Valid for a single buy-sell cycle The problem asks for the maximum possible profit from one purchase and one sale. A dynamic programming approach tracks two states for each day: cash_with_stock[i]: the largest amount of cash achievable on day i while holding the stock. cash_without_stock[i]: the lar...
The classic problem of counting distinct paths in a rectangular grid where movement is restricted to right and down steps can be modeled as a binary tree structure, but such an approach leads to exponential time complexity. A more efficient solution uses dynamic programming with a two-dimensional st...
The objective is to partition an m by n rectangular cake into 1x1 unit squares with the least total cost. Each horizontal cut between rows i and i+1 incurs a cost of horizontalCut[i]. Each vertical cut between columns j and j+1 incurs a cost of verticalCut[j]. The total cost is the sum of all cut co...
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...