Dijkstra’s algorithm is a popular algorithm used to find the shortest path between nodes in a graph. When combined with Fibonacci heaps, it achieves a better time complexity compared to traditional implementations using priority queues. In this article, we will explore Dijkstra’s algorithm with Fibonacci heaps and provide a Python example with explanation.

## Understanding Dijkstra’s Algorithm

Dijkstra’s algorithm is a greedy algorithm that works by iteratively selecting the node with the smallest distance from the source node and updating the distances of its neighboring nodes. It maintains a priority queue to keep track of the nodes with the smallest distances.

The algorithm starts by initializing the distances of all nodes to infinity, except for the source node which is set to 0. It then selects the node with the smallest distance from the priority queue and relaxes its neighboring nodes by updating their distances if a shorter path is found. This process continues until all nodes have been visited or the destination node is reached.

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## The Advantages of Fibonacci Heaps

Traditionally, Dijkstra’s algorithm uses a priority queue to keep track of the nodes with the smallest distances. However, the time complexity of traditional priority queues can be a bottleneck in large graphs.

Fibonacci heaps, on the other hand, provide a more efficient way to implement the priority queue. They have a better amortized time complexity for the decrease key operation, which is a crucial step in Dijkstra’s algorithm. This results in an overall improvement in the time complexity of the algorithm.

## Python Example

Let’s take a look at a Python implementation of Dijkstra’s algorithm with Fibonacci heaps:

```
import heapq
class FibonacciHeapNode:
def __init__(self, key, value):
self.key = key
self.value = value
self.degree = 0
self.marked = False
self.parent = None
self.child = None
self.left = self
self.right = self
def __lt__(self, other):
return self.key < other.key
class FibonacciHeap:
def __init__(self):
self.min = None
self.num_nodes = 0
def insert(self, key, value):
node = FibonacciHeapNode(key, value)
self._insert_node(node)
return node
def _insert_node(self, node):
if self.min is None:
self.min = node
else:
self._link(self.min, node)
if node.key < self.min.key:
self.min = node
self.num_nodes += 1
def _link(self, node1, node2):
node1.right.left = node2
node2.right.left = node1
node1.right = node2
node2.left = node1
def extract_min(self):
min_node = self.min
if min_node is not None:
child = min_node.child
while child is not None:
next_child = child.right
self._insert_node(child)
child.parent = None
child = next_child
self._remove_node(min_node)
self.num_nodes -= 1
return min_node
def _remove_node(self, node):
node.left.right = node.right
node.right.left = node.left
if node == node.right:
self.min = None
else:
self.min = node.right
def decrease_key(self, node, new_key):
node.key = new_key
parent = node.parent
if parent is not None and node.key < parent.key:
self._cut(node, parent)
self._cascading_cut(parent)
if node.key < self.min.key:
self.min = node
def _cut(self, node, parent):
node.left.right = node.right
node.right.left = node.left
parent.degree -= 1
if node == parent.child:
parent.child = node.right
if parent.degree == 0:
parent.child = None
self._insert_node(node)
node.parent = None
node.marked = False
def _cascading_cut(self, node):
parent = node.parent
if parent is not None:
if node.marked:
self._cut(node, parent)
self._cascading_cut(parent)
else:
node.marked = True
def dijkstra(graph, source):
distances = {node: float('inf') for node in graph}
distances[source] = 0
heap = FibonacciHeap()
nodes = {}
for node in graph:
nodes[node] = heap.insert(distances[node], node)
while heap.num_nodes > 0:
min_node = heap.extract_min()
current_node = min_node.value
for neighbor, weight in graph[current_node].items():
new_distance = distances[current_node] + weight
if new_distance < distances[neighbor]:
distances[neighbor] = new_distance
heap.decrease_key(nodes[neighbor], new_distance)
return distances
```

In the above code, we define two classes: `FibonacciHeapNode`

and `FibonacciHeap`

. The `FibonacciHeapNode`

class represents a node in the Fibonacci heap, while the `FibonacciHeap`

class implements the Fibonacci heap data structure.

The `dijkstra`

function takes a graph and a source node as input and returns a dictionary of distances from the source node to all other nodes in the graph.

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## Conclusion

Dijkstra’s algorithm with Fibonacci heaps provides a more efficient way to find the shortest path between nodes in a graph. By utilizing the benefits of Fibonacci heaps, we can achieve a better time complexity compared to traditional implementations using priority queues. The Python example provided in this article demonstrates how to implement Dijkstra’s algorithm with Fibonacci heaps (read here).

Remember to import the necessary libraries and use the provided code as a starting point for your own implementations. Happy coding!