Recurrences and Generating Functions in Algorithms

Question

Combinatorics plays an important role in computer science. We frequently utilize combinatorial methods in both analysis as well as design in algorithms. For example one method for finding a $k$-vertex cover set in a graph might just inspect all $\binom{n}{k}$ possible subsets. While the binomial functions grows exponentially, if $k$ is some fixed constant we end up with a polynomial time algorithm by asymptotic analysis.

Often times real-life problems require more complex combinatorial mechanisms which we may define in terms of recurrences. One famous example is the fibonacci sequence (naively) defined as:

$f(n) = \begin{cases} 1 & \text{if } n = 1 \\ 0 & \text{if } n = 0 \\ f(n-1) + f(n-2) & \text{otherwise} \end{cases} $

Now computing the value of the $n$th term grows exponentially using this recurrence, but thanks to dynamic programming, we may compute it in linear time. Now, not all recurrences lend themselves to DP (off hand, the factorial function), but it is a potentially exploitable property when defining some count as a recurrence rather than a generating function.

Generating functions are an elegant way to formalize some count for a given structure. Perhaps the most famous is the binomial generating function defined as:

$(x + y)^\alpha = \sum_{k=0}^\infty \binom{\alpha}{k}x^{\alpha - k}y^k$

Luckily this has a closed form solution. Not all generating functions permit such a compact description.

Now my question is this: how often are generating functions used in design of algorithms? It is easy to see how they may be exploited to understand the rate of growth required by an algorithm via analysis, but what can they tell us about a problem when creating a method to solve some problem?

If many times the same count may be reformulated as a recurrence it may lend itself to dynamic programming, but again perhaps the same generating function has a closed form. So it is not so evenly cut.

Answer 1

Generating functions are useful when you're designing counting algorithms. That is, not only when you're looking for the number of objects having a certain property, but also when you're looking for a way to enumerate these objects (and, perhaps, generate an algorithm to count the objects). There is a very good presentation in chapter 7 of Concrete Mathematics by Ronald Graham, Donald Knuth, and Oren Patashnik. The examples below are from these books (the mistakes and lack of clarity are mine).

Suppose that you're looking for the ways to make change with a given set of coins. For example, with common US denominations¹, the possible coins are $[1], [5], [10], [25], [100]$. To give ¢42 in change, one possibility is $[25][10][5][1][1]$; another possibility is $[10][10][10][10][1][1]$. We'll write $42 \langle [25][10][5][1]^2 \rangle = \langle [10]^4 [1]^2 \rangle$. More generally, we can write a generating function for all the ways to give change: $$H = \sum_{h\ge0} \sum_{q\ge0} \sum_{d\ge0} \sum_{n\ge0} \sum_{p\ge0} [100]^h [25]^q [10]^d [5]^n [1]^p$$ In more technical terms, $H$ is a term in the space of power series over the five variables $[100], [25], [10], [5], [1]$. Define the valuation of a monomial in this space by $$\langle [100]^h [25]^q [10]^d [5]^n [1]^p \rangle = 100 h + 25 q + 10 d + 5 n + p$$ Then the ways to give $v$ cents in change are the number of monomials whose valuation is $v$. We can express $H$ in an incremental fashion, by first writing down the ways $P$ to give change in pennies only, then the ways $N$ to give change in pennies and nickels, and so on. ($I$ means no coin.) $$\begin{gather*} P = I + [1] + [1]^2 + [1]^3 + \ldots = \frac{I}{I - [1]} \\ N = (I + [5] + [5]^2 + [5]^3 + \ldots) P = \frac{P}{I - [5]} \\ D = (I + [10] + [10]^2 + [10]^3 + \ldots) N = \frac{N}{I - [10]} \\ Q = (I + [25] + [25]^2 + [25]^3 + \ldots) D = \frac{D}{I - [25]} \\ H = (I + [100] + [100]^2 + [100]^3 + \ldots) Q = \frac{Q}{I - [100]} \\ \end{gather*}$$ If you want to count and not just enumerate the ways to give change, then there is a simple way to use the formal series we've obtained. Apply the homomorphism $$S: \quad [1] \mapsto X, \quad [5] \mapsto X^5, \quad [10] \mapsto X^{10}, [25] \mapsto X^{25}, [100] \mapsto X^{100} $$ The coefficient of $X^v$ in $S(C)$ is the number of ways to give $v$ cents in change.

