Problem Description

The problem describes an undirected graph with n nodes numbered from 0 to n-1. Each node has an associated value defined in the array vals, where vals[i] represents the value of the ith node. Additionally, we are given a list of undirected edges defined in the array edges. An edge connects two nodes, implying that the nodes are directly reachable from each other.

A star graph is a specific type of subgraph in which one central node is connected to some or all other nodes through edges, but those other nodes are not connected to each other. The star sum is the combined value of the nodes in this subgraph, including the central node and its neighbors.

The task is to find the maximum star sum possible by including at most k edges in the star graph. In other words, what is the highest sum of node values we can achieve when we select a subset of the graph that forms a star graph, with the central node having up to k neighbors.

Intuition

To find the maximum star sum, we should aim to include the nodes with the highest values adjacent to our chosen central node. Since the edges are undirected, we need to consider each node as a potential center of the star graph and calculate the maximum sum obtainable with it as the center.

The given solution follows these steps:

1. Create an adjacency list, g, using defaultdict(list) from Python's collections module, to store the nodes (a) and their corresponding connected nodes' values (vals[b]) if their value is greater than zero.

2. For all edges (a, b), do the following:

   • If vals[b] is positive, append it to g[a] (connected nodes' values for a).
   • Conversely, if vals[a] is positive, append it to g[b].

3. Sort the lists of connected nodes' values in descending order for each node in g. It ensures that the highest valued neighbor is considered first.

4. Calculate the star sum for each node as the sum of its value, v, and the sum of at most k highest valued neighbors from the respective list in g. This is accomplished by iterating through all nodes enumerated along with their values (enumerate(vals)) and computing the sum of a node's value and the total of its k highest connected nodes' values.

5. Finally, the maxStarSum function returns the maximum star sum obtained from all possible center nodes.

The key to the solution is ensuring that the highest valued neighbors are included in the star graph to maximize the star sum, constrained by the limit of at most k edges.

Learn more about Greedy, Graph, Sorting and Heap (Priority Queue) patterns.

Solution Approach

The solution implements a simple but clever approach to maximize the star sum by leveraging a few common data structures and algorithms, as follows:

Data Structures Used:

1. Adjacency List: A defaultdict of lists is used to create an adjacency list g. It stores the values of connected nodes for each node (excluding values that are zero, as they do not contribute to the sum).

2. List: Python lists are used to store connected nodes' values and to sort these values in descending order.

Algorithm Steps:

1. Creating Adjacency List (Undirected): The solution loops through each edge (a, b) in edges:

   • For node a, if vals[b] > 0, it appends this value to the list g[a].
   • For node b, similarly, if vals[a] > 0, it appends this value to the list g[b].

   This is because the graph is undirected, meaning either a or b can be the center of a star graph with the other being its neighbor.

2. Sorting Neighbors' Values: For each node's list of connected nodes in g, the values are sorted in reverse (descending) order. It ensures that when we choose up to k neighbors, we're choosing the ones that maximize our sum.

3. Calculating Star Sum: With the sorted lists for each potential center node, the star sum is calculated by taking the value of the current node v and adding the sum of the k highest neighboring node values from its list in g. This is done using a list comprehension:

   return max(v + sum(g[i][:k]) for i, v in enumerate(vals))

   • g[i][:k] fetches at most the k largest values from node i's list of neighbors' values.
   • sum(g[i][:k]) calculates the sum of these up to k values.
   • v + sum(g[i][:k]) adds the node i's value to this sum to get the star sum for this particular node i as the center.
   • max(...) then finds the maximum star sum achievable from all potential center nodes.

Complexity Analysis:

• Time Complexity: O(V + E log E) where V is the number of nodes, and E is the number of edges. This is because E edges are iterated to build the adjacency list, and each neighbors list could potentially be sorted up to E times (in the worst case, where all edges are connected to a single node). Therefore, the bottleneck is the sorting operation.
• Space Complexity: O(E), as additional space is required to store neighbors' values for each node, and in the worst case, each node could hold values related to all other nodes.

Patterns Used:

• Greedy Algorithm: By sorting the neighbors' values in descending order and picking the top k elements, the solution uses a greedy approach where it always picks the local optimal choice (highest value) at each step to maximize the star sum in the end.

