hdu 4598 Difference(spfa+差分约束，4级)

1. Problem description

The problem HDU 4598 Difference involves solving a system of difference constraints using the Shortest Path Faster Algorithm (SPFA) and difference constraints. The problem is at the 4th level of HDU's problem difficulty scale. In this article, we will discuss the problem in detail and provide a step-by-step solution to solve it.

2. Problem analysis

The problem provides a directed weighted graph with n nodes and m edges. Each edge represents a difference constraint of the form: xj - xi <= cij. The task is to find the minimum possible value of the difference between the maximum and minimum values of the nodes in the graph. The graph may contain negative cycles, and if it does, the problem is considered to be unsolvable.

This problem can be solved using the Shortest Path Faster Algorithm (SPFA) with difference constraints. SPFA is a variant of the Bellman-Ford algorithm, which can handle negative edge weights and detect negative cycles. The difference constraints can be solved by adding additional slack variables and transforming the original problem into a system of equations.

3. Solution approach

To solve the problem, we need to perform the following steps:

3.1 Graph initialization:

We initialize an empty directed graph with n nodes and no edges. Each node is assigned an initial value of infinity.

# Graph representation

graph = [[] for i in range(n)]
node_values = [float('inf')] * n

3.2 Input processing:

We read the input values and create edges in the graph based on the given difference constraints.

for i in range(m):

    xi, xj, cij = map(int, input().split())
    graph[xj].append((xi, cij))

For each difference constraint xi - xj <= cij, we add an edge from node xi to node xj with weight cij.

3.3 SPFA algorithm:

We use the Shortest Path Faster Algorithm (SPFA) to find the shortest path from each node to all other nodes in the graph. SPFA can handle negative edge weights and negative cycles.

def spfa(graph, node_values):

    queue = []
    in_queue = [False] * n
    # Start from a random node
    start_node = 0
    queue.append(start_node)
    in_queue[start_node] = True
    node_values[start_node] = 0
    while queue:
        current_node = queue.pop(0)
        in_queue[current_node] = False
        for neighbor, weight in graph[current_node]:
            if node_values[current_node] + weight < node_values[neighbor]:
                node_values[neighbor] = node_values[current_node] + weight
                if not in_queue[neighbor]:
                    queue.append(neighbor)
                    in_queue[neighbor] = True
    return node_values
node_values = spfa(graph, node_values)

The SPFA algorithm updates the node values by relaxing the edges in the graph. It continues until no more updates can be made. The final node values represent the shortest distance from the start node to each node in the graph.

3.4 Difference computation:

After obtaining the node values using the SPFA algorithm, we can calculate the minimum and maximum values of the nodes in the graph and compute the difference.

min_value = min(node_values)

max_value = max(node_values)
difference = max_value - min_value

3.5 Output:

We output the calculated difference, which represents the minimum possible value of the difference between the maximum and minimum values of the nodes in the graph.

print(difference)

4. Conclusion

In this article, we discussed the problem HDU 4598 Difference and provided a detailed solution using the Shortest Path Faster Algorithm (SPFA) with difference constraints. We showed how to initialize the graph, process the input, apply the SPFA algorithm, and compute the difference between the maximum and minimum node values. This solution can be used to solve similar problems that involve solving a system of difference constraints.