How can I randomly distribute water into buckets without the buckets overflowing?

Question

I asked this question a week ago but it ended up being closed for not being specific enough. I am going to try and define everything as mathematically as possible in this one, so please let me know if there is anything ambiguous.

How can I randomly distribute W litres of water into n empty buckets without them overflowing (i.e. without putting more water into the bucket then it can hold)

There are $n$ buckets of water with the $i^{th}$ bucket containing $w_i$ litres of water.

1) $W = w_1 + w_2 + ... + w_n =$ Total amount of water

2) $w_i$ $\epsilon$ $[0, x]$ for $i = 1, 2, ..., n$ where $x$ $\epsilon$ $\Re$

Given a natural number $n$ and real number value for $x$ and $W$, what process can I use to assign a random value to each $w_i$, such that (1) and (2) remain true?

To clarify what I mean by random, I am thinking of a function which returns a value in the interval $[0, 1]$ similar to the .NextDouble() method in the Random class (And yes, I know this is pseudo-random and not truly random).

Example:

For $n$ = 3, $W$ = 1 and $x$ = 1 one possible solution is:

$w_1$ = 0.22 $w_2$ = 0.43 $w_3$ = 0.35

Since $w_1 + w_2 + w_3 = 0.22 + 0.43 + 0.35 = 1 = W$ and $w_1$, $w_2$ and $w_3$ are in the interval $[0,1]$

Answer 1

As clarified in the comments, you want to uniformly sample the set of all points $\vec w = (w_1,\dots,w_n)$ such that $$w_1+\dots+w_n = W,\\ 0\le w_i\le x\ \text{for all $i=1,\dots,n$.}$$

Here are two possibilities.

1. You can use rejection sampling: Uniformly generate a point satisfying $w_1+\dots+w_n=W$, $0\le w_i\ \forall i$; if it violates the capacity constraint $w_i\le x$ for any $i$, reject it and try again.

   The first step can be done easily by generating $n-1$ uniform random numbers between $0$ and $W$, sorting them so they are $s_1,\dots,s_{n-1}$ in increasing order, and setting $w_1=s_1-0$, $w_2=s_2-s_1$, . . . , $w_n = W-s_{n-1}$. This is the stick-breaking model alluded to by The Count, since it is equivalent to breaking a stick of length $W$ into $n$ pieces by making cuts at locations $s_1,\dots,s_{n-1}$.

   Rejection sampling is guaranteed to give you a uniform distribution, but the downside is that you may have to try many many times before obtaining a valid sample, especially if $x\ll W$.

2. Tim Seguine on MathOverflow recommends hit-and-run sampling: Start at an arbitrary point $\vec w$ inside your set; pick a direction $\vec d$ uniformly at random; find the range of values $t$ for which $\vec w+t\vec d$ lies in the set; pick a $t$ uniformly at random from this range; replace $\vec w$ with the new point $\vec w+t\vec d$ and repeat. (This is essentially Seguine's description with the variables renamed.)

   For the initialization, the choice $\vec w=(W/n,\dots,W/n)$ is as good as any. To pick a direction $\vec d$ uniformly, a simple way is to choose $n$ numbers from a standard normal distribution, subtract the mean, form them into a vector and normalize it. To find the range of values for $t$, compute all the critical values $t_{i,1} = -w_i/d_i$ and $t_{i,2} = (x-w_i)/d_i$ where $\vec w+t\vec d$ crosses a constraint boundary; since we already know $\vec w$ is in the set, $t=0$ must be valid, so $t$ must lie between the largest negative value and the smallest positive value.

   Hit-and-run sampling converges to a uniform distribution as you repeat the process, but if you stop it after a finite number of iterations, the distribution may only be approximately uniform. Nevertheless, I did some experiments for low $n$ and the distribution appeared to approach uniformity quite rapidly. My totally uninformed conjecture is that $n$ iterations is probably "good enough", at least for your problem.

Both methods are easy to implement, so you should give them both a try. Thanks for asking this question; I wouldn't have learned about hit-and-run sampling otherwise.

Answer 2

The exact method

The formula $W = w_1 + w_2 + ... + w_n$ describes an $(n-1)$-dimensional hyperplane in $\mathbb R^n.$ The intersection of this hyperplane with the $n$-dimensional hypercube $[0,x]^n$ is a convex $(n-1)$-dimensional polytope embedded in $\mathbb R^n.$ Find that polytope and sample uniformly over it. In fact you can take the image of that polytope in $\mathbb R^{n-1}$ when you remove the $n$th coordinate, sample the first $n-1$ coordinates uniformly over that polytope, and solve for the $n$th coordinate.

Admittedly, this is just changing the problem to the problem of describing and sampling the polytope for a given $W$ and $x,$ which also can be a complicated problem.

Rejection sampling

As suggested already in another answer, you can use uniform sampling over the polytope $W = w_1 + w_2 + ... + w_n \cap [0,\infty)^n$ (that is, where all the $w_i$ are non-negative).

If $W \leq x$ this is a complete solution.

If $x < W \leq \frac12 nx,$ sample uniformly over $W = w_1 + w_2 + ... + w_n \cap [0,\infty)^n$ but reject the outcome if there is any value of $i$ for which $w_i > x$ in that outcome.

If $W > \frac12 nx,$ change the problem to one of randomly distributing $nx - W$ "empty space" among $n$ buckets, each of which can hold no more than $x$ "empty space". Then fill the rest of each bucket with water.

This avoids the extreme rejection rates that could otherwise occur when $W$ is close to $nx.$ In fact, for $(n-1)x \leq W \leq nx$ this gives a solution with no rejection.

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Either of these methods can be adapted to the more general problem in which each bucket's capacity is individually specified, that is, where the constraint is $0 \leq w_i \leq x_i.$

Answer 3

I think what you are asking is can I empty W, given x and n. That is I start with $W$ and pour into each $w_i$ for $i \in [1,n]$ until it's capacity of between $[0,x]$ is full. Then if after filling $w_n$ we have water left over, we cannot do it. So we want: $$P(\sum_{i=1}^{n}w_i- W=0|w_1=x) =P(\frac Wn\le x|w_1=x)= {1, \ W\le nx \brace 0, \ W>nx}$$

You could make it more interesting by having $w_i \in [0,x_i]$. That way at least you assign a "random value" to $w_i$ so each bucket is different.

Sorry if I misinterpreted your question.

Answer 4

EDIT: I had a different answer, but Darq pointed out that it wasn't correct.

Discretize the problem, say all values are integers. (If they're rational to begin with, you can multiply by a common denominator.) Repeatedly do the following: take one unit of water and randomly add it to any non-full bucket.