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A graph has vertices (nodes) connected by edges. Used to model networks, maps, dependencies.
Types:
- Directed / Undirected
- Weighted / Unweighted
- Cyclic / Acyclic (DAG = Directed Acyclic Graph)
Representations:
- Adjacency Matrix: 2D array. O(1) edge lookup, O(V^2) space.
- Adjacency List: dict/list of lists. O(V+E) space. Most common.
# Adjacency List representation
graph = {
0: [1, 2],
1: [0, 3, 4],
2: [0],
3: [1],
4: [1]
}
--- BFS (Breadth-First Search) ---
Explore level by level using a queue. Finds shortest path in unweighted graphs.
Time: O(V+E). Space: O(V).
from collections import deque
def bfs(graph, start):
visited = set([start])
queue = deque([start])
order = []
while queue:
node = queue.popleft()
order.append(node)
for neighbor in graph[node]:
if neighbor not in visited:
visited.add(neighbor)
queue.append(neighbor)
return order
graph = {0:[1,2], 1:[0,3,4], 2:[0], 3:[1], 4:[1]}
print(bfs(graph, 0)) # Output: [0, 1, 2, 3, 4]
--- DFS (Depth-First Search) ---
Explore as deep as possible before backtracking. Uses stack (or recursion).
Time: O(V+E). Space: O(V).
# Iterative DFS
def dfs_iterative(graph, start):
visited = set()
stack = [start]
order = []
while stack:
node = stack.pop()
if node not in visited:
visited.add(node)
order.append(node)
for neighbor in graph[node]:
if neighbor not in visited:
stack.append(neighbor)
return order
# Recursive DFS
def dfs_recursive(graph, node, visited=None):
if visited is None:
visited = set()
visited.add(node)
result = [node]
for neighbor in graph[node]:
if neighbor not in visited:
result += dfs_recursive(graph, neighbor, visited)
return result
print(dfs_iterative(graph, 0)) # Output: [0, 2, 1, 4, 3]
print(dfs_recursive(graph, 0)) # Output: [0, 1, 3, 4, 2]
--- Shortest Path: Dijkstra's Algorithm ---
Find shortest path from source to all nodes in a weighted graph (non-negative weights).
Time: O((V+E) log V) with min-heap. Space: O(V).
import heapq
def dijkstra(graph, start):
# graph: {node: [(neighbor, weight), ...]}
dist = {node: float('inf') for node in graph}
dist[start] = 0
heap = [(0, start)] # (distance, node)
while heap:
d, u = heapq.heappop(heap)
if d > dist[u]:
continue
for v, w in graph[u]:
if dist[u] + w < dist[v]:
dist[v] = dist[u] + w
heapq.heappush(heap, (dist[v], v))
return dist
wgraph = {
'A': [('B', 1), ('C', 4)],
'B': [('C', 2), ('D', 5)],
'C': [('D', 1)],
'D': []
}
print(dijkstra(wgraph, 'A'))
# Output: {'A': 0, 'B': 1, 'C': 3, 'D': 4}
--- Cycle Detection in Directed Graph ---
Use DFS with a recursion stack to detect back edges.
def has_cycle_directed(graph):
visited = set()
rec_stack = set()
def dfs(node):
visited.add(node)
rec_stack.add(node)
for neighbor in graph.get(node, []):
if neighbor not in visited:
if dfs(neighbor):
return True
elif neighbor in rec_stack:
return True
rec_stack.remove(node)
return False
for node in graph:
if node not in visited:
if dfs(node):
return True
return False
g1 = {0: [1], 1: [2], 2: [0]} # cycle
g2 = {0: [1], 1: [2], 2: []} # no cycle
print(has_cycle_directed(g1)) # True
print(has_cycle_directed(g2)) # False
--- Topological Sort (Kahn's Algorithm) ---
Linear ordering of vertices in a DAG such that for every edge u->v, u comes before v.
Time: O(V+E). Space: O(V).
from collections import deque
def topological_sort(graph, num_nodes):
in_degree = [0] * num_nodes
for u in graph:
for v in graph[u]:
in_degree[v] += 1
queue = deque([i for i in range(num_nodes) if in_degree[i] == 0])
order = []
while queue:
node = queue.popleft()
order.append(node)
for neighbor in graph.get(node, []):
in_degree[neighbor] -= 1
if in_degree[neighbor] == 0:
queue.append(neighbor)
return order if len(order) == num_nodes else [] # empty = cycle
dag = {0: [1, 2], 1: [3], 2: [3], 3: []}
print(topological_sort(dag, 4)) # Output: [0, 1, 2, 3] or [0, 2, 1, 3]
--- Minimum Spanning Tree: Kruskal's Algorithm ---
Connect all nodes with minimum total edge weight. Uses Union-Find.
Time: O(E log E). Space: O(V).
def kruskal(num_nodes, edges):
# edges: [(weight, u, v), ...]
edges.sort()
parent = list(range(num_nodes))
def find(x):
while parent[x] != x:
parent[x] = parent[parent[x]]
x = parent[x]
return x
def union(x, y):
px, py = find(x), find(y)
if px == py:
return False
parent[px] = py
return True
mst_cost = 0
mst_edges = []
for w, u, v in edges:
if union(u, v):
mst_cost += w
mst_edges.append((u, v, w))
return mst_cost, mst_edges
edges = [(4,0,1),(2,0,2),(3,1,2),(1,1,3),(5,2,3)]
print(kruskal(4, edges))
# Output: (6, [(1, 3, 1), (0, 2, 2), (1, 2, 3)])
--- Bellman-Ford Algorithm ---
Shortest path with negative weights. Detects negative cycles.
Time: O(V*E). Space: O(V).
def bellman_ford(graph, num_nodes, start):
# graph: [(u, v, weight), ...]
dist = [float('inf')] * num_nodes
dist[start] = 0
for _ in range(num_nodes - 1):
for u, v, w in graph:
if dist[u] + w < dist[v]:
dist[v] = dist[u] + w
# Check for negative cycles
for u, v, w in graph:
if dist[u] + w < dist[v]:
return None # negative cycle
return dist
edges = [(0,1,4),(0,2,5),(1,2,-3),(2,3,4)]
print(bellman_ford(edges, 4, 0))
# Output: [0, 4, 1, 5]
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