Freaks City

Introduction Consider a problem where N piles of stones are arranged in a line, each with a certain weight. The task is to merge these N piles into a single pile, with the constraint that only adjacent piles can be merged at each step, and the cost of each merge equals the combined weight of the two piles being merged. The objective is to find ...

Problem Analysis Given an n x 3 grid, each cell must be colored with one of three colors (red, yellow, green) such that adjacent cells (sharing horizontal or vertical edges) have different colors. The goal is to count the total number of valid colorings modulo 10^9 + 7. For a single row (n = 1), there are 12 valid colorings. As n increases, the ...

After the "Ladder Competition" (天梯赛), a company's human resources department requested a list of outstanding student recommendations from the organizing committee. This task was assigned to "Jingjing-jie" (静静姐). The company's recommendation process follows these criteria: Only students scoring 175 or above in the Ladd ...

Quadrilateral inequality optimization represents a powerful technique for optimizing certain types of dynamic programming solutions, particularly thoce involving 2D1D state transitions. The fundamental form of such DP equations is: [f_{l,r}=\min_{k=l}^{r-1}{f_{l,k}+f_{k+1,r}}+w(l,r) ] When the function w(l,r) satisfies specific properties, we c ...

Given a string s, the goal is to locate the longest cnotiguous sequence of characters that reads identically forward and backward. The input string length is capped at 1000 characters. Problem Examples Example 1: Input: "babad" Output: "bab" Note: "aba" is also a valid solution. Example 2: Input: "cbbd&quot ...

AGC010F Tree Game Let \( val[u] \) represent the number of stones located at node \( u \). A useful heuristic arises when considering a node \( u \) with a single child \( v \). If \( val[u] \le val[v] \), and the first player moves the token to \( v \), the second player can immediately move it back to \( u \). This traps the first player, lea ...

Knapsack Problem Overview The knapsack problem involves selecting items with given weights and values to maximize total value within a weight cnostraint. 0/1 Knapsack Given n items and a knapsack with maximum capacity w, where each item has a weight weight[i] and value value[i], determine which items to include to maximzie value. Each item can ...

01 Knapsack Given N item types and a knapsack with capacity V. Each item can be selected at most once, with volume v[i] and value w[i]. Find the maximum total value without exceeding capacity. Constraints: 0 < N, V <= 1000; 0 < v[i], w[i] <= 1000. The classic 01 knapsack uses a two-state transition approach: either skip the current ...

N-ary Tree Preorder Traversal Given the root of an n-ary tree, return the preorder traversal of its nodes' values. /* class Node { public: int val; vector<Node*> children; Node() {} Node(int _val) { val = _val; } Node(int _val, vector<Node*> _children) { val = _val; children = _children; } }; ...

OSU! Let $E_i$ denote the expected score after the $i$-th operation. The key observation is that $(x+1)^3 - x^3 = 3x^2 + 3x + 1$. To compute this increment in expectation, maintain: $$ \text{exp1}i = (\text{exp1}{i-1} + 1) \cdot p_i $$ $$ \text{exp2}i = (\text{exp2}{i-1} + 2 \cdot \text{exp1}_{i-1} + 1) \cdot p_i $$ Then update the total expect ...