This problem requires an insightful observation into the relationship between interval boundaries and Cartesian tree topology. Key Insight: Interval Transformation First, transform the given intervals by incrementing each $l_i$ and decrementing each $r_i$. These adjusted ranges correspond precisely...
Merging Strings for Maximum Palindromes Given two strings $S1$ and $S2$, we aim to merge them into a single string $S3$ while maintaining the relative order of characters from the original strings. The goal is to find the maximum possible length of a palindromic substring within any such $S3$. A fou...
Matrix is not easy to handle, so consider deriving the next row from the current row. Let the previous row have selected columns at positions (p_1, p_2, \ldots, p_k) as 1. Consider the choices of (x) in the current row. (The following requires drawing a diagram for understanding) Case Analysis Case...
Problem 1: Counting Isosceles Triangles (Easy Variant) Description Given a grid containing characters, determine the total number of isosceles triangles that can be formed by connected asterisk (*) symbols. Solution Strategy Iterate through every cell that contains an asterisk, treating it as the po...
Fibonacci Number (LeetCode 509) Problem Statement The Fibonacci sequence, denoted as F(n), starts with base values F(0) = 0 and F(1) = 1. For all integers n greater than 1, each term is the sum of the two preceding terms, following the formula F(n) = F(n-1) + F(n-2). Given an integer n, return the v...
Jury Compromise Define dp[i][j][k] as whether it's feasible to consider the i-th candidate with a total defense score of j and prosecution score of k. Transition: dp[i][j][k] = dp[i][j][k] OR dp[i-1][j-defense[i]][k-prosecution[i]]. After DP, enumerate absolute differences to check feasibility. Spac...
0/1 Knapsack Model Given a knapsack with capacity V and n items, each item has a value v and weight w. Each item can be taken at most once. Determine the maximum total value that can be placed in the knapsack. There are two states for each item: take or not take, leading too 2^n possibilities. Defin...
Linear Programming Formulation Let (\cdot) denote the dot product, with (b,c,x,y) as row vectors. For matrix (A), vectors (u,v) satisfy (u \leq v) if (\forall i, u_i \leq v_i), and similar for (\geq). The standard linear programming (LP) problem is: [\max c \cdot x \quad \text{s.t.} \quad \begin{cas...
Dynamic programming is a technique that decomposes complex problems in to smaller subproblems, avoiding redundant computations and storing data efficiently to solve multi-stage mathematical problems, thereby improving algorithmic efficiency. In algorithm design, dynamic programming includes methods...
Given a set of vertical pillars of integer heights and unit width, we consider arranging them in a line. Rainwater can be trapped between the pillars. The goal is to determine all possible total volumes of trapped rainwater (in unit volumes) achievable across all possible arrangements of the pillars...