I am working on a flight scheduling app (disclaimer: it's for a college project, so no code answers, please). Please read this question w/ a quantum of attention before answering as it has a lot of peculiarities :(
First, some terminology issues:
You have planes and flights, and you have to pair them up. For simplicity's sake, we'll assume that a plane is free as soon as the flight using it prior lands.
Flights are seen as tasks:
They have a duration
They have dependencies
They have an expected date/time for
beginning
Planes can be seen as resources to be used by tasks (or flights, in our terminology).
Flights have a specific type of plane needed. e.g. flight 200 needs a plane of type B.
Planes obviously are of one and only one specific type, e.g., Plane Airforce One is of type C.
A "project" is the set of all the flights by an airline in a given time period.
The functionality required is:
Finding the shortest possible
duration for a said project
The earliest and latest possible
start for a task (flight)
The critical tasks, with basis on
provided data, complete with
identifiers of preceding tasks.
Automatically pair up flights and
planes, so as to get all flights
paired up with a plane. (Note: the
duration of flights is fixed)
Get a Gantt diagram with the projects
scheduling, in which all flights
begin as early as possible, showing
all previously referred data
graphically (dependencies, time info,
etc.)
So the questions is: How in the world do I achieve this? Particularly:
We are required to use a graph.
What do the graph's edges and nodes
respectively symbolise?
Are we required to discard tasks to
achieve the critical tasks set?
If you could also recommend some algorithms for us to look up, that'd be great.
Here some suggestions.
In principle you can have a graph where every node is a flight and there is an edge from flight A to flight B if B depends on A, i.e. B can't take off before A has landed. You can use this dependency graph to calculate the shortest possible duration for the project --- find the path through the graph that has maximum duration when you add the durations of all the flights on the path together. This is the "critical path" of your project.
However, the fact that you need to pair with planes makes it more difficult, esp. as I guess it is assumed that the planes are not allowed to fly without passengers, i.e. a plane must take off from the same city where it landed last.
If you have an excessive number of planes, allocating them to the flights can be most likely easily with a combinatorial optimization algorithm like simulated annealing. If the plan is very tight, i.e. you don't have excess planes, it could be a hard problem.
To set the actual take-off times for your flights, you can for example formulate the allowed schedules as a linear programming problem, or as a semi-definite / quadratic programming problem.
Here some references:
http://en.wikipedia.org/wiki/Simulated_annealing
http://en.wikipedia.org/wiki/Linear_programming
http://en.wikipedia.org/wiki/Quadratic_programming
http://en.wikipedia.org/wiki/Gradient_descent
http://en.wikipedia.org/wiki/Critical_path_method
Start with drawing out a domain model (class diagram) and make a clear separation in your mind between:
planning-immutable facts: PlaneType, Plane, Flight, FlightBeforeFlightConstraint, ...
planning variables: PlaneToFlightAssignment
Wrap all those instances in that Project class (= a Solution).
Then define a score function (AKA fitness function) on such a Solution. For example, if there are 2 PlaneToFlightAssignments which are not ok with a FlightBeforeFlightConstraint (= flight dependency), then lower the score.
Then it's just a matter for finding the Solution with the best score, by changing the PlaneToFlightAssignment instances. There are several algorithms you can use to find that best solution. If your data set is really really small (say 10 planes), you might be able to use brute force.
This is not a directly programming related question, but it's about selecting the right data mining algorithm.
I want to infer the age of people from their first names, from the region they live, and if they have an internet product or not. The idea behind it is that:
there are names that are old-fashioned or popular in a particular decade (celebrities, politicians etc.) (this may not hold in the USA, but in the country of interest that's true),
young people tend to live in highly populated regions whereas old people prefer countrysides, and
Internet is used more by young people than by old people.
I am not sure if those assumptions hold, but I want to test that. So what I have is 100K observations from our customer database with
approx. 500 different names (nominal input variable with too many classes)
20 different regions (nominal input variable)
Internet Yes/No (binary input variable)
91 distinct birthyears (numerical target variable with range: 1910-1992)
Because I have so many nominal inputs, I don't think regression is a good candidate. Because the target is numerical, I don't think decision tree is a good option either. Can anyone suggest me a method that is applicable for such a scenario?
I think you could design discrete variables that reflect the split you are trying to determine. It doesn't seem like you need a regression on their exact age.
One possibility is to cluster the ages, and then treat the clusters as discrete variables. Should this not be appropriate, another possibility is to divide the ages into bins of equal distribution.
