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Fix some finite graph $G = (V, E)$, and some vertex $x$.

Suppose I generate a random sub-tree of $G$ of size $N$, containing $x$, as follows:

  1. Let $T_0 = \{ x \}$.
  2. For $0 < n \leqslant N$

    i. Let $B_n$ be the set of neighbors of $T_{n-1}$ outside of $T_{n-1}$.

    ii. Form $T_n$ by

    • Sample a pair $(x_n, y_n) \in E(G) \cap \left( V (T_{n-1}) \times B_n \right)$, with probability $q_n (x_n, y_n | T_{n-1} )$,
    • Add $y_n$ to $V(T_{n-1})$, and add $(x_n, y_n)$ to $E (T_{n-1})$.
  3. Return $T_N$.

Suppose also that $q_n ( x_n, y_n | T_{n-1} )$ can be computed easily for all $(T_{n-1}, x_n, y_n)$. I am interested in efficiently and exactly calculating the marginal probability of generating the tree $T_N$, given that I began growing it at $T_0 = \{ x \}$, i.e.

$$P(T_N | T_0 = \{ x \}) = \sum_{x_{1:N}, y_{1:N}} \prod_{n = 1}^N q_n (x_n, y_n | T_{n-1} ).$$

My question is essentially whether I should expect to be able to find an efficient (i.e. polynomial-time) algorithm for this, and if so, what it might be.

Some thoughts:

  • Naively, the sum has exponentially-many terms, which precludes trying to evaluate the sum directly.

  • On the other hand, this problem is also highly-structured (trees, recursion, etc.), which might suggest that some sort of dynamic programming approach would be feasible. I'm not sure of exactly how to approach this.

  • Relatedly, I know how to calculate unbiased, non-negative estimators of $P(T_N | T_0 = \{ x \})$, which have reasonable variance properties, by using techniques from Sequential Monte Carlo / particle filtering. This suggests that the problem is at least possible to approximate well in a reasonable amount of time.

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    $\begingroup$ $B_n$ could be empty, depending on $G$ and the previously chosen vertices. Perhaps you mean to say that $B_n$ is the set of neighbors of $T_{n-1}$ outside $T_{n-1}$? $\endgroup$ – Ariel Jun 2 at 13:22
  • $\begingroup$ @Ariel - that's a good point. I have been tacitly assuming that $B_n$ will always be non-empty for the $N$ I choose (in my application, this is the case), so I will make that more explicit. Re: your suggestion, I'll reformulate things slightly, as your comment has helped me to realise that my problem can be posed slightly more generally than I had thought. $\endgroup$ – πr8 Jun 2 at 14:29
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No. If $q(x_n,y_n|T_{n-1})$ is arbitrary -- there can be an arbitrary dependence on $T_{n-1}$ -- then this requires exponential time.

Consider a tree $T_N$ that has a single root node, $N-1$ leaves, and an edge from the root to each leaf. There are $2^N$ subtrees of $T_N$, and in particular, there are $2^N$ possible values of $T_n$ that can occur in the expression

$$\sum_{x, y} \prod_{n = 1}^N q_n (x_n, y_n | T_{n-1} ).$$

One can use a simple adversary argument to prove that evaluating this expression requires exponential time. Suppose that we evaluate $q_n(x_n,x_n|T_{n-1})$ by querying an oracle with $x_n,y_n,T_{n-1}$. Suppose there is a single tree $T$ that is never queried to the oracle as any $T_{n-1}$. Choose all of the $q_n(\cdots)$ values to be strictly positive. Then since $q_n(x_n,y_n|T)$ wasn't queried during execution, we can choose it after observing the output of the algorithm; but by varying it, we can choose a value that makes the algorithm's output wrong (in particular, the value of the expression depends on $q_n(x_n,y_n|T)$ but the algorithm's output doesn't depend on $q_n(x_n,y_n|T)$, so the algorithm's output cannot correct). We have proven that, to produce the correct output, any correct algorithm must query the oracle for all $2^N$ possible subtrees of $T_N$. It takes at least $O(1)$ time to query an oracle.

In conclusion, this argument proves that any correct algorithm for computing this expression must take $\Omega(2^N)$ time.

I don't know whether it can always be done in $O(2^N)$ time, or whether perhaps $O(N!)$ time might be required.

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  • $\begingroup$ Hi - where have I written $q(x_n, y_n | T_n)$? I agree that it should be $q(x_n, y_n | T_{n-1})$ throughout. Re: the main comment - I think that there are generally much fewer than $N!$ valid orderings, as each time an edge is added, it must be added to an existing edge. For example, if the final tree is a chain graph on $\{ 1, 2, \ldots, N \}$, and one began growing the tree from $\{ k \}$, then there are ${ N \choose k}$ ways in which the vertices could be added. Admittedly, this may still be exponentially-large in $N$ (e.g. take $k \sim \alpha n$). $\endgroup$ – πr8 Jun 3 at 9:11
  • $\begingroup$ One approach which I'm considering is to show that the set of possible 'partial trees' $T_n$ at each $n \in \{ 1, \ldots, N \}$ which extend $\{ x \}$ and which can be extended to $T_N$, is not too large (e.g. at most polynomial in $N$). If there is a 'final tree' $T_N$ and an 'initial vertex' $x$ for which this is not the case, it would show that this is not possible in general. $\endgroup$ – πr8 Jun 3 at 9:18
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    $\begingroup$ Consider $T_{N+1}$ which consists of $x$ and $N$ vertices adjacent to $x$ (a tree of height 1, where $x$ is the root and has $N$ children). There are $N!$ possible constructions (where a construction is a permutation of the edges in the tree) that produces $T_{N+1}$ (any edge can be chosen at any step). While this is a good intuition, this is definitely not a proof. $\endgroup$ – Ariel Jun 3 at 10:06
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    $\begingroup$ A complete binary tree rooted at $x$ with $N$ additional vertices also admits to $N!$ possible constructions (at the first step you have two choices, and each step the number of candidates increases by one). $\endgroup$ – Ariel Jun 3 at 17:41
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    $\begingroup$ @Ariel, great example! See revised answer for an argument that this problem requires exponential time in the worst case. $\endgroup$ – D.W. Jun 3 at 18:06

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