I've been working on on introducing some results from computational complexity into theoretical biology, especially evolution & ecology, with the goal of being interesting/useful to biologists. One of the biggest difficulties I've faced is in justifying the usefulness of asymptotic worst-case analysis for lower bounds. Are there any article length references that justify lower bounds and asymptotic worst-case analysis to a scientific audience?

I am really looking for a good reference that I can defer to in my writing instead of having to go through the justifications in the limited space I have available (since that is not the central point of the article). I am also aware of other kinds and paradigms of analysis, so I am not seeking a reference that says worst-case is the "best" analysis (since there are settings when it very much isn't), but that it isn't completeletely useless: it can still gives us theoretically useful insights into the behavior of actual algorithms on actual inputs. It is also important the writing is targeted at general scientists and not just engineers, mathematicians, or computer scientists.

As an example, Tim Roughgarden's essay introducing complexity theory to economists is on the right track for what I want. However, only sections 1 and 2 are relevant (the rest is too economics specific) and the intended audience is a bit more comfortable with theorem-lemma-proof thinking than most scientists[1].


In the context of adaptive dynamics in evolution, I've met two specific types of resistance from theoretical biologists:

[A] "Why should I care about behavior for arbitrary $n$? I already know that the genome has $n = 3*10^9$ base pairs (or maybe $n = 2*10^4$ genes) and no more."

This is relatively easy to brush-off with the the argument of "we can imagine waiting for $10^9$ seconds, but not $2^{10^9}$". But, a more intricate argument might say that "sure, you say you care about only a specific $n$, but your theories never use this fact, they just use that it is large but finite, and it is your theory that we are studying with asymptotic analysis".

[B] "But you only showed that this is hard by building this specific landscape with these gadgets. Why should I care about this instead of the average?"

This is a more difficult critique to address, because a lot of the tools people commonly use in this field are coming from statistical physics where it is often safe to assume a uniform (or other specific simple) distribution. But biology is "physics with history" and almost everything isn't at equilibrium or 'typical', and empirical knowledge is insufficient to justify assumptions about distributions over input. In other words, I want an argument similar to that used against uniform distribution average-case analysis in software engineering: "we model the algorithm, we can't construct a reasonable model of how the user will interact with the algorithm or what their distribuition of inputs will be; that is for psychologists or end users, not us." Except in this case, the science isn't at a position where the equivalent of 'psychologists or end users' exists to figure out the underlying distributions (or if that is even meaningful).

Notes and related questions

  1. The link discusses cognitive sciences, but the mindset is similar in biology. If your browse through Evolution or Journal of Theoretical Biology, you will seldom see theorem-lemma-proof, and when you do it will typically be just a calculation instead of something like an existence proof or intricate construction.
  2. Paradigms for complexity analysis of algorithms
  3. Other kinds of running time analysis besides worst-case, average-case, etc?
  4. Ecology and evolution through the algorithmic lens
  5. Why economists should care about computational complexity
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    $\begingroup$ Worst-case behavior is impossible to justify … the simplex algorithm has exponential worse-case behavior, and the only people who have ever cared are the theorists. What you need to argue is (a) the average-case asymptotic behavior is important; (b) the average-case asymptotic behavior and worst-case asymptotic behavior are quite often similar; (c) the worst-case asymptotic behavior is often much easier to calculate than the average-case asymptotic behavior (especially since nobody knows what the relevant probability distribution is). $\endgroup$ Jan 25, 2014 at 23:15
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    $\begingroup$ Asymptotics is already a problematic aspect. We all know the story about matrix multiplication algorithms (asymptotic upper bounds are meaningless in practice), and perhaps also the story about choosing parameters in cryptography (asymptotic lower bounds are meaningless in practice; exponential algorithms are sometimes feasible [DES]). If your analysis has actual constants then it is more convincing. $\endgroup$ Jan 26, 2014 at 1:56
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    $\begingroup$ If you think about computation as a game (i.e., war) between the input provider and the algorithm, then the worst case analysis is a standard military approach - you want to know how bad can it be. Secondly, and more importantly, worst case analysis does not allow you to be intellectually lazy and accept solutions that might be good to what you believe the world is (and not what the world is really is). Finally, and maybe most importantly, it provide a uniform way to compare algorithms in a hopefully meaningful way. In short, it is the worst approach, except for all the others. $\endgroup$ Jan 26, 2014 at 3:46
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    $\begingroup$ I think a worst-case lower bound should be seen as putting the ball back in their court. You have shown that there is no algorithm that can solve their problem on all instances in a reasonable time frame. They may reasonably believe that their instances are easy -- but you have just shown that if this is so, it is a non-trivial fact. Their model is therefore incomplete unless they come up with an explanation for why this is so. $\endgroup$
    – Aaron Roth
    Jan 28, 2014 at 21:59
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    $\begingroup$ (This is the approach that seems to work when talking to game theorists. It raises the question -- if markets truly equilibriate quickly -- what special property do real markets have that gets around worst case hardness? It is likely possible to define a plausible such property, and the lower bounds just suggest that doing so is an important research direction) $\endgroup$
    – Aaron Roth
    Jan 28, 2014 at 22:01

3 Answers 3


My personal (and biased) take is that asymptotic worst-case analysis is a historical stepping stone to more practically useful kinds of analysis. It therefore seems hard to justify to practitioners.

