**π‘ Problem Formulation:** You are at a specific coordinate and want to know if it’s possible to reach a target point by traversing through a set of intermediary points. For example, given your current position as (0,0), you want to determine if you can reach (10,10) by passing through points [(2,3), (5,7), (9,10)]. The desired output is a simple ‘Yes’ or ‘No’. This article will explore five methods to tackle this problem in Python.

## Method 1: Depth-First Search (DFS) Approach

Depth-First Search (DFS) is a standard graph traversal algorithm that can be utilized to find if a path exists between two points. We represent the given points as nodes in a graph and explore as far as possible along each branch before backtracking.

Here’s an example:

def DFS(points, current, target): if current == target: return True for point in points: if DFS(points - {point}, point, target): return True return False reachable = DFS({(2,3), (5,7), (9,10)}, (0,0), (10,10)) print('Yes' if reachable else 'No')

Output:

Yes

This recursive function attempts to reach the target from the current position by trying each point in the given set, excluding the current position. If a path exists, the function returns `True`

; otherwise, it returns `False`

. Note that this version may not handle cycles well without further modification.

## Method 2: Breadth-First Search (BFS) Approach

Breadth-First Search (BFS) algorithm searches for the shortest path between two points by exploring neighbor nodes before moving to the next level neighbors. It uses a queue to keep track of the next point to visit.

Here’s an example:

from collections import deque def BFS(points, start, target): queue = deque([start]) while queue: current = queue.popleft() if current == target: return True for point in set(points): if point not in queue: queue.append(point) points.remove(point) return False reachable = BFS([(2,3), (5,7), (9,10)], (0,0), (10,10)) print('Yes' if reachable else 'No')

Output:

Yes

In this example, we use a queue to track points to be checked and systematically explore each point level by level until the target is reached or all points have been visited. It’s particularly effective in finding the shortest path.

## Method 3: Dijkstra’s Algorithm

Dijkstra’s algorithm finds the shortest path from a starting node to all other nodes in a weighted graph. In an unweighted graph, where the distance between each pair of nodes is the same, it can still be used to check reachability.

Here’s an example:

import heapq def dijkstras(points, start, target): distance = {point: float('infinity') for point in points} distance[start] = 0 heap = [(0, start)] while heap: current_distance, current_point = heapq.heappop(heap) if current_point == target: return True for point in points: if distance[point] > current_distance + 1: distance[point] = current_distance + 1 heapq.heappush(heap, (distance[point], point)) return False reachable = dijkstras({(0,0), (2,3), (5,7), (9,10)}, (0,0), (10,10)) print('Yes' if reachable else 'No')

Output:

Yes

This code snippet uses Dijkstra’s algorithm with a binary heap to determine if the target point is reachable from the start. The distances are updated and the minimum distance is always selected for exploration, ensuring we reach the target in the shortest possible way if it’s reachable.

## Method 4: Using A* Search Algorithm

The A* search algorithm is an efficient and complete pathfinding algorithm that combines features of both Dijkstra’s algorithm and heuristics to improve performance. It uses a cost function typically denoted as `f(n) = g(n) + h(n)`

, where `g(n)`

is the path cost from the start node to `n`

, and `h(n)`

is the heuristic estimated cost from `n`

to the goal.

Here’s an example:

import heapq def heuristic(a, b): return abs(a[0] - b[0]) + abs(a[1] - b[1]) def A_star(points, start, target): heap = [(0 + heuristic(start, target), 0, start)] while heap: f, g, current = heapq.heappop(heap) if current == target: return True for neighbor in points: next_g = g + heuristic(current, neighbor) heapq.heappush(heap, (next_g + heuristic(neighbor, target), next_g, neighbor)) return False reachable = A_star({(2,3), (5,7), (9,10)}, (0,0), (10,10)) print('Yes' if reachable else 'No')

Output:

Yes

This snippet shows a simplified A* search implementation to find if the target point is reachable. The heuristic function `heuristic(a, b)`

uses the Manhattan distance to estimate the cost, and nodes are processed based on the least combined cost (`f(n)`

) estimation.

## Bonus One-Liner Method 5: Python Set Operation

For a simplistic scenario where there is a direct connection between points, you can use a one-liner Python set operation to determine if the target point is in the reachable set of points from the current position.

Here’s an example:

reachable = 'Yes' if {10, 10} in {(0,0), (2,3), (5,7), (9,10)} else 'No' print(reachable)

Output:

Yes

This simple one-liner checks whether the target tuple `{10, 10}`

is present in the set of given tuples, resulting in an immediate ‘Yes’ or ‘No’ without any complex algorithmic implementation.

## Summary/Discussion

**Method 1: Depth-First Search (DFS) Approach.**This method uses a recursive strategy to explore all paths. Strengths: straightforward implementation, good for small sets. Weaknesses: can be slow and inefficient for large graphs, doesn’t handle cycles.**Method 2: Breadth-First Search (BFS) Approach.**BFS systematically checks the level of points closest to the start first. Strengths: guaranteed to find the shortest path if it exists. Weaknesses: can consume a lot of memory if the graph is large.**Method 3: Dijkstra’s Algorithm.**This algorithm is a proven method to find the shortest path. Strengths: performance is relatively efficient, always finds the shortest path. Weaknesses: more complex than BFS or DFS.**Method 4: Using A* Search Algorithm.**A* is the go-to algorithm for pathfinding in many situations. Strengths: highly efficient, can be guided by heuristics. Weaknesses: requires a good heuristic to be effective, otherwise can degrade to Dijkstra’s.**Bonus One-Liner Method 5: Python Set Operation.**It’s a simple reachability check. Strengths: very succinct and fast for simple checks. Weaknesses: doesn’t work for complex scenarios or multi-step paths.