A bunch of n wires have been labeled at one end with alphabetic codes A, B… The wire identification problem asks for an efficient procedure to mark the other end of the bunch with the corresponding labels. The wires run underground so you can’t track them individually and any wire is visibly indistinguishable from any other (except for the labeling).

As you might have guessed, the only way to gain information in this setup is to connect some set of wires at one end, walk up to the other end, and test for conductivity — this would give some partial labeling information. The process may have to be repeated many times before a complete labeling can be constructed. And since each such step involves walking across the distance of the wires, we wish to solve the problem with the least number of trips.

Small Cases

It should be easy to convince oneself that this problem cannot be solved when n = 2 — there’s really nothing we can use to break the symmetry of the situation.

For n = 3, we can use the following simple procedure (see the illustration on the right):

• Step 1: Connect the and wires at the labeled end, walk up to the unlabeled end, and test for conductivity. The pair that conducts electricity would be and , although we cannot determine exactly which is or at this point, and the remaining wire is uniquely identified as . For notation, we use lowercase letters for the labelings that we discover, versus uppercase letters for the given labeling.




• Step 2: Connect the newly identified wire with any one of the wires (recall, we don’t know which is or individually, only that one is and the other ). Now walk back to the labeled side and test for conductivity of with the remaining wires. If we find that – is conducting, then the wire we hooked up on the other side was really , otherwise it was .

The case n = 4 can be solved similarly with just two trip: Hook up – on one side and then run conductivity tests on the other side to partition the wires into two groups – and –; then hook up one from each of the two groups together etc.

So the question is: Can the problem be solved for all n with just two trips?

The General Case

The wire identification problem is one of those puzzles that is deceptively simple in its description, whose small cases are easy enough to solve in the head, and yet a fair amount of math goes into proving the general result.

I first read about this fascinating problem in the Don Knuth’s Selected Papers on Discrete Mathematics book — in a short 5-page paper titled The Knowlton–Graham partition problem (arXiv link).

Knuth states the following idea, attributed to Kenneth Knowlton, to solve the general problem:

Partition into disjoint sets in two ways and , subject to the condition that at most one element appears in -set of cardinality and a -set of cardinality , for each and . We can then use the coordinates to identify each element.

By a theorem of Ronald Graham, a solution involving just two trip always exists for wires unless is 2, 5, or 9 (On partitions of a finite set [PDF]).

Knuth in his paper translates the problem of Knowlton-Graham partitions in terms of 0/1-matrices. Lemma 1 in that paper states: Knowlton-Graham partitions exist for iff there’s a 0/1-matrix having row sums and column sums such that and are multiples of and .

Let’s use this lemma to label a set of 6 wires (). Since we can write 6 = 1 + 2 + 3, a rather obvious matrix satisfying the conditions of the lemma is:

, corresponding to the assignment

Guided by this matrix, here’s our two-trip identification procedure:

• On the labeled side, hook up wires in the following groups: , , and .
• Walk over to the unlabeled side, run the conductivity tests, and make the corresponding groups , , and .
• Still on the unlabeled side, hook up the wires in the groups , , and . Note that we still don’t know exactly which wire is, say, ; but we do know the two wires of which one if and the other .
• Walk over to the labeled side, unhook the connection we had made in the first step, and check for conductivity. The sole wire here, say, would correspond to the the sole wire from the group on the other side. Similarly, the connected-pair of — corresponds to the pair on the other side. etc.

Extensions

Navin Goyal, Sachin Lodha, and S. Muthukrishnan consider some variants & generalizations of the wire identification problem in their paper The Graham-Knowlton Problem Revisited. In particular, they consider the restrictions to just hooking up pairs in the identification process (in our examples above, we were free to hook, say, 3 wires).

For the past several days, I’ve been working my way through Prof. N.G. de Bruijn’s book Asymptotic Methods in Analysis; and I want to share some of the fun I’ve had reading it.

