From 7a4b8458dd0a18099fc0c3d71b70da6c7c10612d Mon Sep 17 00:00:00 2001 From: fuxuemingzhu Date: Mon, 27 Sep 2021 14:33:45 +0800 Subject: [PATCH 01/24] =?UTF-8?q?=E8=AE=A4=E9=A2=86=20TopologicalSort?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index acbfbf8..95fc418 100644 --- a/README.md +++ b/README.md @@ -118,7 +118,7 @@ docsify serve . | UnionFind | 已认领 | | | SegmentTree | 已认领 | | | Array | | | -| TopologicalSort | | | +| TopologicalSort | 已认领,wu2meng3 | | | Trie | | | | BinaryIndexedTree | | | | BinarySearchTree | | | From 935c961f64618ed5ff188438c887c8b7303ce6d0 Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Fri, 1 Oct 2021 10:48:22 -0700 Subject: [PATCH 02/24] Add BFS Python template --- TopologicalSort/README.md | 52 +++++++++++++++++++++++++++++++++++++++ 1 file changed, 52 insertions(+) create mode 100644 TopologicalSort/README.md diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md new file mode 100644 index 0000000..15bde93 --- /dev/null +++ b/TopologicalSort/README.md @@ -0,0 +1,52 @@ +# 拓扑排序算法 + +作者:wu2meng3 + +## 拓扑排序的应用 + +## 拓扑排序的两个模板 + +* BFS +```python +class Solution(object): + + def topSort(self, graph: Dict[int, List[int]]) -> List[int]: + """ + @type: 邻接表表示的有向图 + @rtype: 图的结点列表 + """ + import collections + # 计算入度 + inDegree = self.getInDegree(graph) + + # 将入度为0的结点放入队列 queue 和哈希集合 visited 防止重复遍历 + startNodeList = [ node for node in graph if inDegree[node] == 0] + # 用 deque 实现队列 + queue = collections.deque(startNodeList) + visited = set(startNodeList) + # 保存结果的数组 + topoOrder = [] + + # 广度优先搜索 + while queue: + node = queue.popleft() + topoOrder.append(node) + + for neighbor in node.neighbors: + if neighbor in visited: + continue + inDegree[neighbor] -= 1 + if inDegree[neighbor] == 0: + queue.append(neighbor) + visited.add(neighbor) + + return topoOrder +``` + +* DFS + +## 精选例题 + +### 题型一:有向无环图中的最长路径 (课程表) + +## 精选练习 From 077dcf310ea901ce4900c03145505afcb47153c5 Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Fri, 1 Oct 2021 14:49:58 -0700 Subject: [PATCH 03/24] 1. Add BFS and DFS results as plots. 2. 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graph[1].neighbors = [graph[4]] @@ -27,9 +28,7 @@ graph[3].neighbors = [graph[4], graph[5]] 其拓扑排序可以是: `0->1->2->3->4->5`,也可以是 `0->3->2->5->1->4`。 -![BFS](/Users/billywu/Coding/AlgoWiki/AlgoWiki/TopologicalSort/BFS.png) - -![DFS](/Users/billywu/Coding/AlgoWiki/AlgoWiki/TopologicalSort/DFS.png) +![TopoSort](/Users/billywu/Coding/AlgoWiki/AlgoWiki/TopologicalSort/TopoSort.png) ## 拓扑排序的应用 @@ -52,21 +51,21 @@ graph[3].neighbors = [graph[4], graph[5]] 伪代码如下 ```python -L ← 输出列表 -Q ← 包含所有入度为 0 结点的队列 -while ( Q 不为空 ) do - n 出列 - 将 n 放入 L +topoSort ← 输出列表,初始为空 +queue ← 包含所有入度为 0 结点的队列 +while ( queue 不为空 ) do + n 出列 queue + 将 n 放入 topoSort for ( n 的所有邻接结点 m ) do 删除该边,m 的入度减 1 if ( m 入度为 0 ) then - m 入列 + m 入列 queue if (图中存在边) then # 存在环,不可以拓扑排序 return None else - return L + return topoSort ``` 之前说过拓扑排序结果并不唯一,按照字典序 ( lexicographical order )可以将所有可能的输出结果进行排序。如果需要最大或者最小结果,可以将 Kahn 算法中的队列替换成最大堆/最小堆实现的优先队列即可,具体实现参考[精选例题X]。 @@ -78,6 +77,7 @@ else #### 代码 ```python +from typing import * import collections class Solution(object): @@ -117,24 +117,58 @@ class Solution(object): return [] return topoOrder ``` +为了形象的理解 BFS 的过程,我们将代码的执行过程以及对应的队列 `queue` 和 `topoOrder` 的值都表示在下面的图中。 + +![BFS](/Users/billywu/Coding/AlgoWiki/AlgoWiki/TopologicalSort/TopoSort_BFS.png) ### 深度优先搜索( Depth-first search, DFS) #### 思路 -一句话总结就是**拓扑排序就是图后序遍历的倒序输出**。我们现在将这句话展开来理解。根据《算法导论》第 22 章,我们可以在 DFS 的过程中给每个结点盖两个时间戳,分别是发现这个结点的时间 $d[u]$和遍历完该结点所有邻接结点的时间 $f[u]$,其中 $u \in V$ 。对于一个结点 $u$,DFS 算法遍历完其所有子结点之后才会判断其遍历结束,更新时间戳 $f[u]$。可以证明如果图中存在一条从 $u$ 指向 $v$ 的边,那么 $f[v] < f[u]$ 。所以,按照 $f[u]$ 的逆序输出结点就可以得到拓扑排序之后的序列。该算法有点类似回溯算法,我们需要在递归之前和递归之后更新每个结点的状态。每个结点有三种的可能的状态,分别是: +一句话总结就是**拓扑排序就是图后序遍历的倒序输出**。我们现在将这句话展开来理解。根据《算法导论》第 22 章,我们可以在 DFS 的过程中给每个结点盖两个时间戳,分别是发现这个结点的时间 `discoverTime[u]` 和遍历完该结点所有邻接结点的时间 `finishTime[u]` ,其中 `u` 代表一个结点。对于结点 `u`,DFS 算法遍历完其所有子结点之后才会判断其遍历结束,更新时间戳 `finishTime[u]` 。可以证明如果图中存在一条从 `u` 指向 `v` 的边,那么 `finishTime[v] < f[u]` 。所以,按照 `finishTime[u]` 的逆序输出结点就可以得到拓扑排序之后的序列。该算法有点类似回溯算法,我们需要在递归之前和递归之后更新每个结点的状态。每个结点有三种的可能的状态,分别是: * `UNVISITED`:未被发现; * `VISITING`:已发现,但是邻接结点还未遍历完; * `VISITED`:已发现,邻接结点已经遍历完。 所有结点的初始状态都是`UNVISITED`,在遍历某结点的所以邻接结点进入下一层递归之前要将该结点的状态设为 `VISITING`,所有邻接结点都遍历完递归返回之后将其状态更新为 `VISITED`。 -**时间复杂度**:在 DFS 过程中,每一个结点和每一条边都会被访问一次,将一个结点按照完成时间 $f[u]$的顺序放入输出数组需要 $O(1)$ 的时间,所以总的时间复杂度是 $O(|V|+|E|)$。 +沿用上面的例子,DFS 算法得到的拓扑排序序列和每个结点的发现/完成时间如下图所示。 + +![DFS_A](/Users/billywu/Coding/AlgoWiki/AlgoWiki/TopologicalSort/TopoSort_DFS_A.png) + +DFS 算法的伪代码如下: + +```python +topoOrder ← 输出列表,初始为空 +while 存在状态为 UNVISITED 的结点 do + 取状态为 UNVISITED 的结点 n + visit(n) + +function visit(node n) + if n 的状态为 VISITING then + stop (not a DAG) + + if n 的状态为 VISITED then + return + + 将 n 的状态标记为 VISITING + + for n 的所有邻接结点 m do + visit(m) + + 将 n 的状态标记为 VISITED + 将 n 加入 topoOrder 的首位 +``` + +**时间复杂度**:在 DFS 过程中,每一个结点和每一条边都会被访问一次,将一个结点按照完成时间 `finishTime[u]` 的顺序放入输出数组需要 $O(1)$ 的时间,所以总的时间复杂度是 $O(|V|+|E|)$。 **空间复杂度**:我们需要 `states` 数组或者字典来保存每个结点的状态,所以空间复杂度是 $O(|V|)$ 。 ```python +from typing import * from enum import Enum +import collections + # 遍历过程中结点的三种状态 class State(Enum): # 《算法导论》中的 WHITE 状态:未被发现 @@ -144,8 +178,14 @@ class State(Enum): # 《算法导论》中的 BLACK 状态:已发现,邻接结点已经遍历完 VISITED = 2 + class Solution(object): + def __init__(self): + self.time = 0 + self.finishTime = collections.defaultdict(lambda: 0) + self.discoverTime = collections.defaultdict(lambda: 0) + def topSort(self, graph: List[DirectedGraphNode]) -> List[int]: numNodes = len(graph) states = {node: State.UNVISITED for node in graph} @@ -172,6 +212,10 @@ class Solution(object): # VISITING 代表该结点已经被发现,但是其邻接结点还未遍历完 states[node] = State.VISITING + # 更新发现时间 + self.time += 1 + self.discoverTime[node] = self.time + # 递归处理子结点,将子结点加入 res 列表 for child in node.neighbors: if (self.hasCycle(graph, child, states, res)): @@ -182,9 +226,17 @@ class Solution(object): # 邻接结点已经遍历结束 states[node] = State.VISITED + # 更新完成时间 + self.time += 1 + self.finishTime[node] = self.time + return False ``` +为了形象的理解 DFS 的过程,我们将上述代码的执行过程以及每个结点的发现时间 `discoverTime` 和 完成时间 `finishTime` 的值都表示在下面的图中。我们用白色代表 `UNVISITED` ,灰色代表 `VISITING`,黑色代表 `VISITED`。 + +![DFS_B](/Users/billywu/Coding/AlgoWiki/AlgoWiki/TopologicalSort/TopoSort_DFS_B.png) + ## 精选例题 ### 题型一:课程表 @@ -201,6 +253,10 @@ class Solution(object): BFS 答案 ```python +from typing import * +import collections + + class Solution(object): def findOrder(self, numCourses: int, prerequisites: List[List[int]]) -> List[int]: @@ -248,8 +304,10 @@ class Solution(object): DFS 答案 ```python +from typing import * from enum import Enum + class State(Enum): UNVISITED = 0 VISITING = 1 @@ -319,6 +377,9 @@ class Solution(object): 以下是 BFS + 最小堆实现的代码。 ```python +from typing import * +import heapq + class Solution(object): def alienOrder(self, words: List[str]): @@ -402,7 +463,8 @@ class Solution(object): | [CF 1385E. Directing Edges](https://codeforces.com/problemset/problem/1385/E) | | -# 参考 +## 参考 + * [Topological sorting @ Wikipedia](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm) * https://oi-wiki.org/graph/topo/ * 《算法导论》第 22 章 diff --git a/TopologicalSort/TopoSort.png b/TopologicalSort/TopoSort.png new file mode 100644 index 0000000000000000000000000000000000000000..09489d60d6a4208d23bf1f6381c86405a7ba6072 GIT binary patch literal 261573 zcmZ^L2RzmP{y!osk&0v%lB{I!Q)I6cG7l9(9NGJjO39vO7qT7M`~&)4&HKB}wAk&#>{!NI{HQ;?U@z`-E|U-%3}c;J!M z#C0X`4cAUX?g36t2i+q0kF&{B1ydCjoIBuWA{>0&>o^3bzXHA_acTem&$76;anAko zIJl0lIS&5+{f;X5e)=Z>e4W1LU*FFq;{M-ngU1r_{{7p88j0ur{WCtp>C03IJGj9& zVjFoKI~<&=h|@1z{PIR{mk=BUnfp(iaaTqOW1pY~_cu%TR0jOM?X>1FYf)fYo9)53>`c7Kc0i=6I1L4WHvUx`0p3T z!L6;TAp9TCd6!T!tP5qMICfX{%~6^ySF$pWBavfSL+cxE z=x|Qo9&M%UH8G#YpCDlOJ^5T{T}bFhtK|Wsxf7FBovc8KXUm8jNwILTB;4sXZNOv;rgkRVCeZ^Y% z8)ST#S(#gqjr{Srw3A4S&dT50{h1Wl=mE144iQt=I|4kgUE( 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z(A7%Kx$W1!>wL}+>!h^`nM?1n8($pnz1#K8pFor|DasMundHdwQaPK4wA@qgNe&ar zo~cd7Ikh-p2bs!wUEEcZ^M;WAvN3^2?*w6F`%9zd*IKYW{#n#v2+bYf@eZs$-6i}U zfhWKj>f_E#feG1yL><8Y3AQriws z@ch*4lY^c=z*xUF*&Lpqo0R>42k~f1FwDSIl=|Se_6c1N*LqRdG;TlTid>F45C5zHIstz}M%Z9=q-bZEAd{L{ z`3_vTm9BvuUmQ;Y;n5KxS#`2Cat4BOzv{DL$V zj3%ZFZU!Fv#&NJ8DAIISy+ciqosuc#LSCHhh{Iu5)4h7Q zTYJEoXN<_EO)3EdVF?hBiY~r9O)NcHUq}8$0a^P^b-W0#s5xj4+g+$ZUQ_ZbY6TUE?DTv!&}59Z{#Nkt01C=Ubn$rxHSGocW)a=n%DXE zyPDH@ZJ{-+2GVJ&bqqYZu&ukLeFDgF3~aOo$-H;%rU?inb~`%NkPJP{8S8b{!oA8) z^J*=)(pPpp8A#`%C0f&B&@Z>d$2xANb#o7P@oL>#i!)etL{03Z+aOaN46FL-XN9C* z$jAxDww`auCy_vz=&ufhWv4)cz5G1YBJ@m9{PzP{GgJGL`2 z{hHEuEl%UYnxjH{Oa)$As{Dt&pQQ2dfm5`1ICAv!Y=s3XgTG?yD#usdfTHCFG@TBd)zhSTZ__{78hXJXz=uc_9hN-_Ll0}%19ja{9C zjjRz86*rnq_%zosF>rWy?{dk~`lI0Seoowav6Gdfp#KBU0dPbB literal 0 HcmV?d00001 From f30a245c393c46e542fb94cee47d417c95fc5ef7 Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Mon, 4 Oct 2021 18:57:33 -0700 Subject: [PATCH 09/24] Cosmetic change --- TopologicalSort/README.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index e15573a..b5ad5b5 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -12,9 +12,9 @@ 例子:如下图所示,我们定义了一幅有向无环图, ```python -class DirectedGraphNode: +class DirectedGraphNode(object): - def __init__(self, x): + def __init__(self, x: int): self.label = x self.neighbors = [] @@ -51,11 +51,11 @@ graph[3].neighbors = [graph[4], graph[5]] 伪代码如下 ```python -topoSort ← 输出列表,初始为空 +topoOrder ← 输出列表,初始为空 queue ← 包含所有入度为 0 结点的队列 while ( queue 不为空 ) do n 出列 queue - 将 n 放入 topoSort + 将 n 放入 topoOrder for ( n 的所有邻接结点 m ) do 删除该边,m 的入度减 1 if ( m 入度为 0 ) then @@ -65,7 +65,7 @@ if (图中存在边) then # 存在环,不可以拓扑排序 return None else - return topoSort + return topoOrder ``` 之前说过拓扑排序结果并不唯一,按照字典序 ( lexicographical order )可以将所有可能的输出结果进行排序。如果需要最大或者最小结果,可以将 Kahn 算法中的队列替换成最大堆/最小堆实现的优先队列即可,具体实现参考[精选例题X]。 From f33257de3549f90f29133b488ada8dff6602a883 Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Mon, 4 Oct 2021 19:06:34 -0700 Subject: [PATCH 10/24] Cosmetic change --- TopologicalSort/README.md | 8 ++++++-- 1 file changed, 6 insertions(+), 2 deletions(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index b5ad5b5..6a0ff11 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -240,12 +240,16 @@ class Solution(object): ## 精选例题 ### 题型一:课程表 + 例题 1: 「力扣」 第 207 题:[课程表(Course Schedule)](https://leetcode-cn.com/problems/course-schedule/) 例题 2: 「力扣」 第 210 题:[课程表 II(Course Schedule II)](https://leetcode-cn.com/problems/course-schedule-ii/) -> 现在你总共有 `numCourses` 门课需要选,记为 `0`到 `numCourses - 1`。给你一个数组 `prerequisites` ,其中 `prerequisites[i] = [a$_i$, b$_i$]` ,表示在选修课程 `a_i` 前**必须**先选修 `b_i` 。 -> 返回你为了学完所有课程所安排的学习顺序。可能会有多个正确的顺序,你只要返回**任意一种**就可以了。如果不可能完成所有课程,返回**一个空数组**。 +> 现在你总共有 `numCourses` 门课需要选,记为 `0`到 `numCourses - 1`。 +> 给你一个数组 `prerequisites` ,其中 `prerequisites[i] = [a$_i$, b$_i$]` ,表示在选修课程 `a_i` 前**必须**先选修 `b_i` 。 +> * 返回你为了学完所有课程所安排的学习顺序。 +> * 可能会有多个正确的顺序,你只要返回**任意一种**就可以了。 +> * 如果不可能完成所有课程,返回**一个空数组**。 **分析** * 这两道题例题就是最经典的拓扑排序应用,我们直接套用上述模板即可。 From 0209b13096963f4e1ab865b7fcd0d208bc199b8c Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Mon, 4 Oct 2021 19:06:34 -0700 Subject: [PATCH 11/24] Cosmetic change. Correct many typos. --- TopologicalSort/README.md | 8 ++++++-- 1 file changed, 6 insertions(+), 2 deletions(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index b5ad5b5..6a0ff11 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -240,12 +240,16 @@ class Solution(object): ## 精选例题 ### 题型一:课程表 + 例题 1: 「力扣」 第 207 题:[课程表(Course Schedule)](https://leetcode-cn.com/problems/course-schedule/) 例题 2: 「力扣」 第 210 题:[课程表 II(Course Schedule II)](https://leetcode-cn.com/problems/course-schedule-ii/) -> 现在你总共有 `numCourses` 门课需要选,记为 `0`到 `numCourses - 1`。