TAXONOMY

The following paper breaks down and explains the innovative techniques that Dr. Levitin uses to get the taxonomy working for him in the classroom. Just as algorithms have patterns and best-practices, so Anany has found that the carefully crafting the taxonomy of a lesson can vastily improve students comprehension of the material by ordering it with flexibility technique grouping in mind. Below you will find the abstract of the paper, the work in full can be ordered from Dr.Dobbs.com.

Do We Teach the Right Algorithm Design Techniques ?

Abstract
Introduction
Four General Design Techniques
A Test of Generality
Further Refinements
Another Example
Conclusion
References

Abstract
Algorithms have come to be recognized as the cornerstone of computing. Surprisingly, there has been little research or discussion of general techniques for designing algorithms. Though several such techniques have been identified, there are serious shortcomings in the existing taxonomy. The paper points out these shortcomings, reevaluates some of the techniques, and proposes a new, hierarchical classification scheme by grouping techniques according to their level of generality. A variety of examples from different areas of computing are used to demonstrate the power and flexibility of the taxonomy being proposed.

1. Introduction

A study of algorithms has come to be recognized as the cornerstone of computer science. The progress in this field to date, however, has been very uneven. While the framework for analysis of algorithms has been firmly established and successfully developed for quite some time, much less effort has been devoted to algorithm design techniques. This comparative lack of interest is surprising and unfortunate in view of the two important payoffs in the study of algorithm design techniques: "First, it leads to an organized way to devise algorithms. Algorithm design techniques give guidance and direction on how to create a new algorithm. Though there are literally thousands of algorithms, there are very few design techniques. Second, the study of these techniques help us to categorize or organize the algorithms we know and in that way to understand them better." [7,p.33]
Despite the dearth of papers dedicated to the subject, primarily through efforts of textbook writers [1,4,5,6,8,13], a consensus seems to have evolved as to which approaches qualify as major techniques for designing algorithms. This list includes: divide-and-conquer, greedy approach, dynamic programming, backtracking, and branch-and-bound. ( In addition, probabilistic and parallel algorithms are two other important design approaches; since they belong to different paradigms, we are not going to discuss them here.)
 
This widely accepted taxonomy has serious shortcoming, however. First, it includes techniques of different levels of generality. For example, it seems obvious that divide-and-conquer is more general than, say, the greedy approach and branch-and-bound, which are applicable to optimization problems only. Second, it fails to distinguish divide-and-conquer and what we will call decrease-and-conquer as two qualitatively different techniques. Third, it also fails to include brute force and transform-and-conquer, which we consider important general design techniques. Fourth, its linear, as opposed to hierarchical, structure fails to reflect important special cases of the techniques. Finally, and most importantly, it fails to classify many classical algorithms (e.g., Euclid's algorithm, heapsort, search trees, hashing, etc.)

The paper seeks to rectify these shortcomings. Section 2 reviews four strategies: brute force, divide-and-conquer, decrease-and-conquer, and transform-and-conquer. Section 3 proposes a test of generality to distinguish more general techniques from less general ones; its application explains why the four techniques discussed in Section 2 were singled out from the rest. Section 4 refines the new classification scheme further. Section 5 gives another convenient educational vehicle for illustrating the major design techniques. Section 6 concludes the paper by summarizing its findings and contains a discussion of the advantages and limitations of the new taxonomy being proposed.

