#1042

Medium

auto_awesomeGraph traversal with depth-first search

LeetCode Problem Workspace

Flower Planting With No Adjacent

In this problem, you are tasked with planting flowers in gardens with specific constraints based on graph traversal principles.

depth first search breadth first search graph

Problem Statement

You are given n gardens, labeled from 1 to n, and a list of bidirectional paths between the gardens. Each path is represented by an array [xi, yi] meaning a direct connection between garden xi and garden yi. The task is to assign one of four types of flowers to each garden in such a way that no two adjacent gardens (i.e., gardens connected by a path) have the same flower type.

The challenge involves ensuring that for every pair of connected gardens, the flowers assigned to them differ. The key observation here is that every garden has at most 3 adjacent gardens, which guarantees that there is always a valid flower assignment for each garden using up to 4 flower types. You are to implement an efficient solution using graph traversal techniques.

Examples

Example 1

Input: n = 3, paths = [[1,2],[2,3],[3,1]]

Output: [1,2,3]

Gardens 1 and 2 have different types. Gardens 2 and 3 have different types. Gardens 3 and 1 have different types. Hence, [1,2,3] is a valid answer. Other valid answers include [1,2,4], [1,4,2], and [3,2,1].

Example 2

Input: n = 4, paths = [[1,2],[3,4]]

Output: [1,2,1,2]

Example details omitted.

Example 3

Input: n = 4, paths = [[1,2],[2,3],[3,4],[4,1],[1,3],[2,4]]

Output: [1,2,3,4]

Example details omitted.

Constraints

1 <= n <= 104
0 <= paths.length <= 2 * 104
paths[i].length == 2
1 <= xi, yi <= n
xi != yi
Every garden has at most 3 paths coming into or leaving it.

Solution Approach

Graph Representation

Represent the gardens as nodes in a graph, and the paths as edges connecting these nodes. This way, the problem becomes a classic graph coloring issue, where each node (garden) must be assigned a color (flower type) that differs from its adjacent nodes (gardens connected by a path).

Depth-First Search (DFS)

Use a DFS traversal to explore each garden and its adjacent gardens. At each garden, assign a flower type that hasn’t been used by its neighboring gardens. Since each garden has at most 3 neighbors, the DFS approach guarantees a valid solution by ensuring that at least one flower type remains available for each garden.

Graph Coloring with Constraints

Graph coloring with a limited number of colors (flowers) is a natural fit for this problem. As each garden can only have up to 3 neighbors, a simple greedy approach combined with DFS is efficient enough to ensure the constraints are satisfied. Each garden can be assigned a flower type that is distinct from its adjacent gardens.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity of this problem depends on the approach used for graph traversal. In the worst case, the solution involves a DFS traversal, which has a time complexity of O(n + m), where n is the number of gardens and m is the number of paths. Space complexity is also O(n + m) due to the storage required for the graph and visited nodes.

What Interviewers Usually Probe

Look for efficient use of graph traversal techniques, especially DFS.
Check for a proper implementation of the greedy coloring algorithm.
Ensure the solution correctly handles edge cases, such as minimal paths or no paths.

Common Pitfalls or Variants

Common pitfalls

Incorrectly assuming there might be no valid flower assignments when there are always enough flower types available.
Not implementing the graph traversal properly, leading to incorrect flower assignments or missed edges.
Overcomplicating the problem with unnecessary data structures when a simple DFS-based greedy approach suffices.

Follow-up variants

Consider modifying the number of flower types to test how the algorithm scales with different constraints.
Explore whether a breadth-first search (BFS) approach could provide advantages in some cases over DFS.
Test the algorithm’s performance on large graphs with up to 10^4 gardens and 2 * 10^4 paths.

FAQ

What is the main algorithmic pattern used in Flower Planting With No Adjacent?

The problem utilizes graph traversal with depth-first search (DFS) to color the graph with a fixed number of colors, ensuring adjacent nodes (gardens) have distinct colors (flower types).

How does DFS help solve the Flower Planting With No Adjacent problem?

DFS allows us to visit each garden and its neighbors, assigning a flower type that hasn’t been used by adjacent gardens. The graph traversal ensures that the constraints are satisfied in an efficient manner.

What are the constraints in the Flower Planting problem?

The problem has a constraint where each garden has at most 3 paths connecting to other gardens, ensuring that there are always enough flower types available to assign to each garden.

Can a breadth-first search (BFS) be used for Flower Planting With No Adjacent?

Yes, BFS can be used as an alternative to DFS for graph traversal. However, DFS is typically simpler to implement for this type of graph coloring problem.

What is the space complexity of the Flower Planting With No Adjacent problem?

The space complexity is O(n + m), where n is the number of gardens and m is the number of paths, as we store the graph structure and visited nodes during traversal.

terminal

Solution

Solution 1: Enumeration

We first construct a graph $g$ based on the array $\textit{paths}$, where $g[x]$ represents the list of gardens adjacent to garden $x$.

class Solution:
    def gardenNoAdj(self, n: int, paths: List[List[int]]) -> List[int]:
        g = defaultdict(list)
        for x, y in paths:
            x, y = x - 1, y - 1
            g[x].append(y)
            g[y].append(x)
        ans = [0] * n
        for x in range(n):
            used = {ans[y] for y in g[x]}
            for c in range(1, 5):
                if c not in used:
                    ans[x] = c
                    break
        return ans

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