2.5 KiB
graphs
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graph is a non-linear data structure consisting of vertices and edges.
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graphs can be represented by adjacent matrices, adjacent lists, and hash table of hash tables.
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in undirected graphs, the edges between any two vertices do not have a direction, indicating a two-way relationship.
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in directed graphs, the edges between any two vertices are directional.
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in weighted graphs, each edge has an associated weight. if the sum of the weights of all edges of a cycle is a negative values, it's a negative weight cycle.
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the degree of a vertex is the number of edges connecting the vertex. in directed, graphs, if the in-dregree of a vertex is
d, there are d directional edges incident to the vertex (and similarly, out-degree from the vertex). -
with
|V|the number of vertices and|E|is the number of edges, search in a graph (either bfs of dfs) isO(|V| + |E|).
traversals
breath first search
def bfs(matrix):
if not matrix:
return []
rows, cols = len(matrix), len(matrix[0])
visited = set()
directions = ((0, 1), (0, -1), (1, 0), (-1, 0))
def traverse(i, j):
queue = deque([(i, j)])
while queue:
curr_i, curr_j = queue.popleft()
if (curr_i, curr_j) not in visited:
visited.add((curr_i, curr_j))
for direction in directions:
next_i, next_j = curr_i + direction[0], curr_j + direction[1]
if 0 <= next_i < rows and 0 <= next_j < cols:
queue.append((next_i, next_j))
for i in range(rows):
for j in range(cols):
traverse(i, j)
depth first search
def dfs(matrix):
if not matrix:
return []
rows, cols = len(matrix), len(matrix[0])
visited = set()
directions = ((0, 1), (0, -1), (1, 0), (-1, 0))
def traverse(i, j):
if (i, j) in visited:
return
visited.add((i, j))
for direction in directions:
next_i, next_j = i + direction[0], j + direction[1]
if 0 <= next_i < rows and 0 <= next_j < cols:
traverse(next_i, next_j)
for i in range(rows):
for j in range(cols):
traverse(i, j)