Given some function and assuming no concern for the time to compute the co-domain for its domain, when might it be preferable to compute and tabulate in advance the co-domain results of the function, and then retrieve it from the table rather than compute it?

For example, if the domain for $f(x)=x^2$ were sufficiently small, why, if ever, would one wish to compute the co-domain and then, for a given x, retrieve the stored value of $f(x)$?

  • $\begingroup$ I can't understand what you are asking. We don't compute the co-domain. Given a function $f:X \to Y$, the set $Y$ is called the co-domain. We don't compute the set $Y$; it's a property of $f$. Instead, given $x$, we compute $f(x)$. So what exactly is your problem/question? Can you give more specifics, more context, ask about a specific function? Also tell us what research you've done, your thoughts, and what possible answers/factors you've considered so far. $\endgroup$ – D.W. Jun 24 '15 at 2:15
  • $\begingroup$ I see how I miswrote: thank you for your clarification. (Somehow Luke was able to guess at my meaning, however.) $\endgroup$ – gallygator Jun 24 '15 at 12:19

This sort of computation can be used for optimization purposes.

A classic simple example is computing the Fibonacci sequence. Apart from the base cases, $f(n) = f(n - 1) + f(n - 2)$, but then $f(n+1) = f(n) + f(n-1)$ and $f(n+2) = f(n+1) + f(n)$, and so on. So the value of $f(n)$ for each $n$ is used repeatedly. The effect is more dramatic if you use the naïve recursive algorithm.

This computation can be significantly optimized if we keep the values of $f$ that we have already computed and simply access the result as we need it.

If all we want to do is compute $f(n)$ for some $n$ once, this is already an observable improvement in the asymptotic running time of the algorithm (and it shows up very quickly in practice too). If we need to repeatedly use values of $f$ in a larger program, the improvement is even more effective.

Doing things like this form part of a technique called memoization, which is closely related with dynamic programming.

  • $\begingroup$ 'Memoization' is the useful reference, especially as the linked resource refers directly between a relationship between 'time cost' and 'space cost' in the context of 'computational complexity'. $\endgroup$ – gallygator Jun 24 '15 at 12:18
  • $\begingroup$ There is a bit more to the question. I do not know whether the parralel-computing was inteded for that. When you CPU, or some other CPU is idle, it can indeed compute values of the function in advance, and tabulate them in case they might be needed. $\endgroup$ – babou Jun 27 '15 at 21:52
  • $\begingroup$ "which is closely related with dynamic programming" -- only in one direction. $\endgroup$ – Raphael Jun 28 '15 at 9:37
  • $\begingroup$ Some 'bit more to the question' is related to parallelization, perhaps. My understanding is that parallelization benefits from functional decomposition. So, if some component of several functions tabulated for their own independent purposes had common need for the result of some shared component, perhaps it should be stored in memory for retrieval. $\endgroup$ – gallygator Jun 29 '15 at 13:35

If your domain $X$ is ordered and small, the values in $f(X)$ are small as well and $f$ is expensive to compute, then there's a simple reason: efficiency.

For instance, if $X$ is a range of natural numbers, you can store the image of $f$ in an array and obtain $O(1)$-time "comptutation" of $f$ at the cost of $\Theta(|X|)$ memory (assuming that the size of $f(x)$ is bounded by some constant).

Of course, this only pays off if you need values of $f$ often, $f$ is particularly expensive, and/or you can access the values quickly¹. A compromise can sometimes be to compute the values lazily, but you'd still have to pay with memory up-front.

Note how the scenario parameters that enable you to resolve the trade-off can change over time.

  1. In times without (universal) computers at every workstation, pre-computed tables for often used functions were indispensible. Logarithm tables, for example, were used in engineering disciplines.
  2. With ever-faster computers in most offices, storage (both analogue and digital) was more sensitive. Efficient algorithms allow to re-compute values quickly enough.
  3. Today, fast storage is cheaper than time and energy. Big players keep their data (which they concurrently update, all the time) in memory at all times in order to fulfill client queries as fast as possible. On the other end of the spectrum, mobile devices have to use their limited amount of energy conservatively, so storing (or pre-computing in times of energy abundance) beats re-computing.

So it's the same as ... all the time: inspect your situation, define your priorities, and pick the job for the tool.

  1. Consider memory hierarchy. If you traverse the values and they lie unordered on the heap, you are in for a hellpit of cache misses which potentially nullifies the advantage (or turns it around on you).
  • $\begingroup$ Thanks for this. So, what is the standard technique of 'trading off' tabulation expense vs memory expense? How is a breakeven in terms of cost coefficients for the tabulation and memory terms expressed? $\endgroup$ – gallygator Jun 29 '15 at 13:28
  • $\begingroup$ @gallygator The point about a trade-off is that there is no general/standard rule. The answer depends on the application context. $\endgroup$ – Raphael Jun 29 '15 at 14:30
  • $\begingroup$ Too bad. I hoped I could represent tabulation vs. retrieval on a graph and play with functions by their complexity -- even if just by renting the x-, y-axis resources on the cloud. $\endgroup$ – gallygator Jun 29 '15 at 18:12
  • $\begingroup$ @gallygator If there's a simple plot like this for anything connected to the real world, it's probably lying. (You can do this, but it won't tell you much in most cases.) $\endgroup$ – Raphael Jun 29 '15 at 19:12
  • $\begingroup$ @gallygator I tend to concur with Raphael that there is no general answer regarding the trade-off. The structure cosen may matter. Then the relative costs of memory and computation. But there may be other factors such as the choice of what is being tabulated Logarithms were tabulated not so much for themselves, but as a useful common intermediary for all sorts of other computations, which considerably increased the tabulation pay-off. Your question turned out to raise interesting issues, and I hope it gets more upvotes . $\endgroup$ – babou Jun 29 '15 at 22:40