One technique that could work very well for your purposes is, instead of clustering or partitioning the ages directly, cluster or partition the average age per name. That is to say, generate a list of all of the average ages, and work with this instead. (There may be some statistical problems in the classifier if you the discrete categories here are too fine-grained, though).
However, the best case is if you have a clear notion of what age range you consider appropriate for 'young' and 'old'. Then, use these directly.
New answer
I would try using regression, but in the manner that I specify. I would try binarizing each variable (if this is the correct term). The Internet variable is binary, but I would make it into two separate binary values. I will illustrate with an example because I feel it will be more illuminating. For my example, I will just use three names (Gertrude, Jennifer, and Mary) and the internet variable.
I have 4 women. Here are their data:
Gertrude, Internet, 57
Jennifer, Internet, 23
Gertrude, No Internet, 60
Mary, No Internet, 35
I would generate a matrix, A, like this (each row represents a respective woman in my list):
[[1,0,0,1,0],
[0,1,0,1,0],
[1,0,0,0,1],
[0,0,1,0,1]]
The first three columns represent the names and the latter two Internet/No Internet. Thus, the columns represent
[Gertrude, Jennifer, Mary, Internet, No Internet]
You can keep doing this with more names (500 columns for the names), and for the regions (20 columns for those). Then you will just be solving the standard linear algebra problem A*x=b where b for the above example is
b=[[57],
[23],
[60],
[35]]
You may be worried that A will now be a huge matrix, but it is a huge, extremely sparse matrix and thus can be stored very efficiently in a sparse matrix form. Each row has 3 1's in it and the rest are 0. You can then just solve this with a sparse matrix solver. You will want to do some sort of correlation test on the resulting predicting ages to see how effective it is.
You might check out the babynamewizard. It shows the changes in name frequency over time and should help convert your names to a numeric input. Also, you should be able to use population density from census.gov data to get a numeric value associated with your regions. I would suggest an additional flag regarding the availability of DSL access - many rural areas don't have DSL coverage. No coverage = less demand for internet services.
My first inclination would be to divide your response into two groups, those very likely to have used computers in school or work and those much less likely. The exposure to computer use at an age early in their career or schooling probably has some effect on their likelihood to use a computer later in their life. Then you might consider regressions on the groups separately. This should eliminate some of the natural correlation of your inputs.
I would use a classification algorithm that accepts nominal attributes and numeric class, like M5 (for trees or rules). Perhaps I would combine it with the bagging meta classifier to reduce variance. The original algorithm M5 was invented by R. Quinlan and Yong Wang made improvements.
The algorithm is implemented in R (library RWeka)
It also can be found in the open source machine learning software Weka
For more information see:
Ross J. Quinlan: Learning with Continuous Classes. In: 5th Australian Joint Conference on Artificial Intelligence, Singapore, 343-348, 1992.
Y. Wang, I. H. Witten: Induction of model trees for predicting continuous classes. In: Poster papers of the 9th European Conference on Machine Learning, 1997.
I think slightly different from you, I believe that trees are excellent algorithms to deal with nominal data because they can help you build a model that you can easily interpret and identify the influence of each one of these nominal variables and it's different values.
You can also use regression with dummy variables in order to represent the nominal attributes, this is also a good solution.
But you can also use other algorithms such as SVM(smo), with the previous transformation of the nominal variables to binary dummy ones, same as in regression.
I need help selecting or creating a clustering algorithm according to certain criteria.
Imagine you are managing newspaper delivery persons.
You have a set of street addresses, each of which is geocoded.
You want to cluster the addresses so that each cluster is assigned to a delivery person.
The number of delivery persons, or clusters, is not fixed. If needed, I can always hire more delivery persons, or lay them off.
Each cluster should have about the same number of addresses. However, a cluster may have less addresses if a cluster's addresses are more spread out. (Worded another way: minimum number of clusters where each cluster contains a maximum number of addresses, and any address within cluster must be separated by a maximum distance.)
For bonus points, when the data set is altered (address added or removed), and the algorithm is re-run, it would be nice if the clusters remained as unchanged as possible (ie. this rules out simple k-means clustering which is random in nature). Otherwise the delivery persons will go crazy.
So... ideas?
UPDATE
The street network graph, as described in Arachnid's answer, is not available.
I've written an inefficient but simple algorithm in Java to see how close I could get to doing some basic clustering on a set of points, more or less as described in the question.
The algorithm works on a list if (x,y) coords ps that are specified as ints. It takes three other parameters as well:
radius (r): given a point, what is the radius for scanning for nearby points
max addresses (maxA): what are the maximum number of addresses (points) per cluster?
min addresses (minA): minimum addresses per cluster
Set limitA=maxA.
