Andersson, Arne

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Theoretical Computer Science 106 (1992) 391-393 
Elsevier 
391 
Letter to the Editor 
Comments on “On the balance 
property of Patricia tries: External 
path length 
. . .m 
viewpoint” 
Arne Anderson 
Department of Computer Science, Lund University, Lund, Sweden 
Communicated by M. Nivat 
Received June 1992 
The method of analyzing data structures and algorithms by assuming that 
the treated items are randomly and independently drawn from some kind of 
distribution, known or unknown, is widely used among computer scientists. 
Although real-world data are very seldom generated this way, such analysis 
gives a lot of help in understanding the behaviour of the algorithms and, 
therefore, it is well motivated. However, some care has to be taken when 
conclusions are drawn from the analysis. 
In the paper “On the balance property of Patricia tries: External path length 
viewpoint” by Kirschenhofer, Prodinger and Szpankowski (Theoretical Com- 
puter Science 68 ( 1 ), pp. l- 17), it is argued that from a practical point of view 
the average case behaviour of a data structure is often more interesting than 
the worst case behaviour. The authors state that “the algorithmic design has 
often to be targeted at coping with quite unrealistic, if not pathological, inputs 
and the possibility is neglected that a simpler algorithm might perform just 
as well, or even better, in practice.” Motivated by this argument they present 
Correspondence to: Department of Computer Science, Lund University, Box 118, S-22100 Lund, 
Sweden. Email: ame@dna.lth.se. 
0304-3975/92/$05.00 @ 1992 - Elsevier Science Publishers B.V. All rights reserved 
392 A. Andersson 
an elegant average case analysis of the variance of the external path length 
of Patricia tries under the model of random input drawn from the uniform 
distribution. From this analysis the authors conclude that the external path 
length in a Patricia trie “is asymptotically equal to n log, n with probability one 
(i.e. almost surely)“. These results are used to show that “from the practical 
(average) viewpoint, the Patricia tree does not need to be reconstructed in or- 
der to keep it balanced”, and, hence, “the Patricia is a very well-balanced tree, 
and in most practical cases it does not need to be additionally rebalanced.” 
To me, it seems that when using reasonable meanings of the terms “patholog- 
ical inputs” and “most practical cases”, the conclusions drawn by Kirschenhofer 
et al. are not correct. Note that I do not claim that their mathematical analysis 
is wrong. Indeed, the analysis is interesting and with more modest claims it 
would have been a very nice paper. 
First, I agree with the statement that a good average behaviour is often suffi- 
cient in practical applications. However, when studying the practical behaviour 
of a data structure, what input could be more “pathological” than randomly 
chosen, uniformly distributed data? The analysis in the paper is heavily depen- 
dent on this model, and the authors do not mention how their analysis relate 
to practical situations. 
A scientific investigation of a real-world phenomenon is very often performed 
by examining a model, either theoretically or by laboratory experiments. From 
the behaviour of the model we draw conclusions about the behaviour of nature. 
Examples are: 
(1) We can draw conclusions about how the model behaves, and how nature 
behaves in that particular case when the conditions of the model are exactly 
fulfilled. 
(2) We can draw conclusions about how nature behaves in cases when the 
conditions of the model are approximately fulfilled. We may argue that in 
those cases nature will behave approximately in the same way as our model. 
(3) Finally, we can draw the conclusion that nature will always (or almost 
always) behave like our model. 
The conclusion drawn by Kirschenhofer et al. is of the third category. 
Which type of conclusion should have been drawn? 
The first type is quite modest and always true if the investigation is properly 
made, which seems to be the case in the paper referred to. 
The second type is often realistic, but some care has to be taken. From 
the analysis of the expected external path length in a Patricia trie for random 
independent uniform data we can probably draw some conclusions about which 
external path length we may expect when input is roughly evenly distributed. 
However, it is not clear in what way we can apply the variance for the random 
uniform model to such inputs. This must be clarified and motivated. 
The third type is very strong and in order to draw such a conclusion we 
Letter to the Editor 393 
must show that our model properly reflects nature. This is certainly not made 
in the paper referred to. 
Thus, the first type of conclusion is the one that could have been drawn 
by the authors. Probably, the second type could also have been made, but 
this would require a careful discussion. The third type of conclusion should 
definitely not have been drawn. Indeed, my personal experience is that in most 
practical applications the external path lengths of Patricia tries are significantly 
larger than n log, n, in contrast to what is claimed by Kirschenhofer, Prodinger 
and Szpankowski. 
To sum up, it is of course important that scientific investigations-empirical 
and theoretical-are properly performed. However, in my opinion, even more 
important is that the conclusions are properly related to the results of the 
investigations. The cited paper is an example where this is not the case. 


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