A harder example: suppose that you want to study all the ways to tile rectangles with 2×1 dominoes. For example, there are two ways to tile a 2×2 rectangle, either with two horizontal dominoes or with two vertical dominoes. Counting the number of ways to tile a $2 \times n$ rectangle is fairly easy, but the $3 \times n$ case quickly becomes nonobvious. We can enumerate all the possible tilings of a horizontal band of height 3 by sticking dominoes together, which quickly yields repetitive patterns: $$\begin{cases} U = \mathsf{o} + \mathsf{L} V + \mathsf{\Gamma} \Lambda + \mathord{\equiv} U \\ V = \substack{\mathsf{I}\\\strut} U + \substack{=\\\:-} V \\ \Lambda = \substack{\strut\\\mathsf{I}} U + \substack{\:-\\=} \Lambda \\ \end{cases}$$ where the funny shapes represent elementary domino arrangements: $\mathsf{o}$ is no domino, $\mathsf{L}$ is a vertical domino on top of the left part of a horizontal domino, $\substack{\strut\\\mathsf{I}}$ is a vertical domino aligned with the bottom of the band of height 3, $\substack{\:-\\=}$ is a horizontal domino aligned with the top of the band plus two horizontal dominoes below it and one step to the right, etc. Here, multiplication represents horizontal concatenation and is not commutative, but there are equations between the elementary patterns that form variables in this power series. As before with the coins, we can substitute $X$ for every domino and get a generating series for the number of tilings of a $3 \times (2n/3)$ rectangle (i.e. the coefficient of $X^{3k}$ is the number of ways to tile a rectangle of area $6k$, which contains $3k$ dominoes and has the width $2k$). The series can also be used in more versatile ways; for example, by distinguishing vertical and horizontal dominoes, we can count the tilings with a given number of vertical and horizontal dominoes.

Again, read Concrete Mathematics for a less rushed³ presentation.

¹ _{I know my list is incomplete; assume a simplified US suitable for mathematical examples.²}
² _{Also, if it comes up, assume spherical coins.}
³ _{And better typeset.}

Answer 2

I remember a problem I had to solve during a student programming contest in 2001. The problem was this one:

Given masses of 1, 7, 13, ... (I don't remember which masses, but there was a finite, determinate set of masses), design a function that determines whether a given weight can be weighed on a scale with this set of masses.

I started with nested loops, but quickly hit a wall. Then I came to realize that I had to start with enumerating what can be done with the lighter masses before going on with the heavier ones. I could solve the problem with a lot of unnested loops.

Hadn't I been youthfully arrogant and self-sufficient at that time (and had I known about and practiced generating functions), I might have defined the problem with generating functions as such :

Define $f(x)$ as the OGF for the number of ways that a weight $n$ can be weighed given the set of masses.

What weight on the right pan can I weigh given a single mass of 1?

Three possibilities:

• If I put the mass on the left pan, I can weigh 1.
• If I put the mass on the right pan, I can weigh -1.
• If I don't use the mass, I can weigh 0.

So there is one way to weigh $-1$, one way to weigh $0$, and one way to weigh $1$. The generating function for this mass is something like $x^{-1} + 1 + x$, which corresponds to:

$$ \frac{1 - x^3}{x(1-x)} $$

The generating function for a single mass $m$ is $x^{-m} + 1 + x^m$, which is :

$$ \frac{1 - x^{3m}}{x^m(1-x^m)} $$

Given a multiset $M$ of masses, $f$ is expressed as the product of the single mass generating functions:

$$ f(x) = \frac{\prod_{m \in M}(1 - x^{3m})}{x^{\sum_{m \in M}m}\prod_{m \in M}(1 - x^{m})} $$

Now, given a package that can perform operations on polynomials, you just have to :

• Calculate both products.
• Perform the division of these products, starting with the lowest degree. (which terminates)
• Shift the polynomial (euclidian division by $x^k$, keeping the quotient and dumping the remainder)

And you're done. Now your polynomial has the number of ways to weigh $w \ge 0$ at index $w$. The only input is the multiset of masses $M$.

I designed the algorithm using mathematically sound components. The main part of the algorithm, which is a polynomial division with lowest degree first, is linear and can be implemented by an off-the-shelf package. It may not be optimal, but it certainly performs better than what I did at the contest, and in a less error-prone way.

If you look closely at the division process, you quickly see that the remainder can be seen as the "current hidden state" at every state of the process, and the quotient as the result. The process terminates when the "current hidden state" reaches zero everywhere.

You can implement polynomials as arrays or, if they are really really sparse, as index-coefficient ordered lists, and this won't change the algorithm.

Answer 3

While developing an algorithm for monotone submodular maximization over a matroid, we had to solve the recurrence $$ \ell \gamma^{(m)}_{\ell+1} = (2\ell-m) \gamma^{(m)}_\ell + (m - \ell + 1) \gamma^{(m)}_{\ell-1}, \quad \gamma^{(m)}_0 = 1, \quad \gamma^{(m)}_{m+1} = e. $$ After noticing that $\gamma^{(m)}_\ell = m(\gamma^{(m-1)}_\ell - \gamma^{(m-1)}_{\ell-1})$, we reduced the problem to computing some universal sequence $\gamma^{(0)}$. The latter was accomplished using generating functions, and from there we got an explicit formula for $\gamma^{(m)}$, again using generating functions. You can find the solution in the paper if you're curious, though we never bothered to include this derivation.

Answer 4

Perhaps the extensive study of Quicksort, and its many variants, is the clearest example. There combinatoric considerations governed the consideration of alternatives, and analyzing the solutions to quite complex equations shows performance advantages (or not) of them.