Main iteration:
Initialize empty list possibleSolutions.
Outer iteration: for every point p in ps.
Initialize empty list pclusters.
A worklist of points wps=copy(ps) is defined.
Workpoint wp=p.
Inner iteration: while wps is not empty.
Remove the point wp in wps. Determine all the points wpsInRadius in wps that are at a distance < r from wp. Sort wpsInRadius ascendingly according to the distance from wp. Keep the first min(limitA, sizeOf(wpsInRadius)) points in wpsInRadius. These points form a new cluster (list of points) pcluster. Add pcluster to pclusters. Remove points in pcluster from wps. If wps is not empty, wp=wps[0] and continue inner iteration.
End inner iteration.
A list of clusters pclusters is obtained. Add this to possibleSolutions.
End outer iteration.
We have for each p in ps a list of clusters pclusters in possibleSolutions. Every pclusters is then weighted. If avgPC is the average number of points per cluster in possibleSolutions (global) and avgCSize is the average number of clusters per pclusters (global), then this is the function that uses both these variables to determine the weight:
private static WeightedPClusters weigh(List<Cluster> pclusters, double avgPC, double avgCSize)
{
double weight = 0;
for (Cluster cluster : pclusters)
{
int ps = cluster.getPoints().size();
double psAvgPC = ps - avgPC;
weight += psAvgPC * psAvgPC / avgCSize;
weight += cluster.getSurface() / ps;
}
return new WeightedPClusters(pclusters, weight);
}
The best solution is now the pclusters with the least weight. We repeat the main iteration as long as we can find a better solution (less weight) than the previous best one with limitA=max(minA,(int)avgPC). End main iteration.
Note that for the same input data this algorithm will always produce the same results. Lists are used to preserve order and there is no random involved.
To see how this algorithm behaves, this is an image of the result on a test pattern of 32 points. If maxA=minA=16, then we find 2 clusters of 16 addresses.
(source: paperboyalgorithm at sites.google.com)
Next, if we decrease the minimum number of addresses per cluster by setting minA=12, we find 3 clusters of 12/12/8 points.
(source: paperboyalgorithm at sites.google.com)
And to demonstrate that the algorithm is far from perfect, here is the output with maxA=7, yet we get 6 clusters, some of them small. So you still have to guess too much when determining the parameters. Note that r here is only 5.
(source: paperboyalgorithm at sites.google.com)
Just out of curiosity, I tried the algorithm on a larger set of randomly chosen points. I added the images below.
Conclusion? This took me half a day, it is inefficient, the code looks ugly, and it is relatively slow. But it shows that it is possible to produce some result in a short period of time. Of course, this was just for fun; turning this into something that is actually useful is the hard part.
(source: paperboyalgorithm at sites.google.com)
(source: paperboyalgorithm at sites.google.com)
What you are describing is a (Multi)-Vehicle-Routing-Problem (VRP). There's quite a lot of academic literature on different variants of this problem, using a large variety of techniques (heuristics, off-the-shelf solvers etc.). Usually the authors try to find good or optimal solutions for a concrete instance, which then also implies a clustering of the sites (all sites on the route of one vehicle).
However, the clusters may be subject to major changes with only slightly different instances, which is what you want to avoid. Still, something in the VRP-Papers may inspire you...
If you decide to stick with the explicit clustering step, don't forget to include your distribution in all clusters, as it is part of each route.
For evaluating the clusters using a graph representation of the street grid will probably yield more realistic results than connecting the dots on a white map (although both are TSP-variants). If a graph model is not available, you can use the taxicab-metric (|x_1 - x_2| + |y_1 - y_2|) as an approximation for the distances.
I think you want a hierarchical agglomeration technique rather than k-means. If you get your algorithm right you can stop it when you have the right number of clusters. As someone else mentioned you can seed subsequent clusterings with previous solutions which may give you a siginificant performance improvement.
You may want to look closely at the distance function you use, especially if your problem has high dimension. Euclidean distance is the easiest to understand but may not be the best, look at alternatives such as Mahalanobis.
I'm presuming that your real problem has nothing to do with delivering newspapers...
Have you thought about using an economic/market based solution? Divide the set up by an arbitrary (but constant to avoid randomness effects) split into even subsets (as determined by the number of delivery persons).
Assign a cost function to each point by how much it adds to the graph, and give each extra point an economic value.
Iterate allowing each person in turn to auction their worst point, and give each person a maximum budget.
