# A ‘Reasonable Amount of Time’

The concept of a ‘reasonable amount of time’ figures a fair bit in abstract computational complexity theory; but what is a ‘reasonable amount of time’ in practice?  This post outlines the problem of balancing between the two competing ideals of determinism and adaptability and offers a flexible working definition.  (Not to be taken too seriously: it’s summer vacation time.)

A standard text on combinatorial problems and optimisation algorithms – perhaps discussing the TSP, for example – might read something like:

“… so we tend not to be as interested in particular complexity values for individual problem instances as how these complexities change as the input problem size (n) increases.  Suppose then, that for a given problem, we can solve a problem instance of size n = L in a reasonable amount of time.  What then happens if we increase the size of the problem from n = L to n = L+1?  How much harder does the …?”

or, filling in a few gaps:

“… so we tend not to be as interested in particular complexity values for individual problem instances as how these complexities change as the input problem size (n) increases.  Suppose then that we can solve a TSP of 20 cities on a standard desktop PC in a reasonable amount of time.  What then happens if we increase the number of cities from 20 to 21?  How much longer does …?”

All good stuff, and sensible enough, but what’s this ‘reasonable amount of time’?

Well, this is a problem because it depends entirely on the circumstances.  Extreme examples are:

• We’re planning a space launch for twelve years’ time.  We need to have the trajectories planned seven years before launch.  A ‘reasonable amount of time’ is five years.  (Possibly on some thumping HPC system)
• Packets are being switched, via routing tables/algorithms, across the Internet.  A ‘reasonable amount of time’ is a fraction of a second.  (On a rusty old router in a tunnel somewhere)

And we’re going to hit the same issue if we try to fix a definition: it won’t work for everyone.  Not only are situations not the same; nor is the available equipment; nor are people.  Even in the research lab, some things (and some people) can have patience; others can’t (or won’t).  We need something definedbut not rigidly defined.

Define a ‘Grout-reasonable’ amount of time to be the time it takes for any given  academic/researcher to:

1. Press [ENTER] (or similar) to start the algorithm;
2. Walk to their preferred refreshment outlet;
3. Order a coffee/tea/other drink;
4. Drink it (possibly in conversation with others)

The algorithm has finished in Grout-reasonable time if it’s finished by the time the academic returns.  If it finishes exactly as the researcher approaches their desk, the process is Grout-optimal since it maximises the problem size solvable in Grout-reasonable time.

This definition has the advantage of being flexible enough to fit the requirements of any individual academic and their habits, whilst providing a base unit for effective comparison if needed.  It also implicitly defines the computational power available as being that on the desk in any given case.  Any necessary adaptation/translation/conversion can then proceed as follows:

• “The algorithm isn’t Grout-reasonable for n = 15 for me: more like 4 Grout Units (GUs), I think”
• “Ah, but my GUs are worth five of yours (my computer is faster but our coffee shop is further away and I walk and drink slower) so it should be Grout-reasonable for me”
• etc.

You get the idea!  Extensive research at various UK universities suggests an individual’s personal GU to be anything between nine minutes and three and a half hours!  (No names – no pack drill!)

Happy holidays!