We need to determine the minimum cost to obtain at least (H) pounds of hay given (n) suppliers. Each supplier offers a bundle weighing (p_i) pounds at cost (c_i), and bundles can be purchased unlimitedly. Approach 1: State Transition with Conditional Bounds Standard knapsack formulations maximize va...
Canvas Painting The problem involves finding the maximum number of effective operations to reduce distinct colors. With n initial colors, each operation can reduce one color by making two positions share the same color. The answer is n minus the maximum effective operations. Algorithm: Group paintin...
139. Word Break Given a non-empty string s and a dictionary wordDict containing a list of non-empty words, determine if s can be segmented into a space-separated sequence of one or more dictionary words. Note: The same word in the dictionary may be reused multiple times in the segmentation. You may...
A. [POI2004] SZP Description Given a directed graph with cycles, where each node has exactly one outgoing edge (non-self-loop), select the maximum number of nodes such that every selected node has at least one immediate predecessor that is not selected. Solution Observe that each weakly connected co...
This problem can be modeled as a dynamic programming problem where decisions are made annually to maximize profit. Each year, we have a certain amount of capital and can choose to invest in d different options. Each option j requires an initial investment of a[j] units and yields a profit of b[j] un...
Given an array (a_1, a_2, \ldots, a_n) and three variables (P, Q, R) initially set to zero, process each element sequentially. For each element (a_i), choose one of three operations: (P := P \oplus a_i) (Q := Q \oplus a_i) (R := R \oplus a_i) After each operation, at least two of (P, Q, R) must be e...
This article summarizes several challenging competitive programming problems, showcasing various algorithmic techniques from dynamic programming and data structures to number theory and tree algorithms. Each problem explores distinct optimization strategies and mathematical insights. Problem 1: Reso...
Divisibility and Digit Extraction The objective is to determine if a given integer $N$ is either divisible by 3 or contains the digit '3'. A direct approach involves checking the modulo and then iteratively extracting digits. The time complexity for this is $O(\log_{10} N)$. bool is_lucky(long long...
This article examines several algorithmic problems featuring dynamic programming, state management, and graph traversal techniques. Problem 1: Frozen Dumplings Consider a scenario where daily dumpling prices vary, and storage costs accrue for each day a dumpling is kept. Given prices for n days and...
String Character AnnihilationWhen two distinct characters in a string annihilate each other, the specific identity of the characters is irrelevant. As long as multiple distinct characters exist, they can be paired off. The goal of minimizing the remaining characters can be modeled as a two-pile stac...