Problem Description

This problem involves n gardens, each needing to be planted with one of 4 types of flowers. The gardens are connected by bidirectional paths, and each garden can have at most 3 paths connected to it. The crux of the problem is to choose the type of flower for each garden in such a way that no two gardens connected by a path have the same type of flower. The goal is to find any one configuration that satisfies this condition. The final output should be an array where the element at the i-th position refers to the type of flower in the (i+1)th garden. You're assured that there is at least one solution to this problem.

Flowchart Walkthrough

To analyze LeetCode 1042. Flower Planting With No Adjacent using the provided algorithm flowchart, let's go through the steps:

1. Is it a graph?

   • Yes: The problem involves placing different flowers in adjacent garden nodes without any two adjacent gardens having the same type of flower. This structure can naturally be represented as a graph where gardens are nodes and edges indicate adjacency.

2. Is it a tree?

   • No: The graph is a generic graph without any specifics that it is a tree. It potentially has cycles, multiple connections, and isn't necessarily hierarchical or acyclic.

3. Is the problem related to directed acyclic graphs (DAGs)?

   • No: The problem setup involves an arbitrary adjacency relationship between gardens, not necessarily directed or acyclic.

4. Is the problem related to shortest paths?

   • No: The objective is to color the graph such that no two adjacent nodes share the same color, there is no pathfinding component to the problem.

5. Does the problem involve connectivity?

   • Yes: Even though the problem's primary goal isn't about discovering connected components, the solution must consider connectivity in the sense that connected gardens cannot have the same flower type. The graph is unweighted as there is no weight associated with connections between gardens; they are merely connected or not.

6. Is the graph weighted?

   • No: As previously assessed, the graph is unweighted.

7. Does the problem have small constraints?

   • No: It does not explicitly state small constraints about the number of nodes or that an exhaustive search is feasible. Since this doesn't fit into a small constraints scenario, and the typical solution involves visiting all nodes ensuring no two adjacent ones have the same color, using DFS for a graph coloring problem (chromatic number) is a suitable method.

Considering the answers and relevant paths in the flowchart, Depth-First Search emerges as an appropriate algorithm for solving this problem. DFS is efficient for exploring all nodes and their connections, ensuring that all garden constraints (no two adjacent nodes share the same flower) are met, favoring backtracking if a contradiction is found. This is a typical application of DFS in graph coloring problems.

Intuition

Imagine each garden as a node in a graph and each path as an edge connecting two nodes. Since a garden can have at most 3 paths, a node can have a maximum of 3 edges. To solve this, we can iterate through each garden and choose for it the lowest numbered flower type which hasn't been used by its connected neighbors.

The intuition behind this approach is that since each garden can have at most 3 paths, there is always at least one type of flower (out of the 4 available types) that is not used by any neighbor garden. By iterating through each garden, we keep track of the types used by its neighbors and then assign the first available type.

The solution uses a graph represented as an adjacency list, where the key is the garden and the value is a list of connected gardens. As it iterates through each garden, it dynamically updates the assigned flower types and maintains a set of used flower types for the current garden's neighbors. By ensuring that no adjacent gardens share a flower type, we respect the constraint and construct a valid configuration for planting flowers.

Learn more about Depth-First Search, Breadth-First Search and Graph patterns.

Solution Approach

The implementation employs a graph-based approach and makes use of Python's built-in data structures:

1. Default Dictionary for Adjacency List: The graph is represented using a defaultdict from Python's collections module, which provides a convenient way to store the adjacency list. Every garden is keyed to a list of neighbor gardens that are connected by paths.

2. Iterating Over Gardens: We iterate sequentially over each garden. Here, the garden's index is treated as its label (decremented by 1 as indices in Python are 0-based, but gardens are labeled from 1 to n).

3. Tracking Used Flower Types: A set is created for each garden to keep track of the flower types used by its neighboring gardens.

4. Assigning Flower Types: For each garden, we loop over the possible flower types (1 through 4). We pick the first flower type that is not present in the set of used types and assign this flower type to the current garden.

5. Updating the Answer: After a valid flower type is chosen for a garden, it is saved to the ans list, which initially is filled with zeroes.

6. Returning the Solution: After all gardens have been assigned a flower type, we return the ans list, which now contains the type of flower for each garden satisfying the problem's constraints.

