Posts Tagged ‘random walk’

New paper on arXiv: indistinguishable sceneries on the Boolean hypercube

January 28, 2017

I’m happy to say that fellow student Uri Grupel and I uploaded a paper to the arXiv recently under the title “Indistinguishable sceneries on the Boolean hypercube” (https://arxiv.org/abs/1701.07667). We had great fun working on it, and most of the theorems are actually pretty simple and do not use heavy mathematical machinery, so I’d like to share the motivation and results with you below. I’ll try to write a fairly simple exposition, without most of the technical terms and details. If you find this too simple, just read the paper!

Our paper shows that the scenery reconstruction problem for the $n$-dimensional hypercube is impossible for $n \geq 4$. It does this using the combinatorial properties of what we call “locally biased” and “locally balanced” functions, some properties of which the paper investigates.

Don’t know what scenery reconstruction is? Want to learn about and play around with locally biased and locally balanced functions? Read on.

We’ll start with a simpler-to-state problem. Suppose you are given a cycle graph, that is, a collection of vertices arranged in a cycle. For example, here is a cycle with $13$ vertices:

Each vertex is colored either black or white. There are many different such possible colorings: if there are $n$ vertices, then there are $2^n$ different colorings. However, the cycle has some symmetries, and some of these colorings can be considered to be the same. For example, in the following figure, if we rotate the center cycle 5 places clockwise, we get the cycle on the left. Similarly, if we reflect the center cycle around a vertical line, we get the cycle on the right. Such colorings are called “isomorphic”, and we will treat them all as “the same” for all intents and purposes.

Suppose now we place a person at one of the vertices at random. The person then performs a random walk: at each step, she randomly goes to the vertex either to her left or to her right, with each possibility equally likely to occur. Then she tells us the color of the vertex she is currently at. The result is a sequence of colors, one for each time step, which constitute the trace of the random walk on the cycle. Here is a short example of such a trace:

The idea you should keep in your mind is this: the coloring of the cycle is a “scenery”, and the agent, walking randomly along the cycle, tells you what she sees but *not* where she is going. The scenery reconstruction on the $n$-cycle is then framed as follows: suppose that you are given the length of the cycle $n$ and also an infinitely long trace of scenery observations by your agent. Can you reconstruct the scenery itself? That is, can you find the original color of each vertex of the cycle?

As an easy example, if the original cycle was colored all black, then no matter where the agent goes, she will always report “black” at every time step. So if she reports “BBBBBBB…”, we can be confident that the original scenery is the all black cycle. Similarly, if the agent reports “BWBWBWB…”, we can be confident that the original scenery is the alternating black-white cycle (we can assume that in the long run, the agent will never be stuck in a “back and forth between two vertices only path). But what about more complicated sceneries, like the one given below? Can we always reconstruct even very bizzare colorings?

The (perhaps surprising) answer is: yes! It was shown that if you are given a long enough series of observations, you can always reconstruct the original coloring! (up to isomorphism, meaning, you cannot differentiate between colorings which are shifts or reflections of one another, but we already mentioned above that weview these to be “the same”). So while this basic question is closed, there are other open questions still to be answered. For example, how many observations do we need so that we will be able to reconstruct the coloring with very high probability?

Variants of the $n$-cycle reconstruction problem abound. What if, in our random walk, we don’t go left or right with equal probability? What if we can jump other distances? What if we color and walk on another graph other than the cycle?

The latter question is what Uri’s and my paper addresses. Instead of the $n$-cycle, we investigate the scenery reconstruction problem on the Boolean hypercube in dimension $n$. In dimension 1, the cube is just a line. In dimension 2, a square. In dimension 3, an actual cube. In higher dimensions visualizations start to get messy. One way to think about it is as follows: the $n+1$-dimensional hypercube is composed of just two identical $n$-dimensional hypercubes, with edges connecting matching vertices in the two cubes.

So again, we place an agent at one of the vertices at random, and she starts a random walk. This time, there are $n$ possible choices of where to go in each step. Again, she reports only the color of the vertex she is currently at. Now can we reconstruct the original coloring?

