Cut Off Trees for Golf Event

The problem requires a traversal of the forest matrix, cutting trees in order of their height and finding the minimum steps needed to cut all the trees. Here’s an approach to solve this problem:

1. Identify the Trees: Identify the positions and heights of all trees and sort them by height. This can be done by going through the entire grid and noting the coordinates of all the cells where the value is greater than 1.

2. Find Shortest Paths between Consecutive Trees: You can then use BFS (Breadth-First Search) to find the shortest path between consecutive trees in the sorted order. You’ll have to ensure that you only walk through cells that are either empty or have a tree (values greater than 0).

3. Calculate Total Steps: Sum the shortest paths between all consecutive trees to find the total steps required to cut all the trees.

4. Handle Inaccessibility: If at any point a tree is inaccessible from another tree, you return -1.

Here is Python code implementing the above approach:

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from collections import deque

class Solution:
    def cutOffTree(self, forest: List[List[int]]) -> int:
        if not forest or not forest[0]:
            return -1

        m, n = len(forest), len(forest[0])
        trees = sorted((v, r, c) for r, row in enumerate(forest) for c, v in enumerate(row) if v > 1)

        def bfs(sr, sc, tr, tc):
            seen = [[False] * n for _ in range(m)]
            seen[sr][sc] = True
            queue = deque([(sr, sc, 0)])
            while queue:
                r, c, steps = queue.popleft()
                if r == tr and c == tc:
                    return steps
                for dr, dc in ((-1, 0), (1, 0), (0, -1), (0, 1)):
                    nr, nc = r + dr, c + dc
                    if 0 <= nr < m and 0 <= nc < n and forest[nr][nc] > 0 and not seen[nr][nc]:
                        seen[nr][nc] = True
                        queue.append((nr, nc, steps + 1))
            return -1

        sr = sc = 0
        total_steps = 0
        for _, tr, tc in trees:
            steps = bfs(sr, sc, tr, tc)
            if steps == -1:
                return -1
            total_steps += steps
            sr, sc = tr, tc

        return total_steps

This code first extracts and sorts the trees and then applies a BFS algorithm to calculate the minimum steps needed to traverse from one tree to the next. If at any point it’s not possible to reach the next tree, the function returns -1.

“Cut Off Trees for Golf Event” involves the breadth-first search (BFS) technique and grid traversal. Here are 10 problems to prepare:

1. 200. Number of Islands: This problem helps you understand grid traversal using depth-first search (DFS), which can be adapted to BFS.

2. 994. Rotting Oranges: This is a classic BFS problem where you have to infect all fresh oranges from the rotten ones.

3. 279. Perfect Squares: This problem involves a BFS approach to find the least number of perfect square numbers that sum to a target.

4. 127. Word Ladder: This problem requires you to find the shortest transformation sequence from one word to another.

5. 286. Walls and Gates: This problem involves a multi-source BFS from multiple gates to empty rooms in a 2D grid.

6. 542. 01 Matrix: This is a BFS problem that requires you to calculate the distance of each cell to the nearest zero.

7. 207. Course Schedule: Although this problem uses a depth-first search (DFS), understanding how it’s solved can help with the understanding of BFS as well.

8. 210. Course Schedule II: A follow-up to the previous problem, this question requires you to find a valid order in which to take all the courses.

9. 752. Open the Lock: This problem is a BFS problem with a lock (or a cube) as a graph.

10. 785. Is Graph Bipartite?: This problem involves coloring a graph using BFS.