给你一个数组 `prerequisites` ,其中 `prerequisites[i] = [a$_i$, b$_i$]` ,表示在选修课程 `a_i` 前**必须**先选修 `b_i` 。 -> 返回你为了学完所有课程所安排的学习顺序。可能会有多个正确的顺序,你只要返回**任意一种**就可以了。如果不可能完成所有课程,返回**一个空数组**。 +> 现在你总共有 `numCourses` 门课需要选,记为 `0`到 `numCourses - 1`。 +> 给你一个数组 `prerequisites` ,其中 `prerequisites[i] = [a$_i$, b$_i$]` ,表示在选修课程 `a_i` 前**必须**先选修 `b_i` 。 +> * 返回你为了学完所有课程所安排的学习顺序。 +> * 可能会有多个正确的顺序,你只要返回**任意一种**就可以了。 +> * 如果不可能完成所有课程,返回**一个空数组**。 **分析** * 这两道题例题就是最经典的拓扑排序应用,我们直接套用上述模板即可。 From 1a8d31a202d990989d4a8c261c635eb4bedde9ec Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Mon, 4 Oct 2021 19:46:48 -0700 Subject: [PATCH 12/24] cosmetic change --- TopologicalSort/README.md | 111 +++++++++++++++++++++----------------- 1 file changed, 61 insertions(+), 50 deletions(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index 6a0ff11..6e7c8e0 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -7,7 +7,7 @@ 1. 每个结点只出现一次。 2. 若存在一条从结点 A 指向结点 B 的路径,则在序列当中 A 出现在 B 之前。 -若一幅图存在环路,则无法进行拓扑排序。另外,对于一幅图,其拓扑排序结果并不唯一。我们采用《算法导论》第22章的规范来定义图:$G=(V,E)$,其中 $V$ 是代表结点的集合,$E$代表边的集合,$|V|$ 和 $|E|$ 分别代表结点和边的个数。$(i,j) \in E$ 代表存在一个从结点 $i$ 指向结点 $j$ 的有向边。因为我们本章中主要研究稀疏图 ( sparse graph ),所以我们采用邻接表( adjacency list )的方式来表示图,其空间开销是 $O(|V|+|E|)$。 +若一幅图存在环路,则无法进行拓扑排序。另外,对于一幅图,其拓扑排序结果并不唯一。我们采用《算法导论》第 22 章的规范来定义图:$G=(V,E)$,其中 $V$ 是代表结点的集合,$E$ 代表边的集合,$|V|$ 和 $|E|$ 分别代表结点和边的个数。$(u,v) \in E$ 代表存在一个从结点 $u$ 指向结点 $v$ 的有向边。因为我们本章中主要研究稀疏图 ( sparse graph ),所以我们采用邻接表( adjacency list )的方式来表示图,其空间开销是 $O(|V|+|E|)$。 例子:如下图所示,我们定义了一幅有向无环图, @@ -38,21 +38,17 @@ graph[3].neighbors = [graph[4], graph[5]] ## 拓扑排序的两个模板 -使用以下两个模板,我们可以判断一幅图是否可以进行拓扑排序,如果无法进行拓扑排序,输出空列表,如果可以进行拓扑排序则输出一种可能的拓扑排序之后的序列。 +使用以下两个模板,我们可以判断一幅图是否可以进行拓扑排序,如果无法进行拓扑排序,则输出空列表;如果可以进行拓扑排序,则输出一种可能的拓扑排序之后的序列。 ### 宽度优先搜索 ( Breadth-first search, BFS ): #### 思路 -该算法也叫 [Kahn 算法](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm)。首先,我们需要介绍**入度**( indegree )的概念。在有向图当中,一个结点的入度表示有多少条边指向它。对应的还有一个概念叫**出度** ( outdegree ),代表从一个结点出发有多少条边,但是在拓扑排序当中我们只会用到入度。入度为 0 的点的集合将作为 BFS 的起点。如果没有入度为零的点,则无法进行拓扑排序。BFS 由先进先出 (First In, First Out, FIFO)的队列 ( queue )数据结构来实现。BFS 算法实现拓扑排序的文字描述如下: +该算法也叫 [Kahn 算法](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm)。首先,我们需要介绍**入度**( indegree )的概念。在有向图当中,一个结点的入度表示有多少条边指向它。对应的还有一个概念叫**出度** ( outdegree ),代表从一个结点出发有多少条边,但是在拓扑排序当中我们只会用到入度。入度为 0 的点的集合将作为 BFS 的起点。