This is actually a centuries-old technique that was largely killed by computers, but could possibly revive or still exist in some technical niche. It is even known to have been used by ancient Greek mathematicians and physicists.

The question asks why one would prefer to tabulate the results of a function, in a table indexed by its parameters, so as to replace computation by table lookup.

Of course, the minimum to expect is that table lookup is cheaper than computation of the result, but that is often the case with properly chosen data-structures.

The answer given by Luke Mathieson describes the best known case of function tabulation, which is memoisation, i.e. simply the preservation in a table of results that had to be computed previously, in case they are needed again.

Raphael argues that a systematic precomputing for a small domain can bring more efficiency to computations when the values are needed often, though, as he proposes lazy computation of the table, the difference with the previous answer is not too clear.

In a comment, I also suggested filling the table in advance, even with values not yet needed, when there is free/cheap computer time available for it.

But all this seems somehow restricted to a single program, which limits the usefulness of the effort.

However, the problem should probably be considered in a more general context, and has been in the past, before computers existed.

Complex calculations are an ancient problem in mathematics, sciences and engineering, and for a long time it was done by hand. These computations were used for all kinds of purposes, including astronomical (and astrological) predictions (including discovery of planets), computing tides, cartography and triangulation, compounded interest rates. In particular, logarithms were used to replace multiplication by addition, which implied using exponentiation on the results.

All these computations made use of hard to compute functions, such as trigonometric functions, logarithm, and others. So it became a business (and a very tedious job) to create precise tables for these functions, that were printed as books and sold to engineers and all people who needed them to conduct calculations. These were functions on the reals, and the tables were designed to reach a given precision, with some improvements using interpolation techniques. The tables were also designed so that they could be used reversibly: the same table could be used for logarithm and for exponential. These books were extremely valuable tools that one would keep around all the time. And they survived until the 1980's, when the microcomputers and especially sophisticated hand-held calculators became available, i.e. not much more than twenty to thirty years ago.

Using efficiently these tabulated functions was part of standard engineering curricula in universities and engineering schools.

Another way of tabulating functions was using a graphical form, called nomogram, nomograph, chart or abaque. They were tabulating a variety of complex functions used in exact sciences, either computed or obtained experimentally.

The formerly ubiquitous slide rule of engineers was yet another way of tabulating functions.

It could be that there are still useful functions that are too costly to compute with good precision, even with a standard computer. Then it can make sense to have them computed with powerful machines, then tabulated and made accessible, either on some memory device, or through the Internet. But I would not be enough in that kind of scientific work to know, and my search on the Internet was not fruitful.

  • $\begingroup$ So, what is a 'technical niche' for this in contemporary computing? $\endgroup$ – gallygator Jun 29 '15 at 13:30
  • $\begingroup$ As I said, it is only a possibility. I did a fast search without convincing results. I would suspect that some useful functions are very hard to compute and might justify such a treatment, but I do not have time right now to investigate. I also depends on what you call a tabulated function. For example, wikipedia has a (limited) table of the minimal values of the Gamma function (for negative values). Some people have tabulated the decimals of Pi, usable as random number generator (I think). It is probably easier than recomputing them. I do not have all answers, only some leads. $\endgroup$ – babou Jun 29 '15 at 13:53
  • $\begingroup$ I don't think this is stricly historical. For instance, Google is notorious for keeping all kinds of tables (that are precomputed and dynamically maintained) in memory, since response time is absolutely critical for them (and memory can be bought). On mobile devices (think mobile sensors...) memory is cheaper than processing time (which costs battery). So maybe, tables were inopportune while computers remained stationary but grew, but in recent years the trade-off parameters have shifted again. $\endgroup$ – Raphael Jun 29 '15 at 14:32
  • $\begingroup$ @Raphael This is very possible. The distinction that interested me was between memoisation and precomputation, though the difference can be blurred. Indeed, in the case of Google, a universe on its own, is it memoization in a huge program, or precomputation in the google universe. A nice topic for investigation ... and possibly improve teaching on the subject. - addition to your answer? $\endgroup$ – babou Jun 29 '15 at 15:26
  • $\begingroup$ @babou I think the practical, engineering perspective is mostly offtopic here, even though it's very interesting. The theory "was always right": even though some aspects have remained understudied, it's always been clear that most problems resp. the choice of data structures and algorithms are subject to trade-offs. $\endgroup$ – Raphael Jun 29 '15 at 17:07

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