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question:You are given an infinite grid with coordinates ranging from (-∞, -∞) to (∞, ∞). Initially, there is a robot at the coordinate (0, 0). The robot can move according to a list of commands. Each command is one of the following four: - "U" (up) which increases the y-coordinate by 1. - "D" (down) which decreases the y-coordinate by 1. - "L" (left) which decreases the x-coordinate by 1. - "R" (right) which increases the x-coordinate by 1. You are given a list of commands executed by the robot. Your task is to determine the minimum number of commands that need to be removed from this list so that the robot returns to the origin (0, 0) after executing the remaining commands. If it is not possible for the robot to return to the origin, output -1. Input - The first line contains an integer `n` (1 ≤ n ≤ 100,000) representing the number of commands. - The second line contains a string of `n` characters, each being one of "U", "D", "L", "R". Output - Output a single integer which is the minimum number of commands to be removed, or -1 if it is impossible for the robot to return to the origin. Examples Input 6 UUDLRL Output 2 Input 4 UDUD Output 0 Input 7 UUDDLRL Output -1 Explanation - In the first example, removing the last two commands "RL" would make the robot return to the origin. - In the second example, the robot already returns to the origin without removing any command. - In the third example, it is impossible to make the robot return to the origin regardless of which commands are removed.
answer:def min_commands_to_remove(n, commands): x, y = 0, 0 for command in commands: if command == 'U': y += 1 elif command == 'D': y -= 1 elif command == 'L': x -= 1 elif command == 'R': x += 1 if (x + y) % 2 != 0: return -1 return abs(x) + abs(y)
question:Write a function `balance_arrays(arr1, arr2)` that takes two lists of integers `arr1` and `arr2`. Determine if you can make `arr1` and `arr2` identical by swapping at most one integer from `arr1` with one integer from `arr2`. If it's possible, return True; otherwise, return False. If either list is empty, return False.
answer:def balance_arrays(arr1, arr2): Determines if you can make arr1 and arr2 identical by swapping at most one integer from arr1 with one integer from arr2. If it's possible, returns True; otherwise, returns False. If either list is empty, returns False. if not arr1 or not arr2: return False if sorted(arr1) == sorted(arr2): return True sum_arr1 = sum(arr1) sum_arr2 = sum(arr2) target_diff = sum_arr1 - sum_arr2 possible_swaps = set(arr1) for num in arr2: required_num = num + target_diff // 2 if (target_diff % 2 == 0) and (required_num in possible_swaps): return True return False
question:You are given an array of integers, `arr`, of size `n` and a positive integer `k`. Define the "k-th distinct prime factor count" of an integer as the number of distinct prime factors of the k-th largest integer in the subarrays of length `k` of the given array. Your task is to determine the maximum "k-th distinct prime factor count" for any valid subarray of length `k`. Input Format: - The first line contains two space-separated integers, `n` (1 ≤ n ≤ 10^5) and `k` (1 ≤ k ≤ n). - The second line contains `n` space-separated integers describing the elements of `arr` (1 ≤ arr[i] ≤ 10^5). Output Format: Print a single integer denoting the maximum "k-th distinct prime factor count" for any valid subarray of length `k`. Sample Input 0: 5 3 10 15 21 24 30 Sample Output 0: 3 Sample Input 1: 8 4 7 11 13 2 3 5 14 20 Sample Output 1: 2 Explanation: For Sample Input 0: - We have `k = 3` which means we need to consider subarrays of length 3. - The possible subarrays of length 3 from `arr` are: [10, 15, 21], [15, 21, 24], [21, 24, 30]. - The third largest integers in these subarrays are: 10 (from [10, 15, 21]), 15 (from [15, 21, 24]), and 21 (from [21, 24, 30]). - The prime factor counts for these numbers are: - 10 has prime factors {2, 5} making it 2. - 15 has prime factors {3, 5} making it 2. - 21 has prime factors {3, 7} making it 2. - Hence, the maximum k-th distinct prime factor count is 3, and for the subarray [10, 15, 21], the third largest integer 21 has 2 prime factors. For Sample Input 1: - We have `k = 4` which means we need to look at subarrays of length 4. - The possible subarrays of length 4 are: [7, 11, 13, 2], [11, 13, 2, 3], [13, 2, 3, 5], [2, 3, 5, 14], [3, 5, 14, 20]. - The fourth largest integers for these subarrays are: 2 (from [7, 11, 13, 2]), 2 (from [11, 13, 2, 3]), 3 (from [13, 2, 3, 5]), 3 (from [2, 3, 5, 14]), 5 (from [3, 5, 14, 20]). - The prime factor counts for these numbers are: - 2 has prime factor {2}, making it 1. - 2 has prime factor {2}, making it 1. - 3 has prime factor {3}, making it 1. - 3 has prime factor {3}, making it 1. - 5 has prime factor {5}, making it 1. - Thus, the maximum k-th distinct prime factor count for these subarrays is 2.
