Erdős’s Book and the Asymptotic Religion

In an undergraduate algorithms class we learn that an algorithm is a high level way to describe a computer program. The running time of the algorithm is the number of operations it takes on inputs of a particular size- the smaller the better. So, as even Barack Obama knows, if you implement Quick-Sort, with its running time of {O(n\log n)}, it would run faster than the {O(n^2)}-time algorithm Bubble-Sort. However, if you pick up a more recent algorithms paper, things become a little murkier. Many papers feature algorithms with running times such as {O(n^6)} or even {O(n^{20})}, and even some linear-time algorithms have hidden constants that make you want to run to your bed and hide under the covers. Asymptotic analysis does become more applicable as technology improves and our inputs grow but some people may still question the practical relevance of, say, an {O(n^{10})}-time algorithm, which seems no more feasible than the trivial {2^n}-time brute force algorithm.

So why do we care about getting asymptotically good algorithms? Every TCS graduate student should be able to recite the “party line” answer that it has happened again and again that once a problem has been shown to be in polynomial time, people managed to come up with algorithms with small exponents (e.g. quadratic) and reasonable leading constants. There is certainly some truth to that (see for example the case of the Ellipsoid and Interior Point algorithms for linear programming) but is there an underlying reason for this pattern, or is it simply a collection of historical accidents?

Paul Erdős envisioned that “God” holds a book with the most elegant proof for every mathematical theorem. In a famous quote he said that “you don’t have to believe in God, but you have to believe in the Book”. Similarly, one can envision a book with the “best” (most efficient, elegant, “right”) algorithms for every computational problem. The {2^n}-time brute force algorithm is in this book, and the essence of the {\mathbf{P}\neq\mathbf{NP}} conjecture is that this exhaustive search algorithm is in fact the best algorithm for some problems in {\mathbf{NP}}. When a researcher shows a cumbersome and impractical {O(n^{20})}-time algorithm A for a problem {X}, the point is not that you should code up A and use it to solve the problem {X} in practice. Rather, the point is that A‘s existence proves that  {X} is not one of those problems for which brute force is optimal. This is a very significant piece of information, since it tells us that the “Book algorithm” for {X} will be much faster than brute force, and indeed once a non-trivial algorithm is given for a problem, further improvements often follow as researchers get closer and closer to the true “Book algorithm” for it. We’ve seen enough of the Book to know that not all algorithms in it are either {O(n^2)} or {2^{\Omega(n)}} time, but there should still be an inherent reason for the “Book algorithm” to have a non-trivial exponent. If the best algorithm for some problem {X} is, say, {n^{\log n}}-time, then by reading that page of the Book we should understand why this is the right answer (possibly because solving {X} boils down to running the brute force algorithm on some smaller domain).

The same intuition holds for hardness results as well. The famous PCP theorem can be thought of as a reduction from the task of exactly solving the problem SAT to the task of solving it approximately. But if you apply the original PCP reduction to the smallest known hard instances of SAT (which have several hundred variables) the resulting instance would be too large to fit in even the NSA’s computers. What is the meaning of such a reduction? This issue also arises in cryptography. As some people like to point out, often when you reduce the security of a cryptographic scheme Y to the hardness of some problem X, the reduction only guarantees security for key sizes that are much larger than what we’d like to use in practice. So why do we still care about such reductions? The answer is that by ruling out algorithms that run much faster than the natural exponential-time one, the reduction gives at least some evidence that this particular approximation or crypto-breaking problem might be one of those problems for which the exponential-time algorithm is the “Book algorithm”. And indeed, just like the case for algorithms, with time people have often been able to come up with increasingly more efficient reductions.

Another way to think of this is that the goal of a theory paper is not to solve a particular problem, but to illustrate an idea- discover a “paragraph from the book” if you will. The main reason we care about measures such as asymptotic running time or approximation factor is not due to their own importance but because they serve to “keep us honest” and provide targets that we won’t be able to reach without generating new ideas that can later be useful in many other contexts.

Erdős’s quote holds true for theoretical computer scientists as well. Just as physicists need to believe in a “book” of the laws of nature to make inferences based on past observations, the Book of algorithms is what makes TCS more than a loose collection of unrelated facts. (For example, this is why in his wonderful essay, Russell Impagliazzo was right to neglect some “non-Book” scenarios such as \mathbf{P}=\mathbf{NP} with an n^{100}-time algorithm.) Ours is a young science, and we only know very few snippets from the Book. I believe that eventually, even statements such as {\mathbf{P}\neq\mathbf{NP}} would be seen as mere corollaries of much grander computational “laws of nature”.

[Updated 2/25 with some minor rephrasing]

9 thoughts on “Erdős’s Book and the Asymptotic Religion

    1. Thank you for the pointer. I think some people take the “Book” metaphor slightly differently than what I mean above. I see the “Book” as the mathematical analog to Occam’s Razor – we have certain observations, for example that a problem is NP hard or that a problem has a non-trivial algorithm, and believing in the Book corresponds to believing that there is a beautiful/economical explanation for these observations – i.e., that the optimal algorithm is elegant and simple. (Of course “elegance” and “simplicity” are not well defined terms, and an important part of our progress in science can be thought of as “learning God’s measures for elegance”; for example as people got used to quantum mechanics and understand it better they started appreciating its mathematical elegance even if it wasn’t immediately apparent.)

      The Book algorithm is not the first paper on the subject but the last one. So to me results such as the AKS primality testing or Gentry’s Fully Homomorphic Encryption are amazing breakthroughs but they are *not* the Book algorithms.

      1. I didn’t see your reply till just now.

        In my first reading of your post, I interpreted your notion of an entry in the Book as a view into “explanations from the Book” rather than the strict notion of “proofs from the Book”. Although, I imagine that the distinction between explanation and proof is a debatable one, especially since a proof that one imagines as being in the Book should have explanatory value.

        The cstheory.stackexchange entries are quite different, with the notion of algorithms from the Book, and a lot of the entries were of the succinct and clever variety, but not necessarily of the type that exposed a fundamental underlying structure.

  1. One remark on this nice post: Of course, the asymptotic lens is a very powerful paradigm, both in mathematics and in the theory of algorithms. One typical flaw in application of asymptotic thinking is in cases where there are more than one parameter. With certain problems depending (say) on two parameters m and n there are cases where mainly the asymptotic behavior for fixed m and n tending to infinity is studied, and that insights for this regime are interpreted too widely. It is useful to remember that with several variables there could be many interesting mutual asymptotic behaviors and when you consider related questions or even applications it is a subtle issue to find the relevant asymptotics.

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