This post is not a review or anything (here’s an image of the back cover with some reviews). Below are just fragments of what I’ve read so far and found fascinating. You could also consider this my attempt at trying to persuade you to get a copy of this beautiful little book! Sadly, new copies appear not to be available at most online bookstores, so it might be hard to get one.

Prof. de Bruijn, his cycles, and average height of trees

This book is by the same Prof. de Bruijn well-known, among other things, for his cycles, which are defined as following: An -ary de Bruijn cycle is a sequence of radix- digits such that every combinations of digits occurs consecutively in the cycle. For example, one possible binary de Bruijn cycle of length is:

Treating this sequence as a cycle, notice that each of the 16 possible 4-bit patterns (0000 … 1111) appear exactly once in it. Knuth describes at least three interesting algorithms for generating such cycles in TAOCP Section 7.2.1.1 (printed in Volume 4, Fascicle 2 and downloadable as pre-fascicle 2a).

There’s surely lots to say about Prof. de Bruijn’s work (including his cycles), but I’ll reserve them for future posts. Take a look, for example, at the index of TAOCP Vol I (perhaps also in other volumes) and you’ll see references to many of his results, sometimes framed as exercises. And if you enjoy asymptotic calculations and analyses of algorithms, you would find it interesting to read his co-authored paper (with Knuth and Rice) where it is shown that the average height of planted plane trees is:

.

As an interesting note, Knuth in his musings on The Joy of Asymptotics (30 May 2000) said:

I’ve dedicated this book [Selected Papers on Analysis of Algorithms] to Prof. de Bruijn … in the Netherlands because he has been my guru … Ever since I got my PhD, I considered him my post-graduate doctoral advisor. Every time I had a problem that I got stuck on, I would write to him, and he would write me back a letter that would help me get unstuck. And that’s been going on for over fourty years now. So I dedicated this book to him. And what I want to talk to you about today is one of the things he has spent most of his life on — asymptotic methods.

By the way, that paper deriving the cool formula for the height of trees is included in these Selected Papers (Ch. 15).

The Book: A Pragmatic Exposition

I had got myself a copy of this book several years ago after first reading about it in the section on asymptotic calculations in TAOCP (Sec 1.2.11.3). But unfortunately, the book suffered the same fate as countless others — it just kept resting on my bookshelf. But a few days ago, I decided to give it a try.

A brief history. As described in its preface, this book grew out of lectures given in 1954–55 and was first published in 1958. The Dover edition I have is a reprint of the third edition from 1970.

Most computer programmers are familiar with the Big-O notation, or the Bachmann-Landau notation, for analyzing space/time-complexity of an algorithm. Among other things, this book talks of the tricks involved in getting precise asymptotic bounds once we have reduced the problem at hand to, say, a sum or an integral.

This is a pragmatic book — its focus is on demonstrating general techniques by working out several examples in detail (sometimes in more than one way) rather than expounding highly generalized theories of analysis. In this context, I found these lines from the preface reassuring:

Usually in mathematics one has to choose between saying more and more about less and less on the one hand, and saying less and less about more and more on the other. In a practical subject like this, however, the proper attitude should be to say as much as possible in a given situation.

From my reading, at certain places it, of necessity, assumes a basic knowledge of analysis and complex variable theory. Other than that, it is a fairly elementary treatment. (Note: as Feynman made a quip in one of his lectures, “Elementary” means that very little is required to know ahead of time in order to understand it, except to have an infinite amount of intelligence.”)

It’s a small book (200 pages) with very few exercises and so can be read in about two weeks, if you read an hour a day. And since there are lots of cliff-hanging moments (well, to the extent a book on analysis can have!) and the reader is left wondering how a tricky sum/integral would be tamed, your might be compelled to finish it even sooner!

de Bruijn has a good sense of humor too. At the start of the first chapter, when trying to answer the question “What is Asymptotics?”, he says:

The safest and not the vaguest definition is the following one: Asymptotics is that part of analysis which considers problems of the type dealt with in this book.

