Knapsack Problems Knapsack problems represent one of the foundational dynamic programming concepts. Typically, either weight or value is represented as a dimension in the state space, with the smaller dimension usually chosen for optimization. When both weights and values are large, standard approac...
Given $n$ closed intervals $[l_i, r_i]$, the objective is to determine the cardinality of the set ${\sum_{i=1}^{n} x_i^2 \mid x_i \in \mathbb{Z}, l_i \leq x_i \leq r_i}$. Assuming $n \leq 100$ and $r_i \leq 100$, the maximum possible sum is bounded by $10^6$. This permits a state-compression approac...
The problem involves determining the lexicographically largest binary string that can be constructed under constraints derived from a set of pattern strings containing '0', '1', and '-' (wildcard) characters. Each query provides a target string, and the solution must find the maximal answer consiste...
Problem 1: Apple Planting Problem Description There is an n-row by m-column farmland in front of Taotao's house. Some grids can grow apples (represented by 1), while others cannot (represented by 0). Additionally, adjacent grids cannot both grow apples. Taotao wants to know how many possible plantin...