AbstractA table with two rows and n columns may be thought of as two vectors with n components. The distance between the two rows then corresponds to the norm of the difference between the rows. We examine the problem of how to collapse the adjacent columns of the table while keeping the norm of the difference as large as possible. First a stepwise algorithm is given which achieves this end with respect to the norm of the vector of differences. After proving the optimality of the stepwise solution we extend the result to the norm which arises from minimizing the number of persons misclassified. The same algorithm suffices

Kleitman, Daniel J.

Lew, Robert A.

Elsevier - Publisher Connector 

Discrete Applied Mathematics 6 (1983) 49-53 
North-Holland Publishing Company 
49 
AN ALGORITHM FOR COLLAPSING SIGN ALTERNATING 
SEQUENCES OF REAL NUMBERS 
Daniel J. KLEITMAN* 
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, 
USA 
Robert A. LEW 
Departments of Nutrition and Biostatistics, Harvard School of Public Health, Boston, MA, 
USA 
Received 25 May 1981 
Revised 20 January 1982 
A table with two rows and n columns may be thought of as two vectors with n components. 
The distance between the two rows then corresponds to the norm of the difference between the 
rows. We examine the problem of how to collapse the adjacent columns of the table while keeping 
the norm of the difference as large as possible. First a stepwise algorithm is given which achieves 
this end with respect to the norm of the vector of differences. After proving the optimality of 
the stepwise solution we extend the result to the norm which arises from minimizing the number 
of persons misclassified. The same algorithm suffices. 
1. Origin of problem 
Statisticians often must collapse large frequency tables into smaller ones in order 
to summarize data in a simple manner. 
For example consider the comparison of survival by age in the following table. 
O-20 21-40 
Age Category 
41-60 61-80 81-100 
Alive 0.4 0.3 0 0.3 0 
Dead 0.1 0.2 0.4 0.1 0.2 
Difference +0.3 +0.1 -0.4 +0.2 -0.2 
Each row total is one. The first line of the table is read “of the persons alive, 40% 
are ages O-20, 30% ages 21-40, and 30% ages 61-80.” We wish to group this data 
into three categories that preserve large differences between rows. 
Note that in the row of differences 0.3 + 0.1 + 0.2= -(-0.4 -0.2), the sum of 
positive differences equals minus the sum of the negative differences. Any collapse 
*Supported in part by ONR Contract No. 00014-76-C-0366. 
0166-218X/83/$03.00 0 1983 North-Holland 
50 D.J. Kleitman, R.A. Lew 
of the table preserves this property. One such collapse is: 
Alive 
Dead 
Difference 
Age Category 
O-40 41-60 61-100 
0.7 0 0.3 
0.3 0.4 0.3 
+0.4 -0.4 0 
With five categories the sum of positive differences is 0.6; with three categories 
the sum is 0.4. Note that merging 41-60 with 61-100 we obtain two categories O-40 
and 41-100 for which the sum is 0.4. 
Our objective is to find an algorithm that collapses adjacent categories of a table 
with n categories into a table with k categories (k given, less than n) while keeping 
the sum of positive differences as large as possible. Merging adjacent categories with 
like signs leaves this positive sum unchanged. The problem becomes harder when 
the signs alternate and mergers bring about cancellation. 
2. The algorithm 
The ‘greedy’ algorithm for this problem proceeds by a sequence of mergings of 
neighboring categories, at each step merging a pair that reduces the sum of positive 
differences least. We shall show that, while the greedy algorithm itself will not 
always produce best solutions, one small modification of it will always produce an 
optimal solution here. 
Consider a sequence of n real numbers A i, . . . , A, with zero sum. We seek a 
partition of the indices into k consecutive blocks that maximizes, over all such 
partitions, the absolute value of the sum of the d’s within each new block. 
We distinguish four cases as follows: 
Case 1. For at least one index j, dj and dj + 1 have the same sign. Excluding 
j= 1 and j=k, let r be the index 2<jln- 1 with minimum Idjl. 
Case 2. Not Case 1, ~A~]I~A,I and ldrl~lA,l. 
Case 3. Not Case 1, Idi] + ldnl I lA,l. 
Case4. Not Case 1, ld,I+ld,/>lA,I and either ld,~~/d,l or IAJsIA~]. 
We define the cost of merging j with j+ 1 as 
cj+l’/djI+Idj+II-Idj+dj+II* 
We define algorithm G, the greedy algorithm, and the optimal algorithm 0, as 
follows: 
Algorithm G(n, k). If n > k choose any index m such that the cost C,, 1 = 
Minj=,,2,...,.-l[C,+ll is minimal. For Case 2 m =r suffices; for Cases 3 and 4 
Algorithm for collapsing sequences 51 
m=l or m=n-1. Merge m with m+l, so that dh=d,+d,+i; dJ=d,+, for 
j>m, Aj=Aj. 
If n =k+ 1 stop; otherwise apply Algorithm G(n - 1,k) to this sequence. 
Algorithm O(n, k). Perform the first step of G(n, k) unless the sequence lies in Case 
4, and n = k (mod 2). In that case set n = r and merge A’s as in Algorithm G. If 
n = k+ 1 stop; otherwise apply O(n - 1, k) to the A sequence. 
At first glance the difference between Algorithms 0 and G may appear peculiar. 
it can be understood as follows: when two adjacent A’s alternate in sign, any 
merging of successive internal indices (not involving 1 or n) necessarily produce two 
adjacent A’s with the same sign; whenever this happens the next merge is free. Thus, 
for 12 -k = 2, an end merge for which this doesn’t happen will be wasted, unless, as 
in Case 3, the other end merger when added to that of the first is cheaper than 
any alternative. This exceptional situation propagates by induction to whenever 
n I k (mod 2). 
Theorem 1. Algorithm O(n, k) will always provide a sequence A; ... A; the sums of 
the absolute values of which are maximal over all partitions possible here. 
Proof. It is straightforward, as already noted, that this algorithm is optimal for 
n - kr2. The order in which mergers are performed in forming a partition is 