Didn’t see that one coming!

And while the book is tiny, it covers lots of tricks & techniques. So, to give my interested readers a taste, I’ve picked a couple of my favorite from the first chapter. All examples are de Bruijn’s.

Uniform Estimates

Uniformity of estimates is this little thing that keeps popping up every so often when analyzing the asymptotics of some characteristic of an algorithm. But it is rarely explained what it means or why uniformity is crucial to the application at hand. Grown up boys and girls are supposed to know what uniformity is, I suppose.

Before looking at uniformity, let’s recap the definition of the -notation. We say:

.

the -notation implies two implicit constants

Returning to the question of uniformity, suppose is a positive integer and and are arbitrary functions. Then:

, because,

the implicit “constants” and depend on the parameter

On the other hand, in the asymptotic relation (with again being a positive integer):

.

Hence if we write , with , we can choose to be 1. In such a case — when the hidden constants in the -notation do not depend on the parameter — the estimate is called uniform. This requirement is much like that of uniform convergence or uniform continuity in analysis.

So why trouble oneself with uniformity? Here’s one case where uniform estimates prove helpful: In “balancing” the asymptotic terms of a sum evenly to achieve a tight bound.

For instance, if we have somehow shown that with and where is a parameter. If the formula holds uniformly, we can get a tight bound by reasoning as follows: When is small, the first part, , is small, but the second part, , is large; the reverse holds if is large. Hence we wish to find a “balance” that minimizes the sum of the terms. And this happens when both the terms are asymptotically equivalent. So to achieve such a balance, we simply equate the two expressions , giving , and we finally get .

Notice that because of uniformity, the four hidden constants (, , , ) of the two do not vary when we vary the parameter . If the two original estimates had not been uniform; that is, had their hidden constants (implied by the -notation) varied with , we wouldn’t have been able to apply this simplistic “balancing” idea. For instance, while for small , the term might have been small, the hidden constant in might have grown, leading to unpredictable overall behavior.

The good thing about uniform estimates is that they are sometimes more useful; but proving uniformity might require a more careful analysis.

Asymptotic Series and Convergence: Some Curious Facts

An asymptotic series or asymptotic expansion for a function is a formula like:

, and,

.

A large class of examples of asymptotic expansions are convergent power series, like:

.

But there could be asymptotic expansions that are not convergent power series. And with such asymptotic expansions, some curious things can happen:

• The series of the asymptotic expansion need not be convergent.
• If the series does converge, its sum need not be equal to
• It is even possible to have and ’s such that the sum of the series does not have the series as its own asymptotic expansion!

The essential reason for these seemingly strange facts, as de Bruijn explains, is:

… that convergence means something for some fixed whereas the -formulas are not concerned with , but with . Convergence of the series, for all , say, means that for every individual there is a statement about the case . On the other hand, the statement that the series is the asymptotic expansion of means that for every individual there is a statement about the case .

For example, when , the asymptotic expansion of is:

,

So…

The book covers lots of useful stuff. In particular, It talks about Lagrange’s inversion formula (using which we can solve for the “tree function”), it has a especially thorough treatment of the saddle-point method. And much much more.

I was going through the first chapter of the book Graphical Enumeration by Frank Harary and Edgar M. Palmer when I chanced upon this perplexing footnote:

Read wrote Wright that both Read [R2]¹ and Wright [W3]² were wrong. So Read and Wright wrote a joint erratum [RW1]³ to set things right. This may be wrong since Wright asserts that Wright wrote Read first.

It appears on Pg. 17 in reference to k-colored graphs. Read and Wright are, of course, the mathematicians Ronald C. Read and Edward M. Wright!

A quick glance at the bibliography confirms that the above footnote describes, even if humorously, a true incident. However, I can’t say what the last sentence means: This may be wrong since Wright asserts that Wright wrote Read first. Does it mean:

• There is a genuine priority dispute as to who figured the error out [I don't think this is the case]. Or,
• Since [RW1] paper has Read as the first author (so, in that sense, “Read wrote first”), Wright couldn’t have written first. Or,
• Since “write” must happen before “read”, Wright wrote first. Or,
• It doesn’t mean anything — the authors are just playing around with words!