This probably matches fairly well how the delivery people would think in real life, as people will find swaps, or will say "my life would be so much easier if I didn't do this one or two. It is also pretty flexible (for example, would allow one point miles away from any others to be given a premium fairly easily).
I would approach it differently: Considering the street network as a graph, with an edge for each side of each street, find a partitioning of the graph into n segments, each no more than a given length, such that each paperboy can ride a single continuous path from the start to the end of their route. This way, you avoid giving people routes that require them to ride the same segments repeatedly (eg, when asked to cover both sides of a street without covering all the surrounding streets).
This is a very quick and dirty method of discovering where your "clusters" lie. This was inspired by the game "Minesweeper."
Divide your entire delivery space up into a grid of squares. Note - it will take some tweaking of the size of the grid before this will work nicely. My intuition tells me that a square size roughly the size of a physical neighbourhood block will be a good starting point.
Loop through each square and store the number of delivery locations (houses) within each block. Use a second loop (or some clever method on the first pass) to store the number of delivery points for each neighbouring block.
Now you can operate on this grid in a similar way to photo manipulation software. You can detect the edges of clusters by finding blocks where some neighbouring blocks have no delivery points in them.
Finally you need a system that combines number of deliveries made as well as total distance travelled to create and assign routes. There may be some isolated clusters with just a few deliveries to be made, and one or two super clusters with many homes very close to each other, requiring multiple delivery people in the same cluster. Every home must be visited, so that is your first constraint.
Derive a maximum allowable distance to be travelled by any one delivery person on a single run. Next do the same for the number of deliveries made per person.
The first ever run of the routing algorithm would assign a single delivery person, send them to any random cluster with not all deliveries completed, let them deliver until they hit their delivery limit or they have delivered to all the homes in the cluster. If they have hit the delivery limit, end the route by sending them back to home base. If they could safely travel to the nearest cluster and then home without hitting their max travel distance, do so and repeat as above.
Once the route is finished for the current delivery person, check if there are homes that have not yet had a delivery. If so, assign another delivery person, and repeat the above algorithm.
This will generate initial routes. I would store all the info - the location and dimensions of each square, the number of homes within a square and all of its direct neighbours, the cluster to which each square belongs, the delivery people and their routes - I would store all of these in a database.
I'll leave the recalc procedure up to you - but having all the current routes, clusters, etc in a database will enable you to keep all historic routes, and also try various scenarios to see how to best to adapt to changes creating the least possible changes to existing routes.
This is a classic example of a problem that deserves an optimized solution rather than trying to solve for "The OPTIMUM". It's similar in some ways to the "Travelling Salesman Problem", but you also need to segment the locations during the optimization.
I've used three different optimization algorithms to good effect on problems like this:
Simulated Annealing
Great Deluge Algorithm
Genetic Algoritms
Using an optimization algorithm, I think you've described the following "goals":
The geographic area for each paper
boy should be minimized.
The number of subscribers served by
each should be approximately equal.
The distance travelled by each
should be about equal.
(And one you didn't state, but might
matter) The route should end where
it began.
Hope this gets you started!
* Edit *
If you don't care about the routes themselves, that eliminates goals 3 and 4 above, and perhaps allows the problem to be more tailored to your bonus requirements.
If you take demographic information into account (such as population density, subscription adoption rate and subscription cancellation rate) you could probably use the optimization techniques above to eliminate the need to rerun the algorithm at all as subscribers adopted or dropped your service. Once the clusters were optimized, they would stay in balance because the rates of each for an individual cluster matched the rates for the other clusters.
The only time you'd have to rerun the algorithm was when and external factor (such as a recession/depression) caused changes in the behavior of a demographic group.
Rather than a clustering model, I think you really want some variant of the Set Covering location model, with an additional constraint to cover the number of addresses covered by each facility. I can't really find a good explanation of it online. You can take a look at this page, but they're solving it using areal units and you probably want to solve it in either euclidean or network space. If you're willing to dig up something in dead tree format, check out chapter 4 of Network and Discrete Location by Daskin.
Good survey of simple clustering algos. There is more though:
http://home.dei.polimi.it/matteucc/Clustering/tutorial_html/index.html
Perhaps a minimum spanning tree of the customers, broken into set based on locality to the paper boy. Prims or Kruskal to get the MST with the distance between houses for the weight.
I know of a pretty novel approach to this problem that I have seen applied to Bioinformatics, though it is valid for any sort of clustering problem. It's certainly not the simplest solution but one that I think is very interesting. The basic premise is that clustering involves multiple objectives. For one you want to minimise the number of clusters, the trival solution being a single cluster with all the data. The second standard objective is to minimise the amount of variance within a cluster, the trivial solution being many clusters each with only a single data point. The interesting solutions come about when you try to include both of these objectives and optimise the trade-off.