The solution effectively combines these data structures and algorithms to efficiently solve the maximum star sum problem while respecting the constraint of at most k edges.

Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we have a graph with 5 nodes (n=5), numbered from 0 to 4. The values associated with each node are given by the array vals = [1, 2, 3, 4, 5], and we are allowed to include at most k=2 edges in our star graph to maximize the star sum. The edges are given by the array edges = [(0, 1), (0, 2), (0, 3), (1, 4), (2, 4)].

Step-by-Step Process:

1. Creating Adjacency List (Undirected): Our adjacency list g would be built as follows by iterating over the edges:

   g[0] -> [2, 3, 4]    # Node 0 is connected to nodes 1, 2, 3 with positive values
   g[1] -> [1, 5]       # Node 1 is connected to nodes 0 and 4
   g[2] -> [1, 5]       # Node 2 is connected to nodes 0 and 4
   g[3] -> [1]          # Node 3 is connected to node 0
   g[4] -> [2, 3]       # Node 4 is connected to nodes 1 and 2

2. Sorting Neighbors' Values: The next step is to sort each list of connected nodes' values in descending order:

   g[0] -> [4, 3, 2]    # Sort 2, 3, 4 in descending order
   g[1] -> [5, 1]       # Sort 1, 5 in descending order
   g[2] -> [5, 1]       # Sort 1, 5 in descending order
   g[3] -> [1]          # Already sorted since there's just one element
   g[4] -> [3, 2]       # Sort 2, 3 in descending order

3. Calculating Star Sum: Now, for each node, we will calculate the star sum by taking its value and adding the sum of its k highest valued neighbors:

   For node 0: Star sum = 1 + (4+3) = 8 (Take the 2 highest values 4 and 3)
   For node 1: Star sum = 2 + (5) = 7 (Only 1 neighbor's value can be taken since k=2 and node 1's value has to be included)
   For node 2: Star sum = 3 + (5) = 8 
   For node 3: Star sum = 4 + (1) = 5 
   For node 4: Star sum = 5 + (3) = 8 (Same reasoning as node 1)

   Note that we only include up to k neighbor's values, which is why even though node 0 has three neighbors, only the highest two (4 and 3) are included in the sum.

4. Finding the Maximum Star Sum: The final step is to find the maximum star sum possible:

   Maximum Star Sum = max(8, 7, 8, 5, 8) = 8

And so, the solution would return 8 as the maximum star sum possible by selecting a subset of the graph that forms a star graph with at most k edges. This example clearly shows how the solution approach effectively uses a greedy method to maximize the star sum.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the code is associated with several operations performed:

1. Construction of the graph g: The for loop that iterates over the edges has a time complexity of O(E) where E is the number of edges because each edge is visited once.

2. Sorting the adjacency lists: Each adjacency list in graph g is sorted in reverse order. In the worst case, if all nodes are connected to all other nodes, it will take O(N log N) time per node to sort, where N is the number of nodes. However, typically the number of edges connected to a node (degree of a node) is much lower than N, so it will more often be O(D log D) where D is the average degree of a node.

3. Calculating the max star sum: The maximum star sum is computed in a single for loop that iterates over the nodes once. The complexity for this part is primarily from the summation operation which takes O(k) time for each node, as it sums up to k elements from the sorted adjacency list. Since it iterates over all N nodes, this results in a O(Nk) time complexity.

The overall worst-case time complexity can be approximated by adding these components together: O(E + N log N + Nk).

However, the sorting step's time complexity will often be dominated by the number of edges (sorting smaller adjacency lists): O(E + D log D + Nk).

Space Complexity

The space complexity involves:

1. Storing the graph g: The graph is stored using adjacency lists, which will require O(N + E) space: O(N) for storing N keys and O(E) for storing the list of neighbor values.

2. Auxiliary space for sorting: Sorting the adjacency lists will require additional space which depends on the sorting algorithm used by Python's .sort() method (Timsort), but since in-place sorting is used, the impact on space complexity should be minimal.

3. Storing sums: The space for storing the sums is not significant as it only stores the sums for each node which is O(1) per node resulting in O(N) in total.

The overall space complexity is O(N + E).

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