For $n=1$, the hypercube is just two points connected by an edge, and there aren’t that many colorings on it, so it’s no big deal. For $n=2$, the hypercube is a square, which is actually a cycle of length $4$, and we already know how to deal with cycles, so that’s no big deal either. For $n=3$, it can be shown that the methods used to solve for the cycle, combined with some case-by-base analysis, again are able to reconstruct the scenery. What about other dimensions?

The (perhaps surprising) answer is: no! Starting from $n=4$, there exist two colorings which are non-isomorphic (that is, they are not obtained by rotations or reflections of the cube), but which nevertheless cannot be distinguished from one another based on the sequence of color observations of our agent alone! So if we are given such a sequence, we can never be sure whether it came from one coloring or the other.

Here is such a pair in dimension $n=4$:

To see that these are indeed two different colorings (meaning, that one is not just a rotation or reflection of the other), note that in the left image there are two small rings of black vertices, each of size $4$, while in the right image there is only one large ring, of size $8$. This difference cannot be due to rotations and reflections alone.

If you look carefuly, you will see an interesting property of these two colorings: every vertex, no matter what color it is, has exactly two neighbors which are black, and exactly two neighbors which are white. It is this property that prohibits us from distinguishing between the two colorings: no matter where our agent is, at the next step she will report “black” with probability $1/2$, and “white” with probability $1/2$, for both colorings. In technical terms, the sequence of observations distributes as a sequence of iid Bernoulli variables with success probability of $1/2$.

We can take this property and generalize it: we call a coloring “locally $p$-biased” if every vertex, no matter what color it is, has a $p$-fraction of its neighbors colored black, and a $1-p$-fraction of its neighbors colored white. If there are two non-isomorphic such colorings on the hypercube, for any $p$, then the scenery reconstruction problem is impossible to solve on that hypercube. In fact, this statement is true for any graph, not just the hypercube, so finding locally biased colorings on different graphs shows that the scenerey reconstruction problem is impossible for those graphs. But we think that such colorings stand alone by their own right, and deserve study regardless of their application to scenery reconstruction; they are simply interesting combinatorially!

Our first result was to classify for which values of $p$ an $n$-dimensional hypercube even has a locally biased coloring. We saw a $1/2$-biased coloring for $n=4$. Obviously, a $1/2$-biased coloring requires the dimension to be even, since each vertex has $n$ neighbors, and exactly $n/2$ of those need to be black, and $n/2$ white. So is there a locally $1/2$-biased coloring on the $6$-dimensional hypercube? What about a $1/3$-biased coloring? You can try it out yourself for a bit and convince yourself that there is no $1/3$-biased coloring on the $3$-dimensional cube. After a bit of work, we came up with the following theorem:

Theorem 1: There exists a locally $p$-biased coloring on the $n$ dimensional hypercube if and only if $p$ is of the form $p = b/2^k$ and $2^k$ divides $n$.

This shows, for example, that there are no locally biased colorings on odd-dimensional hypercubes. But for even-dimensional hypercubes, there do exist such colorings, and then it’s interesting to start counting them: as we said before, if there are more than two colorings for the same $p$ value, then the scenery reconstruction problem is impossible.

Our proof uses elements from coding theory, and is not very complicated. I’ll tell it in short in the following paragraphs: feel free to skip it if you don’t want to go ito the details.

One side, the “only if”, tells you which colorings are impossible. In the $n$-dimensional hypercube, each vertex has $n$ neighbors, and there are $2^n$ vertices overall. Suppose that in a specific locally $p$-biased coloring, a total of $l$ vertices are colored black. Then when you pick a random vertex, it is black with probability $l/2^n$. On the other hand, when the agent takes a random step, it will be black with probability $p = m/n$ for some $m$. These two probabilities are equal, so

$b/n = l/2^n.$

Decomposing $n$ into its prime powers, $n = c2^k$ where $c$ is odd. Then

$l = 2^{n-k} \cdot m / c$

and since $l$ must be a whole integer (and not a fraction), we get that $m = bc$ for some $b$. This gives $p = b/2^k$ after a short calculation.

