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Minimum Path Sum in a Grid Finding the minimum path sum from the top-left to the bottom-right of a grid is a classic dynamic programming problem. To optimize space complexity, we can use a 1D array instead of a 2D matrix to store the DP states, updating the array iteratively as we traverse each row. #include <vector> #include <algorith ...

Made Up Problem Statement Given three sequences A, B, C, count the number of pairs (i, j) such that A[i] = B[C[j]]. Solution Approach Since all values are bounded between 1 and N, we can use frequency counting. For each value v, count how many times it appears in array A (stored in cntA) and how many times it appears as B[C[j]] (stored in cntB) ...

Problem Description Given an array of n positive integers and a positive integer target, find the length of the shortest continuous subarray whose sum is at least target. Return the length of this subarray. If no such subarray exists, return 0. Example 1: Input: target = 7, nums = [2,3,1,2,4,3] Output: 2 Explanation: The subarray [4,3] has the ...

Shortest path problems in graph theory often rely on the concept of relaxation. Relaxation occurs when a path from node u to v can be shoretned by routing through an intermediate node k, i.e., if dist[u][v] > dist[u][k] + dist[k][v], we update dist[u][v] accordingly. Floyd-Warshall Algorithm (All-Pairs Shortest Paths) To compute shortest pat ...

D - Path Traversal A straightforward depth-first search approach can solve this traversal problem. int nodes, edges, max_steps, min_cost, max_cost; vector<int> valid_endpoints; vector<pair<int, int>> graph[MAX_NODES]; void traverse(int current_node, int current_cost, int steps_taken) { if (current_cost > max_cost) retu ...

Task 1 – Factor Splitting Given a positive integer n that is the product of two distinct primes, output the larger one. Input: One integer n (≤ 2·109). Output: The larger prime factor. Observation: The first divisor (other than 1) must be the smaller prime, so the answer is n / i once the smallest i > 1 with n % i == 0 is found. #include &lt ...

Linear DP Core Concepts Dynamic Programming (DP) solves complex problems by breaking them in to overlapping subproblems. The solution to the main problem is derived from solutions to these subproblems. State Representation State are typically represented as dp[i][j] = value, where i and j are indices or variables describing the state, and value ...

The problem models a sequential arrangement where each position $i$ offers two distinct categories of elements. Category X occupies exactly one slot, while Category Y spans two consecutive slots $(i-1, i)$. The objective is to compute the total number of valid configurations modulo $10^9+7$. A two-state dynamic programming approach efficiently ...

A. Apple Tree Ring Given a sequence of integers representing apple counts on trees arranged in a circle, determine the maximum number of distinct values that can be consumed under infinite rotations. Since rotations allow arbitrary reordering over time, the optimal strategy is to consume one unique value per round. Hence, the answer equals the ...

Understanding Dynamic Programming Dynamic programming (DP) is an optimization technique used to solve complex problems by breaking them into simpler subproblems. It stores the results of subproblems to avoid redundant computations, thereby improving efficiency. This article explores fundamental DP concepts and implementations through various al ...