261. Graph Valid Tree

Problem Description

In this problem, we're given a graph that is composed of 'n' nodes which are labeled from 0 to n - 1. The graph also has a set of edges given in a list where each edge is represented by a pair of nodes like [a, b] that signifies a bidirectional (undirected) connection between node a and node b. Our main objective is to determine if the given graph constitutes a valid tree.

To understand what makes a valid tree, we should recall two essential properties of trees:

1. A tree must be connected, which means there should be some path between every pair of nodes.
2. A tree cannot have any cycles, meaning no path should loop back on itself within the graph.

Therefore, our task is to check for these properties in the given graph. We need to verify if there's exactly one path between any two nodes (confirming a lack of cycles) and that all the nodes are reachable from one another (confirming connectivity). If both conditions are met, we return true; otherwise, we return false.

Intuition

To determine if the set of edges forms a valid tree, the "Union Find" algorithm is an excellent choice. This algorithm is a classic approach used in graph theory to detect cycles and check for connectivity within a graph.

Here's why Union Find works for checking tree validity:

1. Union Operation: This is used to connect two nodes. If they are already connected, it means we're trying to add an extra connection to a connected pair, indicating the presence of a cycle.
2. Find Operation: This operation helps in finding the root of each node, generally used to check if two nodes have the same root. If they do, a cycle is present since they are already connected through a common ancestor. If not, we can perform an union operation without creating a cycle.
3. Path Compression: This optimization helps in flattening the structure of the tree, which improves the time complexity of subsequent "Find" operations.

In our solution, we start with each node being its own parent (representing a unique set). Then, we iterate through each edge, applying the "Find" operation to see if any two connected nodes share the same root:

• If they do, we've detected a cycle and return false as a cycle indicates it's not a valid tree.
• If they don't, we connect (union) them by updating the root of one node to be the root of the other.

As we connect nodes, we also decrement 'n' as an additional check. If at the end of processing all edges, there's more than one disconnected component, 'n' will be greater than 1, indicating the graph is not fully connected, thus not a valid tree. If 'n' is exactly 1, it means the graph is fully connected without cycles, so it's a valid tree and we return true.

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

Solution Approach

The solution provided leverages the Union Find algorithm to check the validity of a tree. Here's a step-by-step walkthrough of the algorithm as implemented in the Python code:

1. Initialization:

   • We start by creating an array p to represent the parent of each node. Initially, each node is its own parent, thus p[i] = i for i from 0 to n-1.
   • The variable n tracks the number of distinct sets or trees; initially, each node is separate, so there are n trees.

2. Function Definition - find(x):

   • This function is critical to Union Find. Its purpose is to find the root parent of a node x.
   • The function is implemented with path compression, meaning every time we find the root of a node, we update the parent along the search path directly to the root. This helps reduce the tree height and optimize future searches.

3. Process Edges:

   • We iterate over each edge in the list edges.
   • For each edge [a, b], we find the roots of both nodes a and b using the find function.
   • We check if the roots are equal. If they are, find(a) == find(b), this indicates that nodes a and b are already connected, thus forming a cycle. In this case, we immediately return False, as a valid tree cannot contain cycles.
   • If the roots are different, it means that connecting them doesn't form a cycle, so we perform a union operation by setting the parent of find(a) to find(b).
   • After successfully adding an edge without forming a cycle, we decrement n since we have merged two separate components into one.

4. Final Verification:

   • After processing all edges, we check if n is exactly 1. If it is, it means all nodes are connected, forming a single connected component without cycles, which satisfies the definition of a tree.
   • If n is not 1, it means the graph is either not fully connected, or we have returned False earlier due to a cycle. In either case, the graph does not form a valid tree.

The simplicity of this algorithm comes from the elegant use of the Union Find pattern to quickly and efficiently find cycles and check connectivity. The solution runs in near-linear time, making it very efficient for large graphs.

Example Walkthrough

Let's consider a simple example to illustrate the solution approach using the Union Find algorithm.

Suppose we have a graph with n = 4 nodes labeled from 0 to 3, and we're given an edge list edges = [[0, 1], [1, 2], [2, 3]].

Step 1: Initialization

• We begin by initializing the parent array p with p = [0, 1, 2, 3].

Step 2: Function Definition - find(x)

• We define the find function to find the root parent of a given node x. Additionally, this function uses path compression.

Step 3: Process Edges

• We iterate through each edge [a, b] in the given edges list:
  • The first edge is [0, 1]. We find the roots of 0 and 1, which are 0 and 1, respectively. Since they have different roots, we perform the union operation by setting p[1] to the root of 0 (which is 0). Now p = [0, 0, 2, 3], and we decrement n to 3.
  • The next edge is [1, 2]. The roots of 1 and 2 are 0 and 2. They have different roots, so we union them by setting p[2] to the root of 1 (which is 0). Now p = [0, 0, 0, 3], and we decrement n to 2.
  • The final edge is [2, 3]. The roots of 2 and 3 are 0 and 3. Again, different roots allow us to union them, updating p[3] to 0. Our parent array is now p = [0, 0, 0, 0], and n is decremented to 1.

Step 4: Final Verification

• After iterating over all the edges, we now check if n equals 1. Since that's the case in our example, it means that all nodes are connected in a single component, and since we didn't encounter any cycles during the union operations, the graph forms a valid tree.

So, for this input graph represented by n = 4 and edges = [[0, 1], [1, 2], [2, 3]], our algorithm would return True indicating that the graph is a valid tree.

Solution Implementation

Time and Space Complexity

The given Python code implements a union-find algorithm to determine if an undirected graph forms a valid tree. It uses path compression to optimize the find operation.

Time Complexity:

The time complexity of the validTree function mainly depends on the two operations: find and union.

• The find function has a time complexity of O(α(n)) per call, where α(n) represents the Inverse Ackermann function which grows very slowly. In practical scenarios, α(n) is less than 5 for all reasonable values of n.

• Each edge leads to a single call to the find function during processing, and possibly a union operation if the vertices belong to different components. Thus, with m edges, there would be 2m calls to find, taking O(α(n)) each.

• Also, the loop through the edges happens exactly m times, where m is the number of edges.

Bringing it all together, the overall time complexity is O(mα(n)).

However, since the graph must have exactly n - 1 edges to form a tree (where n is the number of nodes), m can be replaced by n - 1. Therefore, the time complexity simplifies to O((n - 1)α(n)) which is also written as O(nα(n)), since the -1 is negligible compared to n for large n.

Space Complexity:

The space complexity is determined by the storage required for the parent array p, which has length n.

• The space taken by the parent array is O(n).

• Aside from the parent array p and the input edges, only a fixed amount of space is used for variables like a, b, and x.

Thus, the space complexity of the algorithm is O(n).

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