I could not find anything relevant by searching the phrase “Read wrote Wright” on Google.com. I’ll be eager to learn any other clues the reader might have as as to the meaning of the last sentence of the footnote.

Aside: It is sad that the Graphical Enumeration book, which was published by Academic Press, New York (1973), has apparently gone out of print. I’ve added it to the out of print math blog.

1. [R2] R.C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math. 12, 409–413 (1960).
2. [W3] E.M. Wright, Counting coloured graphs, Canad. J. Math. 13, 683–693 (1961).
3. [RW1] R.C. Read and E.M. Wright, Coloured graphs: A correction and extension, Canad. J. Math. 22, 594–596 (1970).

Here’s part (2), the tougher part, of the problem: Values for a random variable are generated independently and uniformly over . By accumulating these values, your job is to reach a sum between and , where is a positive integer, and . You can ask for successive values of this variate as many times as you wish, but you must add each such value to the running total you maintain. If the accumulating sum exceeds , you loose; if you manage to get a sum between and , you win. What is the probability of winning as ?

Earlier that year, I had finished studying Knuth’s Selected Papers on Discrete Mathematics. That is a precious collection for anyone interested in discrete structures, combinatorics, asymptotics, random structures etc. I had especially enjoyed one of the papers, “The first cycles in an evolving graph” (with Flajolet & Pittel as co-authors), not only for the neat results but also for the techniques used to get asymptotic values/bounds of certain integrals. That paper helped introduce me to, and drive home the importance of, the fascinating field I later came to know as Analytic combinatorics (also this). That paper analyzes the evolution of a random graph, where edges are successively added to a set of vertices(under different random graph models). Among other results, it shows that expected length of the first cycle to appear is proportional to and the size of the component having this cycle is — quite fascinating stuff. At any rate, with these analytic techniques still fresh in my mind, I attacked the problem by trying to find a bound on the probability integral. Details follow (with a I-to-we switch).

To begin with, we use the classical Central Limit Theorem to replace the sum of independently and uniformly distributed variates (each of which had mean and variance ) with a Normal variate with mean and variance .

Now, the probability of “hitting” (i.e., achieving a sum lying within the interval) in exactly “moves” is , where is the probability of hitting in exactly moves and is the probability of jumping from to the desired interval in the m^th-move.

When , ; and when , . is the probability density function of the aforementioned normal variate; so

.

Thus the probability of success in precisely moves is

which simplifies to

.

Since we’re interested in a success in any number of moves, we must evaulate , and finish by taking the desired limit .

To evalutate the sum over , we replace the summation by integration (see caveat below) and substitute the variable of integration from to (resulting in an extra factor of ). So our final integral turs out to:

Expanding the integrand (of the first integral) in powers of , the integrand becomes:

When we integrate over and take limit , all terms other than the first go away. As for the first term, it yields since

(where is the Error function).

Whew! That was heavy going. Let us catch our breath. Okay. So the answer to the original problem turns out to be . As a sanity check, it’s value turns out to be 0 at and 1 at . Good.

Caveat: The above derivation is not rigorous (in fact, it is somewhat sloppy) for at least the following reasons:

• Convergence of many integrals/summations is implictly assumed
• The error term has not been calculated when changing a summation to an integration. Euler Summation Formula (also TAOCP 1.2.11.2) should have been used to bound the error terms.
• The Central Limit approximation has been used without being demonstrated that it is being applied only close to the peak of the distribution, and not far into the tails (unless justified)

A question: If you read the publisher solution on Ponder This, it says, “Values between 0 and 1 are equally likely but larger values are more likely to cause the sum to exceed . So as the differential probability that is .” Can the reader explain the reasoning behind this differential probability?