answer:from collections import defaultdict import math # Function to compute prime factors of a number def prime_factors(n): factors = set() while n % 2 == 0: factors.add(2) n = n // 2 for i in range(3, int(math.sqrt(n)) + 1, 2): while n % i == 0: factors.add(i) n = n // i if n > 2: factors.add(n) return factors def max_k_th_prime_factor_count(n, k, arr): max_prime_factors_count = 0 for i in range(n - k + 1): subarray = sorted(arr[i:i + k]) kth_largest = subarray[k - 1] prime_factors_count = len(prime_factors(kth_largest)) max_prime_factors_count = max(max_prime_factors_count, prime_factors_count) return max_prime_factors_count
question:In a distant galaxy, there is an intricate network of planets connected by special wormholes. There are M planets, indexed from 1 to M, in this galaxy. The wormholes form a grid structure with precisely (M-1)^2 connections, such that every planet is connected to its neighboring planets in a square grid fashion. Each wormhole allows bi-directional travel, enabling spaceships to fly to adjacent planets. Each planet in the galaxy generates a unique sequence of energy pulses. These pulses can either influence the wormhole's stability or open new paths to other planets. There are N energy frequencies, indexed from 1 to N, which can activate a wormhole if applied sequentially. You are tasked with finding a way to navigate from planet 1 to planet M using the least number of energy frequencies such that the spaceship can travel from planet 1 to planet M. Assume that no energy frequency is repeated. Determine the minimum number of energy frequencies needed. Input - The first line contains two space-separated integers, M (1 ≤ M ≤ 10^4) – the number of planets and N (1 ≤ N ≤ 10^5) – the number of unique energy frequencies. - The next N lines describe the energy frequencies. Each line contains a single integer e (1 ≤ e ≤ N), representing an energy frequency. Output - You should print a single integer: the minimum number of energy frequencies required to travel from planet 1 to planet M. Example Input 4 3 1 2 3 Output 3 Note In this example, the planets are placed in a 2x2 grid. Thus, M=4 means there are 4 planets {1,2,3,4} connected as follows: 1 - 2 | | 3 - 4 The least number of energy frequencies required to travel from planet 1 to planet 4 is 3 (one for each step in the path 1->2->4). Each energy frequency ensures that the next wormhole is stabilized for safe travel.
answer:def find_min_energy_frequencies(M, N, energy_frequencies): Finds the minimum number of energy frequencies required to travel from planet 1 to planet M. Parameters: M (int): The number of planets. N (int): The number of unique energy frequencies. energy_frequencies (list of int): The list of energy frequencies. Returns: int: Minimum number of energy frequencies required. # Since planets are indexed from 1 to M, the grid is sqrt(M) x sqrt(M) import math grid_size = int(math.sqrt(M)) if grid_size * grid_size != M: raise ValueError("Invalid grid configuration for given M") # Minimum energy frequencies needed to travel from 1 to M is the number # of moves required in that grid from top-left to bottom-right return (grid_size - 1) * 2 # Example usage: M = 4 N = 3 energy_frequencies = [1, 2, 3] print(find_min_energy_frequencies(M, N, energy_frequencies)) # Output should be 2