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from typing import List 
import heapq
import collections
import numpy as np

class Solution:
    def cutOffTree(self, forest: List[List[int]]) -> int:
        m = len(forest)
        n = len(forest[0])

        pq = []
        for x in range(m):
            for y in range(n):
                height = forest[x][y]
                if height > 1:
                    heapq.heappush(pq,(height,x,y))

        def bfs(x,y,nextX,nextY) -> int:
            queue = collections.deque([(x,y,0)])
            seen = {(x,y)}

            while queue:
                x,y,steps = queue.popleft()

                if x == nextX and y == nextY:
                    forest[x][y] = 1
                    return steps

                for dx,dy in [(-1,0),(0,1),(0,-1),(1,0)]:
                    adjX,adjY = x+dx, y+dy
                    if (0 <= adjX < m and 0 <= adjY < n) and (forest[adjX][adjY] >= 1) and ((adjX,adjY) not in seen):
                        queue.append((adjX,adjY,steps+1))
                        seen.add((adjX,adjY))

            return -1

        x,y = 0,0
        steps = 0

        while pq:
            _,nextX,nextY = heapq.heappop(pq)

            shortestPath = bfs(x,y,nextX,nextY)
            if shortestPath == -1:
                return -1
            steps += shortestPath

            x,y = nextX,nextY

        return steps

Problem Classification

The problem falls into the category of “Graph Theory”. It can be considered a form of pathfinding or traversal problem with additional constraints.

‘What’ Components:

1. An m x n matrix representing a forest.
2. The matrix contains three types of cells:
   • ‘0’ represents an unwalkable cell.
   • ‘1’ represents an empty walkable cell.
   • A number greater than ‘1’ represents a walkable cell with a tree. The number indicates the tree’s height.
3. In one step, you can walk in any of the four directions (north, east, south, west).
4. The task is to cut off all trees in ascending order of their height.
5. The operation of cutting turns the cell with the tree into an empty cell (‘1’).
6. You are asked to start from the point (0, 0).
7. The goal is to find the minimum steps needed to cut off all the trees.
8. If it’s impossible to cut off all the trees, return -1.

This problem can be further classified as a pathfinding problem with specific constraints, a variant of the shortest path problem. The main tasks involved are navigating a grid, updating the grid status, and sorting and selecting paths based on additional conditions (tree height). Furthermore, since we deal with changing states of cells (tree or no tree), it can also fall under dynamic programming.

The solution will likely involve some form of breadth-first search or depth-first search algorithm to traverse the matrix, a sorting mechanism to ensure the trees are cut in ascending order of height, and a tracking mechanism to keep track of the number of steps taken and update the matrix as trees are cut.

The problem statement implies several key aspects:

1. Graph Theory / Network Flow: The problem involves a grid, which is a special form of graph. We are traversing through the grid (or graph) to find a specific goal. This typically falls under graph theory or network flow in computer science.

2. Pathfinding / Shortest Path: The problem involves finding the shortest path from one point to another. This is a classic problem in the field of pathfinding and graph theory. Algorithms such as Dijkstra’s or A* are often used for these problems.

3. Priority Queue (Heap): The trees to be cut are organized by their height in a priority queue, which suggests that elements are served by priority.

4. Breadth-first Search (BFS): The search is performed in a breadth-first manner, which is a strategy for graph traversal.

5. Optimization Problem: The objective is to find the minimal total steps to cut all the trees, which makes this an optimization problem.

6. Spatial Reasoning: The problem involves moving through a 2D space, which requires understanding and reasoning about spatial relationships.

This problem involves elements of Graph Theory, Pathfinding, Priority Queue, BFS, Optimization Problems, and Spatial Reasoning.

Identifying Problem Isomorphism

“Cut Off Trees for Golf Event” is about finding the minimum total steps needed to cut down all the trees in a forest for a golf event, where the trees need to be cut in non-descending order of their height.

“Cut Off Trees for Golf Event” has an isomorphic problem: “Swim in Rising Water” (#778 on LeetCode). In this problem, you have a grid filled with water, and the water level in each cell rises over time. The goal is to find the minimum time required to swim from the top-left cell to the bottom-right cell.

Both are similar because they both require finding a path in a grid from one point to another, under certain constraints and conditions. Both problems also require optimizing a certain parameter: in the former, it’s the total distance (steps) travelled, while in the latter, it’s the total time spent swimming. The solving strategies for these problems could involve algorithms such as Dijkstra’s algorithm or other forms of pathfinding algorithms, like A*.

