Freaks City

A bracket string of even length consists of exactly n/2 opening [ and n/2 closing ] characters. The goal is to determine fewest number of arbitrary index swaps required to transform the string into a valid bracket sequence (one where every closing bracket has a matching opening bracket earlier in the string). Pairs of matched brackets can be t ...

Binary Search Implemantation Patterns Two common binary search variations address different optimization scenarios: Maximizing Minimum Value #include <iostream> #include <vector> #include <algorithm> using namespace std; bool validateMin(vector<long>& positions, long min_gap, int removals) { long prev = 0; i ...

Equalizing Card Piles (Greedy Approach) To distribute cards equally among $N$ piles, calculate the average number of cards per pile and subtract this value from each pile's count. This transforms the problem into finding the minimum number of moves to zero out the differences. Iterating from left to right, if a pile has a non-zero discrpeancy, ...

Problem Statement: Given positive integers $n$, $m$, and $m$ triplets $(l_i, r_i, c_i)$, we have an array $a_{1..n}$ initialized with zeros. For each operation $i = 1, ..., m$, perform the following steps: Choose any integer $k \in [0, c_i]$. Add $k$ to all elements $a_j$ where $j \in [l_i, r_i]$. Add $c_i - k$ to all elements $a_j$ where $j \ ...

Resource Redistribution with Asymmetric Operations Given a sequence (a) of length (n) and two positive integers (x, y) where (y \le x). You may repeatedly pick two elements and transfer resources: subtract (x) from one and add (y) to the other, provided the first remains non‑negative. Determine the maximum possible value of any element after op ...

Given an integer array nums, find and return the maximum sum of elements that is divisible by three. Examples Example 1: Input: nums = [3,6,5,1,8] Output: 18 Explanation: Choose numbers 3, 6, 1, and 8. Their sum is 18 (the maximum sum divisible by three). Example 2: Input: nums = [4] Output: 0 Explanation: 4 is not divisible by 3, so no number ...

Factorization into Factorial Divisors Given integers (n) and (m) where (1 \le m \le n!) and (n \le 20), decompose (m) into a sum of at most (n) divisors of (n!). A solution is guaranteed to exist. Define a sequence (d_i = \frac{n!}{i!}) for (i) from 1 to (n). By iterating downwards from (i=n) to (1) and greedily subtracting the largest possible ...

Problem 1: Dumpling Purchase Optimization The problem reduces to a daily deciison: buy dumplings at the current price or rely on an earlier purchase plus storage cost. The total expense for day i if we buy on day j ≤ i is price[j] + c*(i - j). This can be rewritten as (price[j] - c*j) + c*i. Thus we can maintain the minimum value of price[j] - ...

Problem A - 4 Question With (n) horizontal lines and (m) vertical lines drawn on a plane, how many axis-aligned rectangles are formed that contain no interior lines? Solution Consider each dimension independent. Along any straight line, (n) distinct points partition the line into (n - 1) segments. These segments serve as the edges of our rectan ...

Consider a sequence $A = (a_1, a_2, \dots, a_n)$. We want to partition $A$ into contiguous subsequences, and then arrange these subsequences to form a new sequence $f_1, f_2, \dots, f_k$. Each contiguous subsequence $f_1, \dots, f_p$ must satisfy either $f_1 \le f_2 \le \dots \le f_p$ or $f_1 \ge f_2 \ge \dots \ge f_p$. The cost associated with ...