LeetCode 3628 - Maximum Number of Subsequences After One Inserting
The solution partially addresses the exercise, but it does not fully satisfy what was asked. The exercise is not asking for modern algorithms that resemble the work of Sun Tsï, al-Khwārizmī, al-Uqlidisī, Fibonacci, and Recorde.
Difficulty: 🟡 Medium
Topics: String, Dynamic Programming, Greedy, Prefix Sum
Solution
Correctness
The solution partially addresses the exercise, but it does not fully satisfy what was asked.
The exercise is not asking for modern algorithms that resemble the work of Sun Tsï, al-Khwārizmī, al-Uqlidisī, Fibonacci, and Recorde. It asks the reader to study the historical writings and translate their methods as faithfully as possible into precise algorithmic notation.
The proposed solution instead presents modern algorithms that are merely associated with the named authors. In several cases the correspondence is weak or historically questionable.
The algorithms themselves are generally correct as mathematical procedures. However, correctness of the procedures is not the main issue. The exercise is historical and algorithmic: the translation should reflect the original methods. The solution does not demonstrate that the algorithms given are faithful translations of the historical texts.
Gaps and Errors
1. Failure to translate the historical methods faithfully
Type: Critical error.
The exercise explicitly asks for translations of methods found in the original writings.
For Sun Tsï, the solution gives a modern double-loop multiplication algorithm based on positional digits. No evidence is provided that this is the actual procedure described in the Sunzi Suanjing.
For al-Khwārizmī, the solution gives modern long division written with decimal digits and floor functions. No attempt is made to reconstruct the specific procedure appearing in his arithmetic treatise.
For al-Uqlidisī, the solution gives modern column addition. No discussion is given of the distinctive features of al-Uqlidisī's decimal techniques or how the algorithm corresponds to his text.
For Fibonacci, the solution substitutes the Euclidean algorithm for gcd. Although Fibonacci discusses gcd, this is not one of the principal classical arithmetic algorithms emphasized in Liber Abaci. The exercise asks for study of classical arithmetic algorithms, and a faithful translation would ordinarily focus on the arithmetic procedures themselves.
For Recorde, the square-root algorithm is plausible, but no historical correspondence is demonstrated.
The exercise requires historical reconstruction; the solution provides modern replacements.
2. No citations, quotations, or evidence from the sources
Type: Critical error.
The exercise begins with "Study the early history ... by looking up the writings".
The solution contains no references to passages, examples, descriptions, or terminology from the original works. Consequently there is no basis for judging whether the translations are faithful.
3. Unsupported historical claims
Type: Justification gap.
The statement that Sun Tsï's multiplication method corresponds to the presented algorithm is asserted without evidence.
The statement that al-Khwārizmī's division method is represented by the given algorithm is asserted without evidence.
The statement that Fibonacci "explicitly presents practical examples using this algorithm for trade and bookkeeping" is asserted without documentation.
4. Anachronistic notation and operations
Type: Justification gap.
The use of expressions such as
$$q_i=\lfloor R/D\rfloor$$
and
$$R \gets 10R+n_i$$
is mathematically legitimate, but a faithful translation should explain how these modern operations correspond to the historical procedure. No such explanation is provided.
5. Missing comparison with the evolution of arithmetic methods
Type: Justification gap.
The exercise asks the reader to study the early history of classical arithmetic algorithms. The solution lists isolated algorithms but does not discuss how the methods evolved from counting rods to Hindu-Arabic positional notation and later European arithmetic texts.
Summary
The algorithms presented are mostly mathematically correct, but the exercise is historical in nature. The solution does not demonstrate that the algorithms are faithful translations of the methods appearing in the cited sources, and it provides no evidence from those sources. As a result, it does not adequately answer the actual question.
VERDICT: FAIL - the primary flaw is that it replaces the historical algorithms with modern ones instead of faithfully translating the methods found in the original writings.