2. Four General Design Techniques

We will start with a discussion of a well-known design approach that has been missing from the tables of content of textbooks organized around algorithm design techniques: brute force. It can be defined as a straightforward approach to solving a problem, usually directly based on the problem's statement and definitions of the concepts involved. Though very rarely a source of efficient algorithms, the brute-force approach should not be overlooked as an important algorithm design technique in view of the following. First, unlike some of the others, this approach is applicable to a very wide variety of problems. (In fact, it seems to be the only general approach for which it is more difficult to point out problems it cannot tackle.) In particular, it is brute force that is used for many elementary but important algorithmic tasks such as computing the sum of n numbers, finding the largest element in a list, adding two matrices, etc. Second, for some important problems (e.g., sorting, searching, matrix multiplication, string matching), the brute-force approach yields reasonable algorithms of at least some practical value with no limitations on instance sizes. Third, even if too inefficient in general, a brute-force algorithm can be still useful (and an economically sound!) choice for solving small-size instances of a problem. Fourth, a brute-force algorithm can serve an important theoretical or educational purpose, e.g., as the only deterministic algorithm for an NP-hard problem or as a yardstick for more efficient alternatives for solving a problem. Finally, no taxonomy of algorithm design techniques would be complete without it; moreover, as we are going to see below, it happens to be one of only four design approaches classified as most general.

Divide-and-conquer is probably the best known general algorithm design technique. It is based on partitioning a problem into a number of smaller subproblems, usually of the same kind and ideally of about the same size. The sub-problems are then solved (usually recursively or, if they are small enough, by a simpler algorithm) and their solutions combined to get a solution to the original problem. Standard examples include mergesort, quicksort, multiplication of large integers, and Strassen's matrix multiplication; several other interesting applications are discussed by Bentley [3]. Though most applications of divide-and-conquer partition a problem into two subproblems, other situations do arise: e.g., the multiway mergesort [9] and Pan's algorithm for matrix multiplication [14]. As to the case of a single subproblem, it is difficult to disagree with Brassard and Bratley [5] that for such applications, "... it is hard to justify calling the technique divide-and-conquer." [p.223]  Hence, though binary search is often cited as a quintessential divide-and-conquer algorithm, it fits better in a separate category we are about to discuss.

Solving a problem by reducing its instance to a smaller one, solving the latter (recursively or otherwise), and then extending the obtained solution to get a solution to the original instance is, of course, a well-known design approach in its own right. For obvious reasons, we will call it decrease-and-conquer. (Brassard and Bateley [4,5] use the term "simplification" which we are going to use below for a different design technique.) This approach has several important special cases. The first, and more frequently encountered, decreases the size of an instance by a constant. The canonical example here is insertion sort; other examples are provided by Manber [11] who has investigated an intimate relationship between this approach and mathematical induction. Though the size-reduction constant is equal to one for most algorithms of this type, other situations may also arise: e.g., recursive algorithms that have to distinguish between even and odd sizes of their inputs.

The second special case of the decrease-and-conquer technique covers the size reduction by a constant factor. The examples include binary search and multiplication à la russe. Though most natural examples involve a size reduction by the factor of two, other situations do happen: e.g., the Fibonacci search for locating the extremum of a unimodal function (e.g., [2, pp. 153-155]) and the "divide-into-three" algorithm for solving the problem of identifying a lighter false coin with a balance scale.

Finally, the third important special case of the approach covers more sophisticated situations of the variable-size reduction. Examples include Euclid's algorithm, interpolation search, and the quicksort-like algorithm for the selection problem.

Though the decrease-and-conquer approach is well known, most authors consider it either a special case of divide-and-conquer (e.g., [13]) or vice versa [11]. In our opinion, it is more appropriate, from theoretical, practical and especially educational points of view, to consider divide-and-conquer and decrease-and-conquer as two distinct design techniques.

The last technique to be considered here is based on the idea of transformation and will be called transform-and-conquer. One can identify several flavors of this approach. The first one --- we will call it simplification --- solves a problem by first transforming its instance to another instance of the same problem (and of the same size) with some special property which makes the problem easier to solve. Good examples include presorting (e.g., for finding equal elements of a list), Gaussian elimination, and heapsort (if the heap is interpreted as an array with the special properties required from a heap).

The second --- to be called representation change--- is based on a transformation of a problem's input to a different representation, which is more conductive to an efficient algorithmic solution. Examples include search trees, hashing, and heapsort if the heap is interpreted as a binary tree.

