Upgrade Toolkit — Solution¶
Comparing solutions¶
Both implementations of the problem setter’s solution represent type \(i\) by the factor \(1+x+\cdots+x^{c_i}\) , where \(c_i\) is its quantity cap, and multiply factors using ordinary nested-loop convolution. Each term chooses a legal quantity, so expanding the product enumerates exactly all combinations. This is the common counting proof.
Let \(S_i=\sum_{j\le i}c_j\) and \(S=S_n\) . Direct convolution takes \(O(\sum_i(S_{i-1}+1)(c_i+1))\) time; storing all stages as the problem setter’s implementation does costs \(O(n(S+1))\) space. The implementation used by Codex (GPT-6) observes that each new coefficient is a consecutive sum of old coefficients and maintains it with a sliding window. Time becomes \(O(n(S+1))\) and rolling storage is \(O(S+1)\) . Rolling arrays could also be applied separately to the problem setter’s convolution. The time improvement comes from window sums; the counting model is unchanged.
Count by type¶
Type \(i\) allows quantities from \(0\) through \(C_i\) , so the answers are the coefficients of
Let \(f_j\) count plans collecting \(j\) toolkits from the processed types, initially \(f_0=1\) . Adding a type with capacity \(C_i\) gives
Treat out-of-range coefficients as \(0\) . Every plan uniquely splits into a plan for the previous types and a quantity for the new type, so the recurrence counts every plan exactly once.
Sliding window¶
As \(j\) increases, add \(f_j\) and remove \(f_{j-C_i-1}\) from the window sum. Maintaining that sum modulo \(100000007\) makes each coefficient take constant time. Store the current and next arrays and swap them after each type.
A zero-capacity type still occupies its original ID; do not renumber subsequent types. When \(n=0\) , retain the initial array and print \(1\) .
Since \(C_i\le\lfloor1000/i\rfloor\) , the total capacity satisfies
Time complexity is \(O(n(S+1))\) and space complexity is \(O(S+1)\) . Each window update adds or removes one reduced coefficient, so ordinary integer arithmetic with immediate normalization suffices.