Differential constraint system

The differential constraint system refers to a series of inequalities:

$$ x_j-x_i \le C_{ij} (1\le I, J \le N) $$

Of course i,j do not have to take 1-n, and in the case of expansion, for each (I,J), can be a number of $C _{ij}$, but we easily turn them into an inequality.

Before solving the inequality group, we can analyze the nature of this inequality group.

1. This inequality group can be non-solvable, such as:

$ $X _1-x_2 \le-1 \ x_2-x_3 \le-1 \ x_3-x_1 \le-1$$

Otherwise three inequalities are added to get \le-3$.

2. If this inequality group has a solution of $ (x_1, x_2, ..., x_n) $, then for any constant c,$ (x_1 + C, x_2 + C, ..., X_n + C) $ is also the solution.

Because the two variables subtract, the constant c is lost.

If we think of X as a number of points on the axis, then this property means that any set of solutions can be translated as a whole on the axis so that it remains a solution.

3. Will all the solutions overlap after panning? It's certainly not.

Consider an extreme situation, if a $X _k$ in the system does not have any inequality, then the last number of this variable can be taken. This situation is the other point, but this point casually on the axis of the walk. We call this a free point in the system.

Now let's map this system to a single-source diagram.

We find an inequality on the graph: $dis _j-dis_i \le edge_{ij}$, where $dis _i$ represents the length of the shortest path from the source point to I.

So, if we build a diagram that makes $edge _{ij} = c_{ij}$, then $ (dis_1, dis_2, ..., Dis_n) $ is a set of solutions for the system.

There are some properties about this solution:

0. Assuming that the source is S, then $dis _s=0$.

1. The $dis _i$ in this set of solutions happens to be the maximum value $x _i-x_s$ in the possible solution. This proves to look at this http://blog.csdn.net/runninghui/article/details/9137673

2. For a $dis _u = inf$ U, that is, this $x _u-x_s$ no maximum, more than most can.

3. If there is a negative ring from S to this figure, then the shortest path is not, and the condition corresponds to the original differential constraint system without solution.

The nature of the focus of course is the first, so that we can solve the above axis model maximum distance problem.

There are a few points about this model:

1. What if there is an inequality $X _j-x_i \ge C $? So as long as it becomes $X _i-x_j \le-c$.

2. What if we are going to solve the problem of distance minimization? The above process inspires us to build $dis _j-dis_i \ge edge_{ij}$, and the dis here is of course the longest road.

Differential constraint system