irrelevant, and after any one merger, the remaining ones must be optimal starting 
from the resulting n - 1 length sequence. By induction therefore, we need only show 
that for n- kz3 there is an optimal partition with parameters k,n for the given 
sequence for which the pair of indices merged in the first step lie in the same block. 
In Case 1 the first merge involves j and j+ 1 with Aj+l having the same sign. If 
they are in separate blocks in an optimal solution which blocks have A’s the same 
sign, we can merge these blocks at no cost and make an arbitrary split elsewhere to 
get an optimal solution containing the first merged pair together in a block. If one 
block has changed sign, by separating the A in it from the rest and putting it with 
the Aj+l block one actually improves the solution. 
In the remaining cases the A’s alternate in sign. Suppose now that j, j+ 1 form 
a recommended merge according to Algorithm 0, and suppose that optimal parti- 
tion P, which by induction we can assume is formed by use of Algorithm 0 after 
the first merging, has j and j+ 1 in separate blocks. 
In Case 2 (or 4 with n= k (mod 2)) suppose further that lAj/ I lAj+ 1 j; choose 
any merge compatible with P other than an end merge, if possible merging j with 
j- 1. If the latter merge is chosen, then O(n - 1, k) will in its first step merge j with 
j + 1, so that j and j + 1 may be in the same block of an optimal partition. Other- 
wise, if the first merge chosen was s,sf 1 with /ASI < iA,+ 1 / the succeeding free 
merge will link sf 1 with these 2 and the A value of the resulting block will have 
the same sign as A,, 1 and exceed it in magnitude. Thus the j, j + 1 merge will again 
52 D.J. Kleitman, R.A. Lew 
be optimal according to 0, and by the induction hypothesis will appear in some 
optimal partition. 
In the remaining case the d’s alternate in sign. Suppose now that m, m + 1 form 
a recommended merge according to Algorithm 0 while no optimal partition has m 
and m + 1 in the same block. By induction we can assume that an optimal partition 
P is formed by use of Algorithm 0 after the first merging. 
Let s and s+ 1 be a pair of indices which belong to the same block of P. Since 
kr n - 3 we can assume that 
1~s and s+l<n. 
Perform the merging of s with s + 1 and the following free merging of these two with 
either s - 1 or s + 2. After this operation we get an alternate sequence of n - 2 d’s 
each of which is in canonical correspondence with an initial d. Note that the 
absolute values of the new d’s are not smaller than those of the initial ones so that 
either m and m + 1 are already merged or are a recommended merge in O(n - 2, /c). 
A contradiction. 
It is easy to verify that this algorithm can be implemented in a number of steps 
proportional to n log n by partially sorting the d’s by absolute value and updating 
the sorted list with each pairwise merger. 
3. Extensions 
The problem spawns a variety of other questions, some relating to com- 
binatiorics, others to statistics. As an example of the former, consider collapsing n 
categories into k<n categories while minimizing the sum of positive differences. 
This is a version of the knapsack problem that can be handled by a dynamic 
programming algorithm [l]; it seems to require on the order of n3 steps. When the 
categories have no intrinsic order so that adjacency need not be preserved, then the 
original problem becomes trivial. Suppose instead of age categories, we have color 
categories. Then merge together all colors with positive d and, separately, all colors 
with negative d. All such mergers are free and only two categories result. The 
problem is solved. 
For statisticians, questions arise when we alter the cost of mergers. In the age 
category problem suppose a loss results from misclassifying a person. Suppose for 
example we have the following table. 
O-40 
Age Category 
41-60 61-100 
Alive 10 3 8 
Dead 2 4 5 
Difference 8 -1 3 
Minimum 2 3 5 
Algorithm for collapsing sequences 53 
In terms of the ‘raw’ frequencies above, the classification which minimizes loss 
declares persons O-40 and 61-100 ‘alive’, and persons 41-60 ‘dead’. If errors are of 
equal weight in either direction the total loss is the sum of the minima, 2 + 3 + 5 = 10. 
Merging all three categories results in 21 alive and 11 dead so that declaring all to 
be alive yields a loss of 11. One easily shows that in terms of this loss function free 
mergers arise when adjacent dj have the same sign. Also on any step the minimum 
loss occurs when the category with the minimum [dj( is merged with its adjacent 
categories. it follows that in all aspects the problem of reducing n categories to k 
categories while minimizing the loss due to misclassification is in all ways analogous 
to our original problem except that the cost of merger differs. Thus for the 
‘misclassification problem’, we have 
Theorem 2. Algorithm O(n, k) applied to a 2 x n table of raw frequencies with loss 
equul to the number of misclassified persons (the sum of the k resulting category 
minima) yields the k category table with minimum loss. 
The result extends to asymmetric loss functions; that is, a penalty of h units for 
declaring a dead person to be alive and one unit for declaring a living person to be 
dead. The h units lost for a single person misclassified is equivalent o one unit lost 
on h persons. Thus multiplying the number of dead persons by h in the raw 
frequency table restores equal unit losses, the form of the problem resolved by 
Theorem 2. 
Acknowledgement 
The authors would like to thank the referee for helpful suggestions. 
References 
[l] M.R. Carey and D.S. Johnson, Computers and Intractability, A Guide to the Theory of 
NP-Completeness (Freeman, San Francisco, 1980). 


An algorithm for collapsing sign alternating sequences of real numbers 

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An algorithm for collapsing sign alternating sequences of real numbers

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