The other direction, which says that for all $p$ values of the proper form you can build a locally $p$-biased coloring, is a bit harder. We start by building locally $1/n$-biased colorings for $n$ that is a power of two, that is, $n=2^k$. Such colorings have the peculiar property that they induce tiles which tile the hypercube. What do I mean by this? In such a coloring, every vertex, no matter the color, has exactly $1$ black neighbor. In particular, this is true for black vertices. So black vertices always come together in connected pairs, and all the neighbors of such pairs are white. Further, these white neighbors already have their “$1$-black-neighbor” condition satisfied, so all their other neighbors are also white. This is exemplified in the following shape, or “tile”:

If we can “tile the hypercube” with these shapes, that is, if we can cover the hypercube graph with these tiles without any pair of tiles overlapping, we will have a locally $1/n$-biased coloring. Why? Well, these tiles exactly satisfy the properties stated in the above paragraph, and every vertex in such a tiling will have exactly one black neighbor.

The proof constructs such tilings by using perfect codes; without going into detail, these are well-known methods of tiling the hypercube with a half-tile: instead of two black vertices and two rings of white vertices, these methods tile the hypercube with just a single black vertex and a single ring of white vertices. These are actually known as “balls” on the hypercube:

By taking two copies of a perfect code and connecting the black vertices, we obtain the needed tiling! This gives us a locally $1/n$-biased coloring, for $n$ that is a power of two (the perfect codes we need only exist in dimension $2^k-1$, so when we take two copies of the hypercube together we get a hypercube of dimension $n = 2^k$).

To obtain a locally $m/n$-biased coloring, we join together $m$ different disjoint $1/n$-biased colorings. By $m$ disjoint colorings, we mean that if a vertex is colored black in one coloring, then it is necessarily colored white in all the others. Finally, using a bit of algebraic manipulation which I will not go into here, we can extend these colorings into hypercubes of dimension $c \cdot 2^k$ for any $c$, which is exactly what our theorem stated we could do.

The theorem shows existence of some locally $p$-biased colorings, but tells us nothing of the number of such colorings. In order to show that the scenery reconstruction problem is impossible, we need to find at least two different such colorings. That’s where our next two theorems come into play:

Theorem 2.1: As a function of $n$, there is at least a super-exponential number of different locally $1/n$-biased colorings on the $n$ dimensional hypercube.

Theorem 2.2: As a function of $n$, there are at least $C \cdot 2^{\sqrt{n}} / \sqrt{n}$ different locally $1/2$-biased colorings on the $n$ dimensional hypercube, where $C$ is some constant.

Theorem 2.1 is proved by counting the number of perfect codes, which were used in proving the previous theorem; we won’t go into it here.

The proof of Theorem 2.2 is more interesting, and relies on an algebraic treatment of our colorings. It goes as follows.

Up until now, we have treated our colorings as assigning the color either black or white to each vertex. We could have equally well assigned the numbers “$+1$” or “$-1$“, and all the theorems and statements would have stayed the same: all we care is that there are two distinct “colors”. But this description has an immediate advantage: we can now easily perform arithmetic operations on the colorings. For example, if both $f_1$ and $f_2$ are colorings, then their product $f = f_1 \cdot f_2$ is also a coloring, where by multiplication we mean that if $f_1$ assigns the color $f_1(v)$ to a vertex $v$, and $f_2$ assigns the color $f_2(v)$, then $f(v) = f_1(v)\cdot f_2(v)$. In words, what this multiplication means is that $f$ is black precisely when $f_1$ and $f_2$ have the same color. This algebraic representation is often easier to work with than the verbal one.

Now, the $n$-dimensional hypercube can be described as follows: it is the set of points with $n$ coordinates, where each coordinate can be either $+1$ or $-1$ (you can easily verify that this fits in with our notions of $2d$ and $3d$ cubes; it works just the same for higher dimensions). Thus a coloring is just a function

$f(x_1, x_2, \ldots, x_n)$

which returns either $1$ or $-1$, and all the $x_i$‘s are also either $1$ or $-1$.

