I came up with the following problem that I do not know the solutions to nor can I find the 'lookup' term to investigate further.
Say we have N ordered nodes (n_1,n_2....n_N) each with a fixed distance of 1 between them. So dist(n_1,n_N) = N-1. Now we are are allowed to connect any two nodes, thus effectively reducing their distance to 1. Assume we can have k such connections.
The problem is : how do we choose which nodes to connect to minimize the total distance between any two nodes ?
Is this problem a known variant of some well-studied problem ? Does an efficient solution exist for this (or a variant where we only want to minimize the max distance between any two nodes)
Thanks
You may be interested in "On the sum of all distances in a graph or digraph". That paper refers to your "total distance" as the "transmission" of a graph. Your "max distance" is generally called the "diameter" of a graph. It discusses the two, proves some properties of a graph's transmission, and shows that the transmission and diameter are independent of one another.
Naively, you've got n-choose-k options to try. That's pretty bad if n and k are large. Not too bad if one of them is small.
There is work on doing better than that. This Mathoverflow question asks about reducing the mean distance between vertices, which is proportional to the transmission of the graph. There are two answers, neither of which I can vouch for. It also refers to a paper that directly addresses this question.
Minimizing the diameter of a graph is dealt with in this paper.
You might consider addressing this question to the Math stackexchange.
For some project in computer vision I have N points in high-dimensional space. I want to select k of them that will be "the most distinguishable" from each other. For example, it can translate to sum of distances between chosen points is maximum. Or it can be that volume of polyhedron is maximum. But generally anything that has some intuition behind can go.
As expected I want to find these representative points.
There are two questions:
What definition for "the most distinguishable" points is more commonly used? Do they change the algorithm used to find those points?
What is the algorithm to find the points? It highly reminds me maximal weighted clique problem. Is it NP-hard problem? In this case can we make some good approximation against optimal solution?
The way you define "the most distinguishable" will definitely affect the algorithm you'll want to use. for example, you can define "the most distinguishable" as the set with the maximal sum of distances between any two points in the set, but you could also define it as the set with the maximal minimum distance between any two points. these are two completely different problems.
As for algorithms, as I've said, that depends on your definition. If you're looking to find the K farthest points, you should look into this question. This problem is NP-Complete, but you may get some ideas about how to approach the problem.
Problem:
Given a set of 2D points in a plane, find a set of edges E that minimizes both average trip time between any two points and the size of E: ie, by associating a cost r with each unit of trip time and a cost e per edge in the set.
I'm sure that there's a set of algorithms that deal with this problem, but I can't seem to find the right search term. I've considered starting with a complete graph and pruning, but I can't think of an efficient way to calculate the damage done by removing an edge. Any suggestions? Approximate ('good-enough') solutions welcome.
Let me know if my statement of the problem can be improved or clarified.
There is some work in the literature on spanners, which is related to what you describe (the main difference is that spanners control the maximum stretch, while you're concerned with the average). Chew's construction ("There is a planar graph almost as good as the complete graph", SoCG '86) gives an O(1)-approximation for your problem, since the triangulation has less than three times as many edges as a spanning tree (lower bound on the optimum, since the graph must be connected) and adjusts each Euclidean distance by a factor of at most sqrt(10) (hence the sqrt(10) times the average for the optimum).
is there an algorithm for finding all the independent sets of an directed graph ?
From what i've read an independent set represents a set formed by the nodes that are not adjacent.
So for this example I would have {1} {2} {1,3}
So how is possible to find all of them, I am thinking about something recursive but I don't really know the algorithm, if someone could point me in the right direction it would be much appreciated !
Thank you!
Typical way to find independent sets is to consider the complement of a graph. A complement of a graph is defined as a graph with the same set of vertices and an edge between a pair if and only if there is no edge between them in the original graph. An independent set in the graph corresponds to a clique in the complements. Finding all the cliques is exponential in complexity so you can not improve brute force much. Still I believe considering the complement of the graph may make the problem easier to deal with.
Other than complement and finding cliques, I can also think about "Graph Coloring", you color the vertices somehow that no two adjacent vertices have the same color (you can do it with a very simple heuristic algorithm like SL = Smallest Last), and then choose vertices in every color as a subset (as a maximal independent subset).
The only problem is that there are probably too many ways of coloring a graph. You have to keep all the found (maximal) independent sets and move on until you get enough sets!
The Bron–Kerbosch algorithm is commonly used for this problem, see the Wikipedia article for a description and pseudocode that can be turned into a useable program without too much problem. The size of output is, in the worst case, exponential in the number of vertices, but brute force will always be exponential while BK will be polynomial if the output is polynomial. In other words if you know that the output will be reasonable then BK will produce it in a reasonable time. This is an active area of research and there are a number of other algorithms that do the same thing with varying efficiency depending of the type and size of graph. There are applications in several areas, in particular genetics.
I was trying to solve the following problem:
An mn maze is an mn rectangular grid with walls placed between grid cells such that there is exactly one path from the top-left square to any other square.
The following are examples of a 912 maze and a 1520 maze:
Let C(m,n) be the number of distinct mn mazes. Mazes which can be formed by rotation and reflection from another maze are considered distinct.
It can be verified that C(1,1) = 1, C(2,2) = 4, C(3,4) = 2415, and C(9,12) = 2.5720e46 (in scientific notation rounded to 5 significant digits).
