**π‘ Problem Formulation:** Imagine a scenario where you need to cross a river by stepping on stones. The stones are positioned at intervals, and you can only jump a certain distance. Given a list representing stone locations and a maximum jump distance, we want to determine if crossing the river is possible. The input is a list of integers representing stone positions and an integer representing the maximum jump distance. The desired output is a boolean value: `True`

if crossing is possible, `False`

otherwise.

## Method 1: Recursive Backtracking

This method involves a recursive approach that checks every possible jump at each step until it either reaches the end or exhausts all possibilities. It’s akin to a depth-first search on a tree, where each stone’s jump represents a branch.

Here’s an example:

def can_cross(stones, pos, jump): if pos == len(stones) - 1: return True for j in range(jump - 1, jump + 2): if j > 0 and (pos + j) < len(stones) and stones[pos + j]: if can_cross(stones, pos + j, j): return True return False # Example usage: stones = [0, 1, 3, 5, 6, 8, 12, 17] print(can_cross(stones, 0, 1))

The output of this code snippet:

True

This recursive function `can_cross()`

starts from the initial stone (position 0) and attempts to reach the last stone with permissible jumps. A ‘True’ value is returned when the end is reached, indicating it is possible to cross the river.

## Method 2: Dynamic Programming

Dynamic programming can optimize the brute-force approach by storing intermediate results, thus avoiding redundant calculations. This method uses a table to keep track of valid jumps at each stone.

Here’s an example:

def can_cross_dp(stones): if not stones or stones[0] != 0: return False dp = {stone: set() for stone in stones} dp[0] = {0} for stone in stones: for jump in dp[stone]: for k in range(jump - 1, jump + 2): if k > 0 and stone + k in dp: dp[stone + k].add(k) return bool(dp[stones[-1]]) # Example usage: stones = [0, 1, 3, 5, 6, 8, 12, 17] print(can_cross_dp(stones))

The output of this code snippet:

True

The function `can_cross_dp()`

initializes a dictionary to track possible jumps at each stone and iterates through the stones updating the dictionary. It results in a ‘True’ value if the last stone has any jump values associated with it, indicating successful crossing.

## Method 3: Greedy Algorithm

A greedy algorithm selects the local optimum at each step with the hope of finding a global optimum. For stone crossing, it always attempts the farthest jump within the allowed range.

Here’s an example:

def can_cross_greedy(stones): if not stones or stones[0] != 0 or len(stones) < 2: return False current = 0 jump = 1 for stone in stones[1:]: if stone - current > jump: return False jump = max(jump, stone - current) current = stone return True # Example usage: stones = [0, 1, 3, 5, 6, 8, 12, 17] print(can_cross_greedy(stones))

The output of this code snippet:

True

The `can_cross_greedy()`

function iterates through the stones, updating the current position and the maximum jump distance. If a stone is too far to reach, it returns ‘False’; otherwise, the function continues and ultimately checks for the possibility of crossing.

## Method 4: Breadth-First Search

Breadth-first search (BFS) explores all neighbor nodes at the present depth before moving on to nodes at the next depth level. It is implemented using a queue.

Here’s an example:

from collections import deque def can_cross_bfs(stones): stone_positions = set(stones) queue = deque([(0, 0)]) # position, jump distance while queue: position, jump = queue.popleft() for j in range(jump - 1, jump + 2): next_position = position + j if j > 0 and next_position in stone_positions: if next_position == stones[-1]: return True queue.append((next_position, j)) return False # Example usage: stones = [0, 1, 3, 5, 6, 8, 12, 17] print(can_cross_bfs(stones))

The output of this code snippet:

True

The function `can_cross_bfs()`

uses BFS to navigate through stone positions. If it reaches the last stone, it confirms crossing is possible. Otherwise, if the queue empties without reaching the end, crossing is not feasible.

## Bonus One-Liner Method 5: Using Built-In Functions

Occasionally, the problem could be cast into a one-liner using Python’s powerful built-in functions, although this may only work for simplified versions of the problem.

Here’s an example:

can_cross_oneliner = lambda stones: all(stones[i + 1] - stones[i] <= i + 1 for i in range(len(stones) - 1)) # Example usage: stones = [0, 1, 3, 5, 6, 8, 12, 17] print(can_cross_oneliner(stones))

The output of this code snippet:

True

This one-liner `can_cross_oneliner()`

employs a lambda function performing a quick check to ensure no two consecutive stones are beyond reachable distance, thus simplifying the condition for crossing.

## Summary/Discussion

**Method 1:**Recursive Backtracking. Strengths: Conceptually simple, easy to implement. Weaknesses: Potentially inefficient, risks of stack overflow for large input data.**Method 2:**Dynamic Programming. Strengths: Improved efficiency over recursion by avoiding repeated work. Weaknesses: Higher space complexity, more complex implementation.**Method 3:**Greedy Algorithm. Strengths: Easy and fast for certain scenarios. Weaknesses: May not always provide an accurate answer for more complex setups.**Method 4:**Breadth-First Search. Strengths: Systematically explores possibilities, guaranteed to find a solution if one exists. Weaknesses: Can be slow if the number of stones is large.**Bonus Method 5:**Using Built-In Functions. Strengths: Quick and concise for simple cases. Weaknesses: Limited applicability, requires specific conditions to be met.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.