#101 - Optimum Polynomial
If we are presented with the first \(k\) terms of a sequence it is impossible to say with certainty the value of the next term, as there are infinitely many polynomial functions that can model the sequence.
As an example, let us consider the sequence of cube numbers. This is defined by the generating function,
\(u_n = n^3\): \(1, 8, 27, 64, 125, 216, \dots\)
Suppose we were only given the first two terms of this sequence. Working on the principle that "simple is best" we should assume a linear relationship and predict the next term to be \(15\) (common difference \(7\)). Even if we were presented with the first three terms, by the same principle of simplicity, a quadratic relationship should be assumed.
We shall define \(\operatorname{OP}(k, n)\) to be the \(n\)th term of the optimum polynomial generating function for the first \(k\) terms of a sequence. It should be clear that \(\operatorname{OP}(k, n)\) will accurately generate the terms of the sequence for \(n \le k\), and potentially the first incorrect term (FIT) will be \(\operatorname{OP}(k, k+1)\); in which case we shall call it a bad OP (BOP).
As a basis, if we were only given the first term of sequence, it would be most sensible to assume constancy; that is, for \(n \ge 2\), \(\operatorname{OP}(1, n) = u_1\).
Hence we obtain the following \(\operatorname{OP}\)s for the cubic sequence:
| \(\operatorname{OP}(1, n) = 1\) | \(1, {\color{red}\mathbf 1}, 1, 1, \dots\) |
| \(\operatorname{OP}(2, n) = 7n - 6\) | \(1, 8, {\color{red}\mathbf{15}}, \dots\) |
| \(\operatorname{OP}(3, n) = 6n^2 - 11n + 6\) | \(1, 8, 27, {\color{red}\mathbf{58}}, \dots\) |
| \(\operatorname{OP}(4, n) = n^3\) | \(1, 8, 27, 64, 125, \dots\) |
Clearly no BOPs exist for \(k \ge 4\).
By considering the sum of FITs generated by the BOPs (indicated in red above), we obtain \(1 + 15 + 58 = 74\).
Consider the following tenth degree polynomial generating function: \[u_n = 1 - n + n^2 - n^3 + n^4 - n^5 + n^6 - n^7 + n^8 - n^9 + n^{10}.\]
Find the sum of FITs for the BOPs.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=101. Published Friday, 29th July 2005, 06:00 pm. Solved by 13,553 members at time of mirroring.
Why this is useful
Algorithmic Development. Optimal substructure and state-space reasoning are exactly how American-option pricing and optimal execution are solved (Phases 13, 16).
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Prerequisites
Lessons that prepare you:
1.1 Sets, Functions, and Relations · 10.4 Algorithms: Gradient Descent and Newton's Method · 10.1 Convex Sets and Convex Functions · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.7 Dynamic Programming: Memoization and Tabulation · 19.6 Recurrence Relations and Generating Functions · 2.1 Functions, Limits, and Continuity · 2.7 Sequences, Series, Convergence, and Power Series · 2.4 Taylor Series and Local Approximation · 4.2 Linear Maps, Matrices, Rank, and the Null Space
Recommended stepping-stone problems: #30 · #52 · #33
Concepts: algebra optimization sequences-series
Likely techniques: generating-functions memoization polynomial
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- What exactly is the input to problem 101? Is it a bound (216), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single sum.
- Which objects exactly are in scope, and which are excluded by the wording (strict vs non-strict inequality, 'distinct', 'proper', 'below' vs 'up to')?
- What constraint does the bound 216 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 216?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 216 and the cost of testing one.
- Which algebra fact would, if true, collapse the search - and can you state it as a testable claim before you look for a proof?
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- Predict the strategy: in one sentence, what will your solution do? (The classification says algebra / generating-functions - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 216, and the wall-clock time you expect. Write both down now.
- Predict the key data structure: what is stored, keyed by what, and how large will it get at full scale?
- Predict the failure mode: what is most likely to break - an off-by-one on the bound, a definition misread, precision, or memory?
- Predict the output of the small case from rung 3 BEFORE running it (the statement says: "As an example, let us consider the sequence of cube numbers.") - then run it. A surprise here is worth more than an hour of debugging later.
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Which step of your solution were you least confident about, and what evidence would settle it?
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Where did the algebra structure do the real work? Name the single observation that collapsed the search space.
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- Related problem: #30 · #52 · #33
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