Explaining and Controlling Ambiguity in Dynamic Programming

Ambiguity in dynamic programming arises from two independent sources, the non-uniqueness of optimal solutions and the particular recursion scheme by which the search space is evaluated. Ambiguity, unless explicitly considered, leads to unnecessarily complicated, in exible, and sometimes even incorrect dynamic programming algorithms. Building upon the recently developed algebraic approach to dynamic programming, we formalize the notions of ambiguity and canonicity. We argue that the use of canonical yield grammars leads to transparent and versatile dynamic programming algorithms. They provide a master copy of recurrences, that can solve all DP problems in a well-de ned domain. We demonstrate the advantages of such a systematic approach using problems from the areas of RNA folding and pairwise sequence comparison. 1 Motivation and Overview 1.1 Ambiguity Issues in Dynamic Programming Dynamic Programming (DP) solves combinatorial optimization problems. It is a classical programming technique throughout computer science [3], and plays a dominant role in computational biology [4, 10]. A typical DP problem spawns a search space of potential solutions in a recursive fashion, from which the nal answer is selected according to some criterion of optimality. If an optimal solution can be derived recursively from optimal solutions of subproblems [1], DP can evaluate a search space of exponential size in polynomial time and space. Sources of Ambiguity. By ambiguity in dynamic programming we refer to the following facts which complicate the understanding and use of DP algorithms: { Co-optimal and near-optimal solutions: It is well known that the \optimal" solution found by a DP algorithm normally is not unique, and there may be relevant near-optimal solutions. A single, \optimal" answer is often unsatisfactory. Considerable work has been devoted to this problem, producing algorithms providing near-optimal [15, 17] and parametric [11] solutions. { Duplicate solutions: While there is a general technique to enumerate all solutions to a DP problem (possibly up to some threshold value) [21, 22], such enumeration is hampered by the fact that the algorithm may produce the same solution several times { and in fact, this may lead to combinatorial explosion of redundancy. Heuristic enumeration techniques, and post-facto ltering as a safeguard against duplicate answers are employed e.g. in [23]. { (Non-)canonical solutions: Often, the search space exhibits additional redundancy in terms of solutions that are represented di erently, but are equivalent from a more semantic point of view. Canonization is important in evaluating statistical signi cance [14], and also in reducing redundancy among near-optimal solutions. Ambiguity examples. Strings aaaccttaa and aaaggttaa are aligned below. Alignments (1) and (2) are equivalent under most scoring schemes, while (3) may even be considered a mal-formed alignment, as it shows two deletions separated by an insertion. aaacc--ttaa aaa--ccttaa aaac--cttaa aaa--ggttaa aaagg--ttaa aaa-gg-ttaa (1) (2) (3) In the RNA folding domain, each DP algorithm seems to be a one-trick pony. Di erent recurrences have been developed for counting or estimating the number of various classes of feasible structures of a sequence of given length [12], for structure enumeration [22], energy minimization [25], and base pair maximization [19]. Again, enumerating co-optimal answers will produce duplicates in the latter two cases. In [4](p. 272) a probabilistic scoring scheme is suggested to nd the most likely RNA secondary structure { this is a valid idea, but will work correctly only if the underlying recursion scheme considers each feasible structure exactly once. Our main contributions. The recently developed technique of algebraic dynamic programming (ADP), summarized in Section 2, uses yield grammars and evaluation algebras to specify DP algorithms on a rather high level of abstraction. DP algorithms formulated this way can have all the ambiguity problems illustrated above. However, the ADP framework also helps to analyse and avoid these problems. 1. In this article, we devise a formal framework to explain and reason about ambiguity in its various forms. 2. Introducing canonical yield grammars, we show how to construct a \master copy" of a DP algorithm for a given problem class. This single set of recurrences can correctly and e ciently perform all analyses in this problem class, including optimization, complete enumeration, sampling and statistics. 3. Re-use of the master recurrences for manifold analyses provides a major advantage from a software-engineering point of view, as it enhances not only programming economy, but also program reliability. 2 A Short Review of Algebraic Dynamic Programming ADP introduces a conceptual splitting of a DP algorithm into a recognition and an evaluation phase. A yield grammar is used to specify the recognition phase (i. e. the search space of the optimization problem). A particular parsing technique turns the grammar directly into an e cient dynamic programming scheme. The evaluation phase is speci ed by an evaluation algebra, and each grammar can be combined with a variety of algebras to solve di erent problems over the same data domain, for which heretofore DP recurrences had to be developed independently.