💡 Problem Formulation: We are tasked with determining the count of blocks that are traversed exactly ‘k’ times by a walker in a grid-like path. Imagine a grid of streets and a walker takes steps north, east, south, or west. Given the sequence of steps and an integer ‘k’, our goal is to calculate the number of blocks walked over ‘k’ times. For instance, if a walker takes a path represented by “NESW” and k equals 1, the output should be 4, since each block is walked over once.
Method 1: Brute Force Simulation
This method directly simulates the walking process using a sequence of steps. We create a dictionary to track the number of times each block is visited. The function should accept a string representing the walking path and an integer ‘k’, returning the count of blocks visited ‘k’ times.
Here’s an example:
def count_blocks(path, k):
step_dict = {'N': (0, 1), 'E': (1, 0), 'S': (0, -1), 'W': (-1, 0)}
position = (0, 0)
visited = {}
# Simulate the walk
for step in path:
move = step_dict[step]
for i in range(k):
position = (position[0] + move[0], position[1] + move[1])
visited[position] = visited.get(position, 0) + 1
# Count blocks visited exactly k times
return sum(1 for visit in visited.values() if visit == k)
# Example usage
print(count_blocks("NESW", 1))
Output:
4
In the given example, we define a function count_blocks() which accepts a walking path as a string and ‘k’ as the count of times a block must be walked over to be considered. We simulate the walking using a dictionary of moves and update the visited counts in another dictionary. Finally, we sum up all blocks visited exactly ‘k’ times and return the number.
Method 2: Complex Number Tracking
Python’s complex numbers can be used for tracking positions on a 2D grid. This method uses the real and imaginary parts to represent the coordinates. We still use a dictionary to count the visits but with less code complexity.
Here’s an example:
def count_blocks_complex(path, k):
directions = {'N': 1j, 'E': 1, 'S': -1j, 'W': -1}
position = 0+0j
visited = {}
# Simulate the walk
for step in path:
position += directions[step] * k
visited[position] = visited.get(position, 0) + 1
# Count the result
return len([pos for pos, count in visited.items() if count == k])
# Example usage
print(count_blocks_complex("NESW", 1))
Output:
4
This snippet introduces a function count_blocks_complex() which takes the same inputs as the prior example but uses complex numbers to track positions. It’s more concise as each move along the grid simply adds a complex number to our current position. The rest of the logic is similar to Method 1, focusing on counting the blocks visited ‘k’ times.
Method 3: Matrix Implementation
Creating a 2D matrix to represent the path walked, this method initializes a sufficiently large grid to ensure all steps lie within bounds. It proceeds to mark visits within this matrix, then counts the blocks visited ‘k’ times post-simulation.
Here’s an example:
import numpy as np
def count_blocks_matrix(path, k):
size = 100 # Assuming the steps will not go beyond this size
grid = np.zeros((size * 2, size * 2))
position = (size, size)
# Movement dictionary
move_dict = {'N': (-1, 0), 'E': (0, 1), 'S': (1, 0), 'W': (0, -1)}
# Walk simulation
for step in path:
dx, dy = move_dict[step]
for i in range(k):
position = (position[0] + dx, position[1] + dy)
grid[position] += 1
# Count blocks visited exactly k times
return np.sum(grid == k)
# Example usage
print(count_blocks_matrix("NESW", 1))
Output:
4
The count_blocks_matrix() function relies on NumPy to create a 2D grid matrix. Each block’s position gets updated based on the simulated path. After the steps are done, NumPy’s comparison and summation capabilities are used to conveniently count the blocks that have the desired visit count, ‘k’.
Method 4: Optimized Hash Map
Employing a hashing strategy for the step positions can allow for an optimized look-up time for blocks visited. This approach uses coordinate hashing in a single-dimensional dictionary for more efficient space complexity compared to the brute force approach.
Here’s an example:
def count_blocks_hashed(path, k):
move_dict = {'N': (0, 1), 'E': (1, 0), 'S': (0, -1), 'W': (-1, 0)}
position = (0, 0)
visited = {}
# Simulate walking
for step in path:
dx, dy = move_dict[step]
for i in range(k):
position = tuple(map(sum, zip(position, (dx, dy))))
visited[position] = visited.get(position, 0) + 1
# Count blocks visited exactly k times
return sum(val == k for val in visited.values())
# Example usage
print(count_blocks_hashed("NESW", 1))
Output:
4
Within the count_blocks_hashed() implementation, a dictionary is used with tuples representing coordinates. The hashing of steps through tuple mapping ensures faster access and update of visit counts. To obtain the final count, we iterate through the value of the dictionary checking for ‘k’ visits.
Bonus One-Liner Method 5: Python Functional Approach
Leveraging Python’s functional programming features, such as map() and lambda functions, this one-liner method efficiently computes the count of k-visited blocks in a compact form.
Here’s an example:
from collections import Counter
# One-Liner using functional programming
count_blocks_oneliner = lambda path, k: sum(1 for v in Counter(map(lambda x: (x.count('E') - x.count('W'), x.count('N') - x.count('S')), [path[i:i + k] for i in range(len(path) - k + 1)])).values() if v == k)
# Example usage
print(count_blocks_oneliner("NESW", 1))
Output:
4
This concise one-liner function count_blocks_oneliner() utilizes Counter from the collections module to count up the occurrences of each block visited. It uses a lambda function to map a substring of the path of length ‘k’ to its resultant position and then filters the count by ‘k’ visits.
Summary/Discussion
- Method 1: Brute Force Simulation. This method is simple and straight-forward, making it easy to understand and implement. However, its simulation approach can be inefficient for long paths or large ‘k’ values.
- Method 2: Complex Number Tracking. Offers cleaner code and can be more intuitive for those familiar with complex numbers. Its downside is potentially less readability for developers unfamiliar with using complex numbers for grid representation.
- Method 3: Matrix Implementation. Utilizes NumPy’s efficient array operations for quick computation but can be memory-intensive due to matrix size, which must be pre-defined and might lead to issues if the path exceeds the matrix bounds.
- Method 4: Optimized Hash Map. Provides better efficiency in space and lookup complexity than the brute force approach. It requires careful implementation to avoid hashing issues.
- Method 5: Python Functional Approach. This method is effcient and concise, best for one-off computations where code brevity is preferred over readability. It requires a solid grasp of Python’s functional programming features.