You are given a string and a set of queries. For each query, determine if a given substring can be rearranged to form a palindrome after replacing up to a certain number of its characters.

You're given a list of numbers. By deleting a number, you earn its value, but you must also delete all numbers one less and one greater than it. Maximize the total earned value.

Given a string, reorder its characters by repeatedly picking the smallest character, then the largest, and appending them to the result.

Imagine two people picking cherries from a grid, starting and ending at opposite corners. Find the maximum number of cherries they can collect together, with some constraints on their movement and the grid's cherry distribution.

Implement a data structure that can store key-value pairs and calculate the sum of values for keys with a given prefix. You'll need to efficiently handle updates and prefix-based sum queries.

You are given a sequence of numbers and must insert them into an initially empty array. Calculate the cost of inserting each number, defined as the minimum of elements smaller or larger than the number currently in the array, and return the total cost to create the sorted array.

You are given a list of tasks, each requiring a certain amount of energy to start and providing some energy upon completion. Find the minimum initial energy you need to have in order to complete all the tasks in the given order.

You're given an array of numbers. Find the minimum number of distinct integers you need to remove so that the remaining array size is at most half of the original size.

Given an array of unique positive integers, find the total number of different greatest common divisors possible from all of its non-empty subsequences.

Imagine you're watering a garden using strategically placed taps. Find the fewest taps you need to open to water the entire garden, given each tap's watering range.

You're given a network of computers and cables. Determine the minimum number of cables you need to move to connect all the computers, or if it's impossible.

Find the number of distinct sequences possible when rolling a standard six-sided die multiple times, subject to constraints that limit how many times each face can appear consecutively.

You are given a network of nodes and the time it takes to travel between them. Determine the minimum time it takes for a signal to reach all nodes, starting from a given source node.

Find the maximum sum of a subsequence in an array, where the distance between consecutive elements in the subsequence is constrained by a given value.

In a graph where each node has a value, find the maximum sum achievable by selecting a node and up to 'k' of its neighbors. The star sum is the sum of the chosen node's value and its neighbors' values.

You're given the processing times for processors and tasks. Assign each task to a processor to minimize the overall completion time, considering each processor can handle tasks in batches.

Identify the critical connections in a network of computers, where removing a critical connection will disconnect the network. These are the connections that are essential for maintaining the network's connectivity.

Given an undirected graph that contains exactly one cycle, find the shortest distance from each node to any node in that cycle. Nodes that are part of the cycle have a distance of zero.

Given a grid representing land and water, count the number of distinct islands. An island is formed by connected land cells.