They both require understanding of graph theory and pathfinding algorithms. They both have constraints that make the problem more difficult than a straightforward shortest-path problem. “Cut Off Trees for Golf Event” is more complex, because it requires cutting the trees in a specific order.

Language Agnostic Coding Drills

1. Understanding Data Structures: Understand the data structures used in Python. In this problem, we see the use of Lists, Deque, Sets, and Priority Queue.

2. Working with Lists: Learn about accessing elements, iterating over a list, and list methods such as append(), pop(), len().

3. Working with Priority Queues (heapq): Understand the nature of a priority queue. Priority queue is a data structure that holds data and returns it in priority order (smallest item first, heapq.heappop()). Learn to insert data into the queue (heapq.heappush()).

4. Working with Deque and Set: Understand the deque data structure which is a queue with two ends and methods like popleft(). Learn to add elements to a set and check existence in a set.

5. Nested Loop: Learn how to use nested loops to iterate over a matrix.

6. Breadth-First Search (BFS) Algorithm: This is a key concept to solve this problem. BFS is an algorithm for traversing or searching tree or graph data structures. It starts at the tree root (or some arbitrary node of a graph) and explores the neighbor nodes at the present depth prior to moving on to nodes at the next depth level.

7. Understanding the problem and applying BFS: Apply BFS to this problem. The BFS here finds the shortest path between the current location and the next location.

8. Understanding Tuples: A tuple is a collection which is ordered and unchangeable. In Python tuples are written with round brackets.

The problem solving approach step by step would look something like:

1. Parse the 2D list forest and push all tree locations and heights to the priority queue pq where trees with smaller heights have higher priority.

2. Implement the bfs() method to find the shortest path from the current location to the next location with a smaller height. bfs() should also mark visited locations.

3. Keep using the bfs() method to find the shortest path to the next tree and add the path length to the total steps until there are no trees left or until the bfs() method returns -1, meaning there is no valid path to the next tree.

4. Return the total steps if there are no trees left in the priority queue pq, otherwise return -1.

Targeted Drills in Python

Certainly, let’s break this down into drills based on each learning unit:

1. Understanding and Working with Lists:

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   # Create a list list_1 = [1, 2, 3, 4, 5] # Access elements print(list_1[0]) # Output: 1 # Iterate over a list for i in list_1: print(i) # List methods list_1.append(6) print(list_1.pop()) # Output: 6 print(len(list_1)) # Output: 5

2. Working with Priority Queues (heapq):

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   import heapq # Initialize a priority queue pq = [] # Insert data into the queue heapq.heappush(pq, (1, 'one')) heapq.heappush(pq, (3, 'three')) heapq.heappush(pq, (2, 'two')) # Remove and return the smallest element from the queue print(heapq.heappop(pq)) # Output: (1, 'one')

3. Working with Deque and Set:

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   import collections # Initialize a deque dq = collections.deque([1, 2, 3]) # Use popleft method print(dq.popleft()) # Output: 1 # Initialize a set and add elements set_1 = set() set_1.add(1) set_1.add(2) # Check existence in a set print(1 in set_1) # Output: True

4. Nested Loop:

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   for i in range(3): for j in range(3): print((i, j))

5. Breadth-First Search (BFS) Algorithm Drill: For this, create a grid and a start and end point. Write a function to find the shortest path from start to end. You can use deque for the queue and a set to store visited nodes.

6. Understanding and Working with Tuples:

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   # Create a tuple tuple_1 = (1, 2, 3) # Access elements print(tuple_1[0]) # Output: 1 # Iterate over a tuple for i in tuple_1: print(i)

Each of these drills focuses on a different aspect of the problem. By mastering each of these drills, you can then integrate these skills to solve the overall problem.

The problem-specific drill here would be understanding and implementing BFS in the context of this problem, which you can practice by creating different 2D lists and attempting to find the shortest path between different points.