This representation lets us combine two locally $1/2$-biased colorings on small hypercubes together into one locally $1/2$-biased coloring on a larger hypercubes: Suppose that $f_1$ is such a coloring on the $n$ dimensional hypercube, and $f_2$ is such a coloring on the $m$ dimensional hypercube. Then the new coloring

$f(x_1, x_2, \ldots, x_{n+m}) = f_1(x_1,\ldots,x_n) f_2(x_{n+1}, \ldots, x_{n+m})$

is a locally $1/2$-biased coloring on the $n+m$ dimensional hypercube! Our method for counting the number of locally $1/2$-biased colorings then goes as follows: we construct many high dimensional colorings from different lower-dimensional ones. In fact, our “elementary building blocks”, from which we build all other colorings, are basically generalizations of the two colorings of the $4$-dimensional hypercube which we encountered earlier in this post. It can be shown (via Fourier decomposition, a method which we won’t go into here) that taking combinations of different low-dimensional colorings always gives non-isomorphic high-dimensional colorings. In other words, if we use different building blocks, then the resultant colorings will be distinct.

We can then get a bound on the number locally $1/2$-biased colorings by counting in how many distinct ways we can add up low-dimensional colorings into a high-dimensional one. After a bit of arithmetic, we find that there are at least as many colorings as there are solutions to this inequality:

$4a_1 + 8a_2 + \ldots + 4k a_k \leq n,$

where $a_1,\ldots,a_k$ are non-negative integers. The number of solutions to this inequality is at least $C \cdot 2^{\sqrt{n}} / \sqrt{n}$, giving us Theorem 2.2.

This, I think, is quite neat: it’s enough to find just two different colorings in order to show that the scenery reconstruction problem is impossible, but in fact there are many, many more of them. And what we gave was only a lower bound: it’s entirely plausible that there is a more clever way of counting these colorings, which will give an even larger number.

So far, so good. We saw that the scenery reconstruction problem is impossible for even-dimensional hypercubes, and even saw that there are different indistinguishable colorings. What about odd-dimensional cubes?

The answer comes from what we call “locally $p$-balanced” colorings. In these colorings, every vertex, no matter what color it is, has a $p$-fraction of its neighbors colored in its own color, and a $1-p$-fraction of its neighbors colored the opposite color. These colorings also yield identical scenery observations: when our agent tells us the colors she sees, the next color will always be the same as the current one with probability $p$. For example, the following figure shows a locally $2/3$-stable and a locally $1/3$-stable coloring on the $3$-dimensional cube.

Locally balanced colorings are different than locally biased colorings, although the two are related. For example, we saw that there is a strong restriction on the possible $p$ values for locally biased colorings, but it turns out that for every $p$, there is at least one locally balanced coloring. On the other hand, if $p = 1/n$ or $p = (n-1)/n$, then there is exactly one locally balanced coloring, while, as we saw, there are many, many locally biased ones.

The main point, though, is that every locally $1/2$-biased coloring $f$ on the $n$-dimensional hypercube can easily give a locally balanced coloring $g$ on the $n+1$ dimensional hypercube, by:

$g(x_1, \ldots, x_{n+1}) = g(x_1, \ldots, x_n) \cdot x_{n+1}.$

You should verify for yourself that this is indeed a locally balanced coloring, and calculate its $p$ value. This construction gives different locally balanced functions for all odd-dimensional hypercubes of dimension $n \geq 5$, showing that the scenery reconstruction problem is impossible for all $n \geq 4$. And that’s what we promised to show in the beginning of this post.

You should play around with locally biased and locally balanced colorings yourself; they serve as good brain teasers, and indeed we’ll end with a small riddle: here are two locally balanced colorings on the infinite one-dimensional and two-dimensional integer lattices. Can you find one for the three dimensional lattice?