Preconditioning (or Preprocessing) can be considered as yet another variety of the transformation strategy. The idea is to process a part of the input or the entire input to get some auxiliary information which speeds up solving the problem. The examples include the Knuth-Morris-Pratt and Boyer-Moore algorithms for string matching, Winograd's matrix multiplication, and determining ancestry in a tree [5, p.293].
Finally, in the last and most drastic version of the transform-and-conquer approach, an instance of a problem is transformed to an instance of a different problem altogether. Though this idea plays a central role in the NP-completeness theory, practical algorithms based on this idea are relatively rare. It is by this reason that we are not going to include it in the taxonomy chart to be given below.

3. A Test of Generality

In addition to the observations made above, we propose to partition design techniques into two categories: The first one will include most general techniques while the second will contain the remaining, i.e. more limited, approaches. Though one can probably come to a reasonable consensus as to which of these two categories each of the known techniques should belong to, it would clearly be better to have a specific criterion or criteria for making such a determination. For our part, we would like to suggest the following test. In order to qualify for inclusion in the category of most general approaches, a technique must yield reasonable (though not necessarily optimal) algorithms for the two problems: sorting and searching.

We can justify this choice by the importance of the problems selected (" Indeed, I believe that every important aspect of programming arises somewhere in the context of sorting or searching" [9, p.v]) and by noting with satisfaction that it results in partitioning the known techniques in a manner quite supported by intuition. Indeed, only four techniques --- brute force, divide-and-conquer, decrease-and-conquer, and transform-and-conquer --- pass the test (see Table 1); the others --- greedy approach, dynamic programming, backtracking, and branch-and-bound --- fail to qualify as the most general design techniques.
 
SORTING SEARCHING

Table 1: Applicability of design techniques to sorting and searching

TECHNIQUE	SORTING	SEARCHING
Brute force	selection sort	sequential search
Divide & conquer	mergesort	applicable
Decrease & conquer	insertion sort	applicable
Transform & conquer	heapsort	search trees, hashing

 
4. Further Refinements

By identifying above special cases of the decrease-and-conquer and transform-and-conquer, we have made a stride toward a multilevel taxonomy. One can strengthen this line of thinking further. First, it might be useful to distinguish between two types of divide-and-conquer algorithms. For some of such algorithms, the bulk of the work is done while combining solutions to smaller subproblems (e.g., mergesort); for others, it is processing before a partition into subproblems that constitutes the heart of the algorithm in question (e.g., quicksort). We will call these two types of the divide-and-conquer technique divide-before-processing and process-before-dividing, respectively.
Second, Moret and Shapiro [12] have considered the greedy approach as one of two types of a more general approach to optimization problems: "Greedy methods build solutions piece by piece... Each step increases the size of the partial solution and is based on local optimization: the choice selected is that which produces the largest immediate gain while maintaining feasibility. Iterative methods start with any feasible solution and proceed to improve upon the solution by repeated applications of a simple step. The step typically involves a small, localized change which improves the value of the objective function." [p.254]. Moret and Shapiro devote separate chapters to illustrate each of these techniques; more examples can be found by following the references therein. We like this delineation though the name improvement methods seems to be more descriptive of the second type than "iterative methods."

Further, one can point out two types of dynamic programming algorithms as well. The canonical version of this approach is bottom-up: a table of solutions to subproblems is filled starting with the problem's smallest subproblems. A solution to the original instance of the problem is then obtained from the table constructed, with many of the table's entries remaining typically unused for the instance in question. In order to overcome the latter shortcoming, a top-down version of dynamic programming was developed, based on using so called "memory functions" (e.g., [5, sec. 8.8]).

