Freaks City

The objective is to evaluate the following double summation for a sequence $A$ of length $N$: $$ \sum_{L=0}^{N-1} \sum_{R=L}^{N-1} \left( \max_{k \in [L, R]} A[k] - \min_{k \in [L, R]} A[k] \right) $$ A brute-force enumeration of all contiguous segments results in quadratic or cubic complexity, which is insufficient for large inputs. Two linear ...

Subtree Budget Allocation and Optimal Path Reconstruction The core challenge revolves around allocating a fixed budget across a binary tree structure to maximize a specific threshold value. The solution begins by analyzing the root's contribution and propagating constraints downward. We define a dynamic programming state dp_max[u][b] representi ...

A. Omkar and Password Given a sequence of integres, we can merge adjacent disitnct elements into their sum. The goal is to minimize the final sequence length. If all elements are equal, no merges are possible and the result is the original length. Otherwise, we can always reduce the sequence to a single element by repeatedly merging with the ma ...

Problem Description Given a permutation of length n and an integer k, partition the array in to k contiguous segments such that the sum of inversion counts within each segment is minimized. Constraints: n ≤ 25000, k ≤ 25 Solution Approach This problem can be solved using dynamic programming with decision monotonicity and divide-and-conquer opti ...

Introduction During the summer vacation, I systematically studied various operations on trees through Teacher Tuo's sharing and discovered many useful techniques, which I now summarize. Teacher Tuo is amazing! One Problem Type: Batch processing queries of the form f(dep(lca(x,y))), where f(x) is a function of x. Approach: First perform tree ...

1. Gymnastic Team Formation Given the small input constraints, a brute-force approach with backtracking and pruning is suitable. The solution uses depth-first search (DFS) to explore valid permutations while eliminating invalid paths early. #include <iostream> using namespace std; int constraints[11] = {0}; bool used[11] = {false}; int v ...

Given an $n \times m$ grid where each cell is either empty (.) or blocked (*), place the minimum number of $1 \times 2$ or $2 \times 1$ dominoes such that no two adjacent empty cells remain — i.e., it's impossible to place any additional domino. Constraints: $1 \leq n \leq 70$, $1 \leq m \leq 7$. Due to the small value of $m$, a dynamic program ...

Problem 1: Optimized Dynamic Programming with Prefix Sums This problem involves a basic dynamic programming approach where we process from the end to the beginning. The naive solution has a time complexity of O(n²), but we can optimize it using prefix sums and binary search. We maintain a prefix sum array and for each position, use binary searc ...

Potion-making Solution This problem requires solving the equation i/(i+j) = k/100 to find the minimal total ingredients. The solution involves iterating through possiblle values of i and j. #include <iostream> #include <cmath> using namespace std; void solvePotion() { int target_percentage; cin >> target_percentage; ...

Longest Increasing Subsequence (LIS) The Longest Increasing Subsequence problem involves findinng the maximum length of a strictly increasing subsequence from a given sequence of length n. The subsequence elements need not be contiguous in the original sequence. Dynamic Programming Approach (O(n²)) State Representation DP array: Stores the len ...