Cracking Multivariate Recursive Equations Using Generating Functions

In this post, we return back to the combinatorial problem discussed in Introduction to Dynamic Programming and Memoization post. We will show that generating functions may work great not only for single variable case (see The Art of Generating Functions), but also could be very useful for hacking two-variable relations (and of course, in general for multivariate case too).

For making the post self-contained, we repeat the problem definition here.

The Problem

Compute the number of ways to choose $m$ elements from $n$ elements such that selected elements in one combination are not adjacent.

For example, for $n=4$ and $m=2$, the answer is $3$, since from the $4$-element set: $\lbrace 1,2,3,4 \rbrace$, there are three feasible $2$-element combinations: $\lbrace 1,4 \rbrace$, $\lbrace 2,4 \rbrace$, $\lbrace 1,3 \rbrace$.

Another example: for $n=5$ and $m=3$, there is only one $3$-element combination: $\lbrace 1,3,5 \rbrace$.

As we discussed in the first post, there is a nice recursive relation for this problem:

$$ F_{n, m} = F_{n - 1, m} + F_{n - 2, m - 1} + [n=m=0] + [n=m=1] $$

We assume that for any $n < 0$ or $m < 0$, $F_{n,m} = 0$. The indicator $[P]$ gives $1$ if the predicate $P$ is true.

The Generating Function for $F_{n,m}$

Let’s introduce the generating function $\Phi(x,y)$ of two (floating point) variables $x$ and $y$:

$$\Phi(x,y) = \sum_{n,m} F_{n,m} x^n y^m$$

Substituting the definition of $F_{n,m}$ and simplifying the sums, we will get:

$ \Phi(x,y) $ $ = \sum_{n,m} F_{n,m} x^n y^m $ $ = \sum_{n,m} \left(F_{n - 1, m} + F_{n - 2, m - 1} + [n=m=1] + [n=m=0] \right) x^n y^m$ $ = \sum_{n,m} F_{n - 1, m} x^n y^m + \sum_{n,m} F_{n - 2, m - 1} x^n y^m + x \cdot y + 1 $ $ = x \sum_{n,m} F_{n - 1, m} x^{n-1} y^m + x^2 y \sum_{n,m} F_{n - 2, m - 1} x^{n-2} y^{m-1} + x \cdot y + 1 $ $ = x \Phi(x,y) + x^2 y \Phi(x,y) + x \cdot y + 1 $

We have just found the simple representation for the generating function:

$$\Phi(x, y) = \frac{1 + x \cdot y}{1 - x - x^2 y}$$

The Closed Form for $F_{n,m}$

Using the infinite series: $ \frac{1}{1-z} = \sum_{k\geq 0} z^k $, we can make the following transforms:

$\Phi(x, y) $ $ = \frac{1 + x \cdot y}{1 - x - x^2 y}$ $ = (1 + x \cdot y) \sum_{k\geq 0} (x + x^2 y)^k $ $ = (1 + x \cdot y) \sum_{0 \leq i \leq k} {k \choose i} x^{k+i} y^i $ $ = \sum_{0 \leq i \leq k} {k \choose i} x^{k+i} y^i + \sum_{0 \leq i \leq k} {k \choose i} x^{k+i+1} y^{i+1}$

  1. Introducing the new variables: $n=k+i, m=i$, we cat transform the first expression as follows: $ \sum_{0 \leq i \leq k} {k \choose i} x^{k+i} y^i $ $ = \sum_{m \geq 0, n \geq 2m} {n-m \choose m} x^{n} y^m $.

  2. And introducing the new variables: $n=k+i+1, m=i+1$, we can transform the second expression similarly: $ \sum_{0 \leq i \leq k} {k \choose i} x^{k+i+1} y^{i+1} $ $ = \sum_{m \geq 1, n \geq 2m-1} {n-m \choose m-1} x^n y^m $.

The last two transformations give us the closed form:

$F_{n, m} $ $ = {n - m \choose m} + {n - m \choose m - 1}$.

Which actually equals to

$$ F_{n, m} = {n - m + 1 \choose m} $$

Fast Solutions Based on Binomials

Now we can reflect this idea in very trivial Python code:

import math

def f_binom(n, m):
    assert n >= 0 and m >= 0

    if n + 1 < 2*m:
        return 0

    return binom(n - m + 1, m)

def binom(n, m):
    assert 0 <= m <= n

    return math.factorial(n) // math.factorial(m) // math.factorial(n - m)

This implementation overperforms significantly the initial DP and memoization solutions. A naive implementation of math.factorial() might make $n$ multiplications. This could still be faster than doing $\Theta(n)$ additions in DP approach.

The actual implementation of math.factorial() is written in C and probably has precomputed results for some range of $n$ and might even cache the results for bigger $n$.

Implementations Based on scipy and sympy Libraries

Several third party libraries provide a functionality to compute binomial coefficients. Let’s take a look at scipy and sympy.