Proving bounds for the worst case is often easier than proving bounds for even "nice" definitions of average case. Asymptotic analysis is also often much easier than proving reasonably tight bounds. Worst-case asymptotic analysis is therefore a great place to start.

The work of Daniel Spielman and Shanghua Teng on smoothed analysis of Simplex seems to me a harbinger of what can happen when we start to gain a better understanding of the shape of a problem: tackling the worst-case first enables a more nuanced understanding to be developed. Further, as Aaron Roth suggested in the comments, if the "usual" behaviour of a system is significantly different from its worst case, then the system is not yet completely specified and more work is needed to improve the model. So going beyond worst-case generally seems important as a long-term goal.

As far as asymptotic analysis is concerned, it usually serves to keep a long and messy proof clear of distracting details. Unfortunately there currently does not seem to be a way to reward the tedious work of filling in the details to obtain the actual constants, so that seldom seems to get done. (Page limits also work against this.) Careful analysis of the details of an asymptotic bound has led to actual algorithms, with good bounds on the constants, so I personally would like to see more of this kind of work. Perhaps if more proofs were formalised using proof assistant systems, then the constants could be estimated with less additional effort. (Or bounds on the constants, along the lines of Gowers' bound for the Szemerédi Regularity Lemma, might become more routine.) There are also ways to prove lower bounds that are free of constants, by using more explicit machine models (such as deterministic finite-state automata). However, such (near-)exact lower bounds for more general models of computation may require a lot of work, or be out of reach altogether. This seems to have been pursued in ~1958–73 during the first heyday of automata theory, but as far as I can tell has since largely been left alone.

In short: in future, I think we'll see worst-case analysis as just a first step, and I hope we'll see expressions like $O$*$(n^k)$ with galactic hidden constants as just the first step. I find it difficult to justify the current focus on worst-case asymptotic analysis to practitioners, but perhaps it is useful to see this as the beginning steps of a longer journey.

  • $\begingroup$ I don't share your enthusiasm for ditching asymptotics in favor of precise bounds with definite constants. Asymptotics may be imprecise -- but they're usefully imprecise. They abstract over implementation differences for the same machine model. E.g., a sorting algorithm that was quadratic on 1950s hardware will still be quadratic on today's hardware. Furthermore, asymptotic formulas compose nicely. Linears and polynomials are closed under composition, for instance. (Note that arguing for better bounds on the average case compared to worst case is orthogonal from arguing against asymptotics.) $\endgroup$
    – brandjon
    Jan 31, 2014 at 1:05
  • $\begingroup$ You are right in general, but there is a big difference between a small constant and one that is a non-elementary function of a relevant parameter. $\endgroup$ Jan 31, 2014 at 1:13
  • $\begingroup$ I like this answer overall, but I agree with @brandjon that hiding constants is crucial. For me, the reason TCS is useful in biology is because it needs to make much fewer assumptions about micro dynamics than physics. However, if you don't make assumptions about the micro dynamics (I.e. the exact specification of the model of computation) then you can't get the constant factors out. The other useful feature of TCS is rigorous qualitative dichotomies (something that is easier to compare to the more qualitative observations in bio), usually to get these you also have to ditch constants. $\endgroup$ Jan 31, 2014 at 18:13
  • $\begingroup$ Hiding constants is OK up to a point. But when one reads of an algorithm that takes $\tilde{O}(n^{\tilde{O}(1/\epsilon)})$ time (as I often do) there is definitely something missing. Also, I am not entirely convinced that including constants in an analysis will typically require a detailed machine model. For example, the constants in sorting algorithms are now well known and this hasn't required some micro description of a particular machine. $\endgroup$
    – user17100
    Jan 31, 2014 at 19:59
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    $\begingroup$ As a side note, there are examples where worst-case analysis makes sense. For example, when you develop a library of general-purpose subroutines and do not know in what application domains they will be useful: you cannot possibly anticipate all cases of when and why someone will want to compute a minimum cost bipartite matching, for example. Adversarial settings, such as cryptography, are even more clear cut (however, in crypto you'd really like to know the constants when it comes to security parameters). $\endgroup$ Feb 3, 2014 at 2:42

Lower bounds and worst-case analysis don't usually go together. You don't say an algorithm will take at least exponential time in the worst case, therefore it's bad. You say it can take at most linear time in the worst case, and therefore is good. The former is only useful if you are going to run your algorithm on all possible inputs, and not merely an average input.

If you want to use lower bounds to demonstrate badness, then you want a best-case analysis or an average-case analysis. You can simplify things by relying on @PeterShor's point that worst and average are often very similar, and give a laundry list of algorithms for which this is true. (Ex: all the classic sorts besides quicksort.)

