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Commit a13d3c6

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started working on graph algorithms
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‎README.md

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- [二分搜索](./basic_algorithm/binary_search.md)
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- [排序算法](./basic_algorithm/sort.md)
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- [动态规划](./basic_algorithm/dp.md)
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- [图相关算法](./basic_algorithm/graph/)
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### 算法思维 🦁
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‎basic_algorithm/graph/README.md

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### 图相关算法
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Ongoing...
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[拓扑排序](./topological_sorting.md)
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# 拓扑排序
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图的拓扑排序 (topological sorting) 一般用于给定一系列偏序关系,求一个全序关系的题目中。以元素为结点,以偏序关系为边构造有向图,然后应用拓扑排序算法即可得到全序关系。
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### [course-schedule-ii](https://leetcode-cn.com/problems/course-schedule-ii/)
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> 给定课程的先修关系,求一个可行的修课顺序
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**图森面试真题**。非常经典的拓扑排序应用题目。下面给出 3 种实现方法,可以当做模板使用。
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方法 1:DFS 的递归实现
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```Python
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NOT_VISITED = 0
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DISCOVERING = 1
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VISITED = 2
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class Solution:
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def findOrder(self, numCourses: int, prerequisites: List[List[int]]) -> List[int]:
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# construct graph
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graph_neighbor = collections.defaultdict(list)
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for course, pre in prerequisites:
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graph_neighbor[pre].append(course)
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# recursive postorder DFS for topological sort
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tsort_rev = []
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status = [NOT_VISITED] * numCourses
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def dfs(course):
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status[course] = DISCOVERING
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for n in graph_neighbor[course]:
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if status[n] == DISCOVERING or (status[n] == NOT_VISITED and not dfs(n)):
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return False
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tsort_rev.append(course)
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status[course] = VISITED
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return True
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for course in range(numCourses):
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if status[course] == NOT_VISITED and not dfs(course):
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return []
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return tsort_rev[::-1]
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```
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方法 2:DFS 的迭代实现
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```Python
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NOT_VISITED = 0
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DISCOVERING = 1
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VISITED = 2
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class Solution:
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def findOrder(self, numCourses: int, prerequisites: List[List[int]]) -> List[int]:
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# construct graph
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graph_neighbor = collections.defaultdict(list)
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for course, pre in prerequisites:
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graph_neighbor[pre].append(course)
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# iterative postorder DFS for topological sort
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tsort_rev = []
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status = [NOT_VISITED] * numCourses
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dfs = []
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for course in range(numCourses):
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if status[course] == NOT_VISITED:
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dfs.append(course)
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status[course] = DISCOVERING
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while dfs:
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if graph_neighbor[dfs[-1]]:
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n = graph_neighbor[dfs[-1]].pop()
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if status[n] == DISCOVERING:
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return []
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if status[n] == NOT_VISITED:
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dfs.append(n)
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status[n] = DISCOVERING
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else:
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tsort_rev.append(dfs.pop())
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status[tsort_rev[-1]] = VISITED
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return tsort_rev[::-1]
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```
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方法 3:[Kahn's algorithm](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm)
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```Python
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class Solution:
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def findOrder(self, numCourses: int, prerequisites: List[List[int]]) -> List[int]:
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# construct graph with indegree data
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graph_neighbor = collections.defaultdict(list)
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indegree = collections.defaultdict(int)
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for course, pre in prerequisites:
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graph_neighbor[pre].append(course)
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indegree[course] += 1
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# Kahn's algorithm
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src_cache = [] # can also use queue
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for i in range(numCourses):
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if indegree[i] == 0:
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src_cache.append(i)
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tsort = []
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while src_cache:
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tsort.append(src_cache.pop())
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for n in graph_neighbor[tsort[-1]]:
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indegree[n] -= 1
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if indegree[n] == 0:
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src_cache.append(n)
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return tsort if len(tsort) == numCourses else []
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```
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### [alien-dictionary](https://leetcode-cn.com/problems/alien-dictionary/)
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```Python
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class Solution:
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def alienOrder(self, words: List[str]) -> str:
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N = len(words)
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if N == 0:
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return ''
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if N == 1:
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return words[0]
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# construct graph
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indegree = {c: 0 for word in words for c in word}
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graph = collections.defaultdict(list)
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for i in range(N - 1):
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first, second = words[i], words[i + 1]
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len_f, len_s = len(first), len(second)
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find_different = False
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for j in range(min(len_f, len_s)):
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f, s = first[j], second[j]
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if f != s:
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if s not in graph[f]:
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graph[f].append(s)
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indegree[s] += 1
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find_different = True
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break
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if not find_different and len_f > len_s:
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return ''
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tsort = []
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src_cache = [c for c in indegree if indegree[c] == 0]
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while src_cache:
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tsort.append(src_cache.pop())
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for n in graph[tsort[-1]]:
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indegree[n] -= 1
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if indegree[n] == 0:
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src_cache.append(n)
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return ''.join(tsort) if len(tsort) == len(indegree) else ''
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```
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### [sequence-reconstruction](https://leetcode-cn.com/problems/sequence-reconstruction/)
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Kahn's algorithm 可以判断拓扑排序是否唯一。
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```Python
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class Solution:
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def sequenceReconstruction(self, org: List[int], seqs: List[List[int]]) -> bool:
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N = len(org)
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inGraph = [False] * (N + 1)
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graph_set = collections.defaultdict(set)
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for seq in seqs:
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if seq:
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if seq[0] > N or seq[0] < 1:
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return False
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inGraph[seq[0]] = True
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for i in range(1, len(seq)):
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if seq[i] > N or seq[i] < 1:
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return False
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inGraph[seq[i]] = True
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graph_set[seq[i - 1]].add(seq[i])
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indegree = collections.defaultdict(int)
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for node in graph_set:
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for n in graph_set[node]:
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indegree[n] += 1
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num_valid, count0, src = 0, -1, 0
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for i in range(1, N + 1):
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if inGraph[i] and indegree[i] == 0:
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count0 += 1
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src = i
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i = 0
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while count0 == i and src == org[i]:
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num_valid += 1
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for n in graph_set[src]:
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indegree[n] -= 1
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if indegree[n] == 0:
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count0 += 1
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src = n
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i += 1
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return num_valid == N
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```
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