We can install both of them using pip-package manager:

pip install scipy sympy

We can easily write two implementations of $F_{n,m}$ which will call scipy.special.comb() or sympy.binomial() functions:

import scipy.special

def f_sci(n, m):
    assert n >= 0 and m >= 0

    if n + 1 < 2*m:
        return 0

    return scipy.special.comb(n - m + 1, m, exact=True)

The second one is very similar to the first one:

import sympy

def f_sym(n, m):
    assert n >= 0 and m >= 0

    if n + 1 < 2*m:
        return 0

    return sympy.binomial(n - m + 1, m)

We can use the same test() helper function that we defined in Introduction to Dynamic Programming and Memoization. Let’s run it on all the 5 implementatnions:

funcs = [f_mem, f_dp, f_binom, f_sci, f_sym]
test(6000, 2000, funcs)

It will print something similar to following output:

f(6000,2000): 192496093
       f_mem:   6.7195 sec, x 4195.10
        f_dp:   5.3249 sec, x 3324.43
     f_binom:   0.0016 sec, x 1.00
       f_sci:   0.0021 sec, x 1.32
       f_sym:   0.0043 sec, x 2.69

The first two methods, which are based on memoization and DP, are much slower than the last three, which are based on the binomial coefficients.

The Intuition for the Time Complexity Analysis

DP and memoization makes $O(n^2)$ of “addition” operations over long integers. The long integers are bounded by $F_{n,m}$ value, which could be bounded by $2^n$. One “addition” operation takes $O(N)$ time for $N$-digit integer input. For our case $N$ could be bounded by $ O(\log 2^n)$ $ = O(n)$. So the total time complexity is bounded by $ O(n^2 \cdot N) $ $ = O(n^2 \cdot \log 2^n) $ $ = O(n^2 \cdot n) $ $ = O(n^3)$.

The binomial based solutions make $O(n)$ “multiplication” opearations over long integers The long integers could be bounded by $O(n!)$ $ = O(n^n)$. The “multiplication” opertion could be implemented in a naive way which runs $O(N^2)$ in time and is used for a small input. But it also has a more advanced implementation, which takes $O(N^{\log_2 3})$ $ = O(N^{1.59})$ and is used on big integers. Note that here $N$ denotes the number of digits in the input long integers for multiplication. In this case, $N$ is bounded by $ O(\log n!) $ $ = O(\log (n^n)) $ $ = O(n \log n) $. The total time complexity of the binomial based implementations is bounded by $ O\left(n \cdot N^{\log_2 3}\right) $ $ = O\left(n \cdot (\log n!)^{\log_2 3}\right)$ $ = O\left(n \cdot (n \log n)^{1.59}\right)$ $ = O\left(n^{2.59} \cdot (\log n)^{1.59}\right)$.

This is not really a formal proof, but it gives some intuition why the last approach overperforms the former one. In practice, the results of factorial computation are cached, therefor we observe even bigger gap in performance (yeahh again not formal claim, just an intuition).

You can play with running tests on different $n$ and $m$. What I saw that actually there is no clear winner between the last 3 implementations. Probably, the most of the time is spent on the long arithmetic computation.

Modular Arithmetics

In questions where it is required to count some objects, not rearly the answer might be very big even on very small input. In such case, typically it is asked to print the answer modulo some big prime integer, let’s say, $M=1000^3+7$. Since Python has built-in long arithmetics, we can apply modulo on the final result, but executing the entire agorithm with long arithmetics while knowing that only small part of it is really important is very costly, and of course, not that efficient.

Let’s look, briefly, at very simple change we can do for f_binom function that will speed up the computation significantly:

def f_binom_mod(n, m):
    assert n >= 0 and m >= 0

    if n + 1 < 2*m:
        return 0

    return binom_mod(n - m + 1, m)

def binom_mod(n, m):
    assert 0 <= m <= n

    return ((fact_mod(n) * inv_mod(fact_mod(m))) % M * inv_mod(fact_mod(n - m))) % M

def fact_mod(m):
    if m <= 1:
        return 1

    return (m * fact_mod(m - 1)) % M

def inv_mod(x):
    return pow(x, M - 2, M)

As we can see, all the operations are computed modulo $M$. The function fact_mod is recursive but uses Memoization. The most tricky part is how to implemenent modular-division. From Fermat’s little theorem, we know that if $M$ is prime and $0 < x < M$, then $x^{-1} \equiv x^{M-2} \pmod M$. This allows to compute the multiplicative inverse of $x$ using the Python’s built-in function pow.

Let’s test the new approach against other implementations:

fact_mod(10000) # for caching factorials

funcs = [f_binom_mod, f_binom, f_sci, f_sym]

test(10000, 1000, funcs)
test(10000, 2000, funcs)
test(10000, 3000, funcs)

It is not a surprise that taking the benefits of modular computations results in the huge speedup in running-time:

f(10000,1000): 450169549
  f_binom_mod:   0.0000 sec, x 1.00
      f_binom:   0.0073 sec, x 337.60
        f_sci:   0.0011 sec, x 49.33
        f_sym:   0.0076 sec, x 353.22

f(10000,2000): 75198348
  f_binom_mod:   0.0000 sec, x 1.00
      f_binom:   0.0063 sec, x 368.94
        f_sci:   0.0026 sec, x 153.33
        f_sym:   0.0053 sec, x 308.93

f(10000,3000): 679286557
  f_binom_mod:   0.0000 sec, x 1.00
      f_binom:   0.0060 sec, x 361.12
        f_sci:   0.0056 sec, x 338.13
        f_sym:   0.0053 sec, x 319.02