This notebook was prepared by Donne Martin. Source and license info is on GitHub.

- Is the graph directed?
- Yes

- Can we assume we already have Graph and Node classes?
- Yes

- Can we assume this is a connected graph?
- Yes

- Can we assume the inputs are valid?
- Yes

- Can we assume this fits memory?
- Yes

Input:

`add_edge(source, destination, weight)`

```
graph.add_edge(0, 1, 5)
graph.add_edge(0, 4, 3)
graph.add_edge(0, 5, 2)
graph.add_edge(1, 3, 5)
graph.add_edge(1, 4, 4)
graph.add_edge(2, 1, 6)
graph.add_edge(3, 2, 7)
graph.add_edge(3, 4, 8)
```

Result:

- search_path(start=0, end=2) -> True
- search_path(start=0, end=0) -> True
- search_path(start=4, end=5) -> False

To determine if there is a path, we can use either breadth-first or depth-first search.

Breadth-first search can also be used to determine the shortest path. Depth-first search is easier to implement with just straight recursion, but often results in a longer path.

We'll use a breadth-first search approach:

- Add the start node to the queue and mark it as visited
- If the start node is the end node, return True
- While the queue is not empty
- Dequeue a node and visit it
- If the node is the end node, return True
- Iterate through each adjacent node
- If the node has not been visited, add it to the queue and mark it as visited

- Return False

Complexity:

- Time: O(V + E), where V = number of vertices and E = number of edges
- Space: O(V + E)

In [1]:

```
%run ../graph/graph.py
```

In [2]:

```
from collections import deque
class GraphPathExists(Graph):
def path_exists(self, start, end):
if start is None or end is None:
return False
if start is end:
return True
queue = deque()
queue.append(start)
start.visit_state = State.visited
while queue:
node = queue.popleft()
if node is end:
return True
for adj_node in node.adj_nodes.values():
if adj_node.visit_state == State.unvisited:
queue.append(adj_node)
adj_node.visit_state = State.visited
return False
```

In [3]:

```
%%writefile test_path_exists.py
import unittest
class TestPathExists(unittest.TestCase):
def test_path_exists(self):
nodes = []
graph = GraphPathExists()
for id in range(0, 6):
nodes.append(graph.add_node(id))
graph.add_edge(0, 1, 5)
graph.add_edge(0, 4, 3)
graph.add_edge(0, 5, 2)
graph.add_edge(1, 3, 5)
graph.add_edge(1, 4, 4)
graph.add_edge(2, 1, 6)
graph.add_edge(3, 2, 7)
graph.add_edge(3, 4, 8)
self.assertEqual(graph.path_exists(nodes[0], nodes[2]), True)
self.assertEqual(graph.path_exists(nodes[0], nodes[0]), True)
self.assertEqual(graph.path_exists(nodes[4], nodes[5]), False)
print('Success: test_path_exists')
def main():
test = TestPathExists()
test.test_path_exists()
if __name__ == '__main__':
main()
```

In [4]:

```
%run -i test_path_exists.py
```