In more detail, for each garden x, we perform the following steps:

• Get the set used of flower types that have already been used by the neighbors of garden x.
• Iterate through the flower types c from 1 to 4, checking if c is not in the used set.
• When we find such a c, we assign it to ans[x] and break the inner loop since we've found a valid flower for this garden.

This greedy algorithm ensures that since no more than 3 gardens can be adjacent to a garden, there will always be at least one type of flower from the 4 available that we can plant in every garden, thus always providing a valid solution.

Example Walkthrough

Consider a scenario with five gardens and the following paths between them: [1,2], [2,3], [3,4], [4,5], [1,5]. This means garden 1 is connected to gardens 2 and 5, garden 2 is connected to gardens 1 and 3, and so on.

Let's apply the solution approach to this example:

• Start with an empty adjacency list and an ans list filled with zeroes, denoting that no flowers have been planted yet.

• Interpret the paths to fill the adjacency list, which in this case results in:

  1 -> [2, 5]
  2 -> [1, 3]
  3 -> [2, 4]
  4 -> [3, 5]
  5 -> [4, 1]

• Begin iterating from garden 1 to 5.

• For garden 1, check the flower types of its neighbors (gardens 2 and 5 which have not been planted yet) and note that all flower types (1, 2, 3, 4) are available. Choose the lowest number which is 1, and assign it to the garden, so ans[1] = 1.

• Move to garden 2, which is connected to garden 1 (planted with flower type 1) and garden 3 (not yet planted). The available flower types are 2, 3, and 4. Pick the lowest, which is 2, so ans[2] = 2.

• For garden 3, connected to garden 2 (flower type 2) and garden 4 (not yet planted). Available types are 1, 3, and 4. Choose the lowest, so ans[3] = 1.

• Move to garden 4, which has neighbors 3 and 5. Garden 3 has flower type 1, and garden 5 is unplanted. The choices are 2, 3, or 4. Pick the lowest, so ans[4] = 2.

• Finally, garden 5 is connected to gardens 1 and 4, which have flower types 1 and 2, respectively. The remaining flowers are 3 and 4. So we pick 3, and ans[5] = 3.

• The flower assignments for each of the five gardens represented by the ans list are now [1, 2, 1, 2, 3].

This walkthrough demonstrates the straightforward greedy strategy described in the solution approach, reflecting how we can always ensure that each garden has a unique type of flower compared to its immediate neighbors.

Solution Implementation

Time and Space Complexity

The code defines a Solution class with the gardenNoAdj method, which assigns a unique type of flower to each garden given the constraints that no adjacent gardens can have the same type of flower. There are n gardens numbered from 1 to n and at most 4 types of flowers.

Time Complexity:

The time complexity is determined by the following factors:

1. Building the graph: The loop iterates through the paths array, which might contain up to n(n-1)/2 paths in the worst case (a complete graph). This process has a time complexity of O(E) where E is the number of paths (edges).

2. Assigning flowers to gardens: There's an outer loop iterating n times for each garden and, inside it, a nested loop that iterates through the adjacent gardens (up to n-1 times in the worst-case scenario for a complete graph). However, since each garden is limited to 3 edges (paths) to prevent excessive adjacency according to the problem statement, the inner loop has a constant factor, and thus, this part has a time complexity of O(n).

3. Choosing a flower that hasn't been used: We iterate through the 4 types of flowers. This is a constant operation since the number of flower types does not change with n.

Thus, the total time complexity is O(E + n), where E is the number of edges or paths.

Space Complexity:

The space complexity is determined by the following factors:

1. The graph g: In the worst-case scenario (a complete graph), each node connects to n-1 other nodes. Therefore, the space required by this graph is O(E) where E is the number of edges.

2. ans array: This is an array of size n, so it consumes O(n) space.

3. used set: At most, the set contains 4 elements because there are only 4 different types of flowers. This is a constant space O(1).

Considering all factors, the total space complexity of the algorithm is O(E + n). The space taken by the used set is negligible in comparison to the graph and ans array.

Combining both the time and space complexities, we sum them up as O(E + n) for time complexity and O(E + n) for space complexity.

Learn more about how to find time and space complexity quickly using problem constraints.