Freaks City

The Floyd-Warshall algorithm solves the all-pairs shortest path problem in a weighted graph, handling both positive and negative edge weights (with no negative cycles). It uses dynamic programming to iteratively improve shortest path estimates between every pair of vertices. Core Principal Define dist[i][j][k] as the shortest distance from node ...

Problem A: Plus Minus Given the sum and difference of two numbers, we can easily retrieve the original values. The first number is the average of the sum and difference, while the second is half of the difference subtracted from the sum. s, d = map(int, input().split()) print((s + d) // 2, (s - d) // 2) Problem B: DNA Sequence A substring is c ...

This document presents an analysis of several competitive programming problems, outlining their descriptions, solution approaches, and specific implementation details or common pitfalls encountered. The problems cover various domains including number theory, combinatorics, geometry, and dynamic programming. Problem 1: Minimizing Sum with Given ...

Progress: Dynamic Programming - Linear DP, Knapsack, Interval DP Merging Palindromic Substrings Tags: Interval DP Foundation - Longest Palindromic Substring Problem: Given a string (S), find the length of its longest palindromic substring. Approach: A longer palindrome can always be constructed by adding identical characters to both ends of a s ...

Given an integer array representing coin denominattions and a target amount, determine the minimum number of coins required to make up that amount. Return -1 if no valid combination exists. Coins can be used unlimited times. Recursive Approach A recursive function findMinCoins(int[] denominations, int target) calculates the minimum coins needed ...

Problem Overview The problem involves merging a sequence of stones where each stone has a weight. The goal is to merge consecutive stones within a sequence into a single stone with a weight equal to the sum of the merged stones, at a cost equal to that sum. The merging must result in a final number of stones between a given range [L, R], and th ...

Problem Analysis Given (n) where (1 \le n \le 20), we need to count the number of ways to completely cover the interval ([1, n]) using segments where each segment connects two coprime numbers. First, preprocess all coprime pairs ({a, b}) where (\gcd(a, b) = 1) and (a < b). Note that ({1, 1}) is excluded. When (n = 20), there are exactly 127 ...

Sorting and Monotonicity When a problem does not enforce a specific elemant order, applying a sort operation often introduces monotonicity. This property simplifies constraint checking and enables efficient querying through prefix sums combined with binary search. Processing Cumulative Constraints By sorting the input array and computing its pr ...

In this problem, we are given an array $a$ of length $n$ and a target array $b$ of length $m$. Each element $a_i$ has an associated deletion cost $p_i$. We need to find the minimum cost to transform $a$ in to $b$ using a specific "strange function" $f(a)$, or determine if it is impossible. Condition Analysis The function $f(a)$ genera ...

Consider two sequences of non-negative integers \(p_1 \ge p_2 \ge \dots \ge p_n\) and \(q_1 \ge q_2 \ge \dots \ge q_m\) satisfying \(\sum_{i=1}^n p_i = \sum_{i=1}^m q_i\). The Gale-Ryser theorem states that a simple bipartite graph exists with left vertices having degrees \(p_1, p_2, \dots, p_n\) and right vertices having degrees \(q_1, q_2, \d ...