As for demonstrating that asymptotics matter when compared to constant inputs and constant factors, my favorite article on the topic is Jon Bentley's "Programming pearls: algorithm design techniques". He presents four different solutions to a simple array problem, and demonstrates how the linear approach annihilates the cubic one. He calls his table "The Tyranny of Asymptotics", after the term used by physicists for the intractability of the rocket equation. I use this example to motivate the search for better algorithms to pre-college students.

Will a non-computer scientist read through an article that contains code, and know to skip over the low-level details to get the big picture? I don't know. Perhaps there's a better presentation elsewhere. But I think this is a decent resource to cite.

And if they argue that they don't care about arbitrarily large n, have them run recursive un-memoized Fibonacci on 3 * 109 base pairs, and tell them it's O(1) since the size of the DNA sequence is fixed. ;)

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    $\begingroup$ I like the fibonacci example :) $\endgroup$ Jan 28, 2014 at 17:58
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    $\begingroup$ Re: your first paragraph: actually, that's almost exactly what a lot of complexity theory does. If a problem is EXP-complete, that means it requires exponential time on worst-case inputs. This is generally taken as an indication of its overall difficulty (which, to be fair, in practice is often not so bad as a general indicator). This is the de facto standard, called an "infinitely-often" or i.o. lower bound; getting average-case or almost-everywhere lower bounds (that is, for all but finitely many inputs) is a goal sometimes pursued, but often far out of reach compared to i.o. lower bounds. $\endgroup$ Jan 30, 2014 at 21:15
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    $\begingroup$ Let me point out that not only can you give a laundry list of algorithms for which worst-case and average-case analysis are the same, but you can also give numerous examples where they are very different (the simplex algorithm just being the most famous of these). You really need to argue somehow that they are the same for your particular application; experimental testing is a good way to do this. $\endgroup$ Jan 30, 2014 at 21:28
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    $\begingroup$ @JoshuaGrochow Fair enough. How about we revise the statement as follows: Lower bounds on worst-cases are important when you want to demonstrate the absence of a mathematical guarantee of non-horribleness. ;) $\endgroup$
    – brandjon
    Jan 31, 2014 at 0:56

much agreed this an important topic to survey/cover but it seems not to have been much yet. a few refs of varying style/coverage/audience/ formality not exactly as requested but somewhat close (best seen online so far on medium search, hope to hear further of any better ones; more notes below):

  • The complexity of algorithms Atkinson (alas only a single ref to biology in the paper, but may suffice on more general science/engineering terms)

    The modern theory of algorithms dates from the late 1960s when the method of asymptotic execution time measurement began to be used. It is argued that the subject has both an engineering and scientific wing. The engineering wing consists of well-understood design methodologies while the scientific wing is concerned with theoretical underpinnings. The key issues of both wings are surveyed. Finally some personal opinions on where the subject will go next are offered.

  • Complexity and Algorithms J. Diaz. 100 slides. broad; one could excerpt relevant ones in particular.

  • A Gentle Introduction to Algorithm Complexity Analysis Dionysis "dionyziz" Zindros

in other words is there a sort of introduction/survey/overview of the complexity theoretic lens in close combination/conjunction/companion with the advancing algorithmic lens in science, something like "Complexity theory for scientists, engineers, and researchers"?

there are good refs on the former "algorithmic lens" you have cited eg Papadimitriou but it does not seem a highly satisfactory ref by an expert in the field has been written on the latter "complexity lens"... yet (maybe some "elite" member of this site will consider that as their next book or paper project).

note also there are a lot of refs on relevance P vs NP outside of complexity theory & in other scientific fields that could be used somewhat for these purpose. will add them in comments if there is any interest.

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    $\begingroup$ I don't think this really answers the question. $\endgroup$ Jan 29, 2014 at 20:16
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    $\begingroup$ uh huh, did you look at any of the refs? part my answer is that there is not (yet) any ideal/perfect answer :| $\endgroup$
    – vzn
    Jan 29, 2014 at 20:18
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    $\begingroup$ They seem to define asymptotic and worst case analysis rather than focusing on justifying it, but maybe I missed something? $\endgroup$ Jan 29, 2014 at 20:21
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    $\begingroup$ Actually, I think researchers outside TCS could easily dismiss worst-case as "artificially constructed examples that would never occur in practice" and would be (without strong convincing otherwise) much more interested in average-case (despite the fact that it's not clear that the average-case is that much closer to real-world instances). $\endgroup$ Jan 30, 2014 at 2:59
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    $\begingroup$ @vzn: Asymptotic (e.g. big-Oh) and worst-case are somewhat orthogonal. One can do asymptotic worst-case analysis, asymptotic average-case analysis, or even asymptotic easiest-case analysis (though I admit the latter seems somewhat perverse). One could instead do exact worst-case analysis, or exact average-case analysis, and so on, though these would be much more model-dependent and less robust. Justifying the use of asymptotics (and hiding things like constant factors) is entirely distinct from justifying worst-case vs average-case or "real-"case (whatever the latter might mean...). $\endgroup$ Jan 31, 2014 at 22:51

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