Finally, the two techniques---backtracking and banch-and-bound --- both deal with combinatorial problems by constructing so called "state-space trees." The difference between them lies in that backtracking is not limited to optimization problems, while branch-and-bound is not restricted to a specific way of traversing the problem's space tree. So, it would be natural to consider them as special cases of a more general approach. What is this approach to be called? The name "exhaustive search" is often used in the literature. However, there are two problems with this term. First, both backtracking and branch-bound try, in fact, to avoid exhaustive search by pruning a problem's tree. Second, the exhaustive search can be, in fact, an alternative approach to solving a problem (albeit usually an inferior one) by generating  and checking all the candidates for the problem's domain. Therefore it is more natural to consider the latter approach as an application of the brute-force technique. Thus, instead of "exhaustive search",  we will use state-space-tree techniques to refer to the general design approach containing both backtracking and branch-and-bound as its special cases.

Thus, we end up with the following alternative taxonomy of major design techniques:

MAJOR ALGORITHM DESIGN TECHNIQUES
More general techniques	Local search techniques
Brute force	- improvement methods - greedy methods
Divide-and-conquer - divide before processing - process before dividing	Dynamic programming - bottom-up - top-down (memory f.)
Decrease-and-conquer - decrease by a constant - decrease by a constant factor - variable size decrease	State-space-tree technique - backtracking - branch-and-bound
Transform-and-conquer - simplification - representation change - preconditioning

5. Another Example
It is not easy to find problems to which all the four general design strategies are applicable with a reasonable result. Sorting and searching are fortuitous examples exploited above as the criterion for separation of more general techniques from less general ones. Here, we will point out another problem which can play a useful educational role: the exponentiation problem of computing an. (Of course, the problem of computing an mod p is of great practical interest as well because of its importance to public-key encryption algorithms.)
A brute-force algorithm would simply multiply a by itself n-1 times. A divide-and-conquer algorithm would use the formula an=a[n/2]a[n/2]. The decrease-by-one variety of the decrease-and-conquer approach yields an=an-1a; the decrease-by-half variety would be based on the formula an=(a[n/2])2 for even n's and (a[n/2])2a for odd n's. Finally, the transformation strategy can be illustrated by two well-known algorithms that exploit the binary representation of n (e.g., [15, pp. 524-525])
 
6. Conclusion
The existing taxonomy of algorithm design techniques has several important shortcomings pointed out in this paper. We suggested reclassifying, and in a few cases renaming some of the known approaches to algorithm design. The paper further proposed grouping them according to the technique's level of generality; the principal criterion suggested in the paper for separating more general techniques from less general ones is the technique's applicability to sorting and searching. The resulting taxonomy put four techniques --- brute force, divide-and-conquer, decrease-and-conquer, and transform-and-conquer --- in the category of more general techniques, leaving the rest --- local search techniques, dynamic programming, and state-space-tree techniques --- in the second, less general, category. Further, important special types of the major design techniques were also identified.
A few cautionary notes seems to be in order, however. First, no matter how many general design techniques are recognized, there will always be algorithms that cannot be naturally interpreted as an application of one of those techniques. Some algorithms are just based on insights peculiar to the problem they solve and do not allow for a broad and useful generalization.

On the other hand, some algorithms can be interpreted as an application of different techniques. For example, selection sort can be legitimately interpreted both as a brute-force algorithm and as a decrease-and-conquer method. As another example, Horner's rule can be considered both as a decrease-conquer algorithm and as an algorithm based on the transformation strategy.

Finally, some algorithms may incorporate ideas of several techniques. For example, the fast Fourier transform takes advantage of both the transformation and divide-and-conquer ideas. Further, most successful approximation algorithms for the traveling salesman problem are comprised from a greedy heuristic to get an initial approximation followed by one of the improvement procedures (see [10] ).

The above remarks notwithstanding, we see several advantages in this new classification. First, it improves the currently accepted taxonomy by eliminating the shortcomings enumerated above. Second, it better reflects the richness of algorithm design techniques and allows showing them on different levels of detail. Finally, it allows to classify some important algorithms (e.g., Euclid's algorithm, heapsort, search trees, hashing) which the currently accepted taxonomy is incapable of doing.

References

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