Find C(100,500)
Now, there is an explicit formula which gives the right result, and it is perfectly computable. However, as I understand, the solutions to Project Euler problems should be more like clever algorithms and not explicit formula computations. Trying to formulate the solution as a recursion, I could only arrive at a linear system with number of variables growing exponentially with the size of the maze (more precisely, if one tries to write a recursion for the number of mxn mazes with m held fixed, one arrives at a linear system such that the number of its variables grows exponentially with m: one of the variables is the number of mxn mazes with the property given in the declaration of problem 380, while the other variables are numbers of mxn mazes with more than one connected component which touch the boundary of the maze in some specific "configuration" - and the number of such "configurations" seems to grow exponentially with m. So, while this approach is feasible with m=2,3,4 etc, it does not seem to work with m=100).
I thought also to reduce the problem to subproblems which can be solved more easily,
then reusing the subproblems solutions when constructing a solution to larger subproblems(the dynamic programming approach), but here I stumbled upon the fact that subproblems seem to involve mazes of irregular shapes, and again, the number of such mazes is exponential in m,n.
If someone knows of a feasible approach (m=100, n=500) other than using explicit formulas or some ad hoc theorems, and can hint where to look, for me it would be quite interesting.
This is basically a spanning tree counting problem. Specifically, it is counting the number of spanning trees in a grid graph.
Counting Spanning Trees in a Grid Graph
From the "Counting spanning trees" section of the Wikipedia entry:
The number t(G) of spanning trees of a connected graph is a
well-studied invariant. In some cases, it is easy to calculate t(G)
directly. For example, if G is itself a tree, then t(G)=1, while if G
is the cycle graph C_n with n vertices, then t(G)=n. For any graph G,
the number t(G) can be calculated using Kirchhoff's matrix-tree theorem...
Related Algorithms
Here are a few papers or posts related to counting the number of spanning trees in grid graphs:
"Counting Spanning Trees in Grid Graphs", Melissa Desjarlais and Robert Molina
Department of Mathematics and Computer Science, Alma College, August 17, 2012? (publish date uncertain)
"Counting the number of spanning trees in a graph - A spectral approach", from Univ. of Maryland class notes for CMSC858W: Algorithms for Biosequence Analysis,
April 29th, 2010
"Automatic Generation of Generating Functions for Counting the Number of Spanning Trees for Grid Graphs (and more general creatures) of Fixed (but arbitrary!) Width", by Shalosh B. Ekhad and Doron Zeilberger
The latter by Ekhad and Zeilberger provided the following, with answers that matched up with the problem-at-hand:
If you want to see explicit expressions (as rational functions in z)
for the formal power series whose coefficient of zn in its Maclaurin
expansion (with respect to z) would give you the number of spanning
trees of the m by n grid graph (the Cartesian product of a path of m
vertices and a path of length n) for m=2 to m=6, the the input
gives the output.
Specifically, see the output.
Sidenote: Without the provided solution values that suggest otherwise, a valid interpretation could be that the external structure of the maze is important. Two or more mazes with identical paths would be different and distinct in this case, as there could be 3 options for entering and exiting a maze on a corner, where the top left corner would be open at top, top left corner open on left, or open on both left and top, and similar for a corner exit. If trying to represent these maze possibilities as a tree, two nodes may converge on entry rather than just diverging from start to finish, and there would be one or more additional nodes for exit possibilities. This would increase the value of C(m,n).
The insight here comes from the question (emphasis mine)
A .. maze is a rectangular grid with walls placed between grid cells such that there is exactly one path from the top-left square to any other square.
If you think of the dual of the maze, i.e. the spaces one can occupy, it is clear that a maze must form a graph. Not just any graph either, for there to be a singular path the graph must not contain any cycles which makes it a tree. This reduction to a combinatorics problem suggests an algorithm. In the spirit of Project Euler, the rest is left as an exercise to the reader.
SPOILER AHEAD
I was wrong, stating in one of the comments that "Now, there is a general theorem about spanning trees in a graph, but it does not seem to give a computationally feasible way to compute the number sought". The "general theorem", being the Matrix-Tree theorem, attributed to Kirchhoff, and referred to in one of the answers here, gives the result not only as the product of the nonzero eigenvalues of the graph Laplacian divided by the order of the graph, but also as the absolute value of any cofactor of the Laplacian, which in this case is the absolute value of the determinant of a 49999x49999 matrix. But, although the matrix is very sparse, it still looked to me out of reach.
However, the reference
http://arxiv.org/pdf/0712.0681.pdf
("Determinants of block tridiagonal matrices", by Luca Guido Molinari),
permitted to reduce the problem to the evaluation of the determinant of an integer 100x100 dense matrix, having very large integers as its entries.
Further, the reference
http://www.ams.org/journals/mcom/1968-22-103/S0025-5718-1968-0226829-0/S0025-5718-1968-0226829-0.pdf
by Erwin H. Bareiss (usually one just speaks of "Bareiss algorithm", but the recursion which I used and which is referred to as formula (8) in the reference, seems to be due to Charles Dodgson, a.k.a. Lewis Carroll :) ), perimitted me then to evaluate this last determinant and thus to obtain the answer to the original problem.
I would say that finding a explicit formula is a correct way to solve an Euler problem. It will be fast, it can be scaled. Just go for it :)