At the core of the proposed approach is something called a memetic algorithm that is a little like a genetic algorithm, which steve mentioned, however it not only explores the solution space well but also has the ability to focus in on interesting regions, i.e. solutions. At the very least I recommend reading some of the papers on this subject as memetic algorithms are an unusual approach, though a word of warning; it may lead you to read The Selfish Gene and I still haven't decided whether that was a good thing... If algorithms don't interest you then maybe you can just try and express your problem as the format requires and use the source code provided. Related papers and code can be found here: Multi Objective Clustering
This is not directly related to the problem, but something I've heard and which should be considered if this is truly a route-planning problem you have. This would affect the ordering of the addresses within the set assigned to each driver.
UPS has software which generates optimum routes for their delivery people to follow. The software tries to maximize the number of right turns that are taken during the route. This saves them a lot of time on deliveries.
For people that don't live in the USA the reason for doing this may not be immediately obvious. In the US people drive on the right side of the road, so when making a right turn you don't have to wait for oncoming traffic if the light is green. Also, in the US, when turning right at a red light you (usually) don't have to wait for green before you can go. If you're always turning right then you never have to wait for lights.
There's an article about it here:
http://abcnews.go.com/wnt/story?id=3005890
You can have K means or expected maximization remain as unchanged as possible by using the previous cluster as a clustering feature. Getting each cluster to have the same amount of items seems bit trickier. I can think of how to do it as a post clustering step by doing k means and then shuffling some points until things balance but that doesn't seem very efficient.
A trivial answer which does not get any bonus points:
One delivery person for each address.
You have a set of street
addresses, each of which is geocoded.
You want to cluster the addresses so that each cluster is
assigned to a delivery person.
The number of delivery persons, or clusters, is not fixed. If needed,
I can always hire more delivery
persons, or lay them off.
Each cluster should have about the same number of addresses. However,
a cluster may have less addresses if a
cluster's addresses are more spread
out. (Worded another way: minimum
number of clusters where each cluster
contains a maximum number of
addresses, and any address within
cluster must be separated by a maximum
distance.)
For bonus points, when the data set is altered (address added or
removed), and the algorithm is re-run,
it would be nice if the clusters
remained as unchanged as possible (ie.
this rules out simple k-means
clustering which is random in nature).
Otherwise the delivery persons will go
crazy.
As has been mentioned a Vehicle Routing Problem is probably better suited... Although strictly isn't designed with clustering in mind, it will optimize to assign based on the nearest addresses. Therefore you're clusters will actually be the recommended routes.
If you provide a maximum number of deliverers then and try to reach the optimal solution this should tell you the min that you require. This deals with point 2.
The same number of addresses can be obtained by providing a limit on the number of addresses to be visited, basically assigning a stock value (now its a capcitated vehicle routing problem).
Adding time windows or hours that the delivery persons work helps reduce the load if addresses are more spread out (now a capcitated vehicle routing problem with time windows).
If you use a nearest neighbour algorithm then you can get identical results each time, removing a single address shouldn't have too much impact on your final result so should deal with the last point.
I'm actually working on a C# class library to achieve something like this, and think its probably the best route to go down, although not neccesairly easy to impelement.
I acknowledge that this will not necessarily provide clusters of roughly equal size:
One of the best current techniques in data clustering is Evidence Accumulation. (Fred and Jain, 2005)
What you do is:
Given a data set with n patterns.
Use an algorithm like k-means over a range of k. Or use a set of different algorithms, the goal is to produce an ensemble of partitions.
Create a co-association matrix C of size n x n.
For each partition p in the ensemble:
3.1 Update the co-association matrix: for each pattern pair (i, j) that belongs to the same cluster in p, set C(i, j) = C(i, j) + 1/N.
Use a clustering algorihm such as Single Link and apply the matrix C as the proximity measure. Single Link gives a dendrogram as result in which we choose the clustering with the longest lifetime.
I'll provide descriptions of SL and k-means if you're interested.
I would use a basic algorithm to create a first set of paperboy routes according to where they live, and current locations of subscribers, then:
when paperboys are:
Added: They take locations from one or more paperboys working in the same general area from where the new guy lives.
Removed: His locations are given to the other paperboys, using the closest locations to their routes.
when locations are:
Added : Same thing, the location is added to the closest route.
Removed: just removed from that boy's route.
Once a quarter, you could re-calculate the whole thing and change all the routes.