如果没有入度为 0 的点,则无法进行拓扑排序。BFS 由先进先出 (First In, First Out, FIFO)的队列 ( queue )数据结构来实现。BFS 算法实现拓扑排序的伪代码如下: -1. 遍历邻接表中所有结点和边,计算每个结点的入度。 -2. 从入度为 0 的结点出发进行 BFS ,将入度为 0 的结点放入输出列表。遍历入度为 0 结点的邻接结点时将这些邻接结点的入度减 1(相当于删掉已经遍历过的边),如果其入度是 0 ,则放入队列。 - -伪代码如下 ```python topoOrder ← 输出列表,初始为空 -queue ← 包含所有入度为 0 结点的队列 +queue ← 包含所有入度为 0 的结点的队列 while ( queue 不为空 ) do n 出列 queue 将 n 放入 topoOrder @@ -61,18 +57,14 @@ while ( queue 不为空 ) do if ( m 入度为 0 ) then m 入列 queue -if (图中存在边) then +if ( 图中存在边 ) then # 存在环,不可以拓扑排序 return None else return topoOrder ``` -之前说过拓扑排序结果并不唯一,按照字典序 ( lexicographical order )可以将所有可能的输出结果进行排序。如果需要最大或者最小结果,可以将 Kahn 算法中的队列替换成最大堆/最小堆实现的优先队列即可,具体实现参考[精选例题X]。 - -**时间复杂度**:计算入度的循环需要遍历每个结点和每条边,BFS 循环也要遍历每一条边并且将每个结点入列和出列一次,入列和出列操作都是 $O(1)$, 所以总的时间复杂度是 $O(|V|+|E|)$ 。 - -**空间复杂度**:我们需要一个数组或者字典来存储每个结点的入度,还需要一个队列来存储入度为 0 的结点,总的空间复杂度为 $O(|V|)$ 。 +之前说过拓扑排序结果并不唯一,按照字典序 ( lexicographical order )可以将所有可能的输出结果进行排序。如果需要最大或者最小结果,可以将上述算法中的队列替换成最大堆/最小堆实现的优先队列即可,具体实现请参考精选例题。 #### 代码 @@ -83,24 +75,21 @@ import collections class Solution(object): def topSort(self, graph: Dict[int, List[int]]) -> List[int]: - """ - @type: 邻接表表示的有向图 graph - @rtype: 图的结点列表 topoOrder - """ - # 计算入度 + # 计算入度 inDegree = self.getInDegree(graph) # 将入度为 0 的结点放入队列 queue 和哈希集合 visited 防止重复遍历 startNodeList = [ node for node in graph if inDegree[node] == 0] - # 用 deque 实现队列 queue = collections.deque(startNodeList) visited = set(startNodeList) + # 保存结果的数组 topoOrder = [] - # 判断是否存在环 + + # count 用于判断是否存在环 count = len(graph) - # 广度优先搜索 + # BFS while queue: node = queue.popleft() topoOrder.append(node) @@ -113,11 +102,18 @@ class Solution(object): queue.append(neighbor) visited.add(neighbor) + # 存在环,返回空列表 if count != 0: return [] return topoOrder ``` -为了形象的理解 BFS 的过程,我们将代码的执行过程以及对应的队列 `queue` 和 `topoOrder` 的值都表示在下面的图中。 +**复杂度分析** + +* 时间复杂度:计算入度的循环需要遍历每个结点和每条边,BFS 循环也要遍历每一条边并且将每个结点入列和出列一次,入列和出列操作都是 $O(1)$, 所以总的时间复杂度是 $O(|V|+|E|)$ 。 + +* 空间复杂度:我们需要一个数组或者字典来存储每个结点的入度,还需要一个队列来存储入度为 0 的结点,总的空间复杂度为 $O(|V|)$ 。 + +为了形象的理解 BFS 的过程,我们将代码的执行过程以及对应的 `queue` 和 `topoOrder` 的值都表示在下图中。 ![BFS](/Users/billywu/Coding/AlgoWiki/AlgoWiki/TopologicalSort/TopoSort_BFS.png) @@ -125,7 +121,7 @@ class Solution(object): #### 思路 -一句话总结就是**拓扑排序就是图后序遍历的倒序输出**。我们现在将这句话展开来理解。根据《算法导论》第 22 章,我们可以在 DFS 的过程中给每个结点盖两个时间戳,分别是发现这个结点的时间 `discoverTime[u]` 和遍历完该结点所有邻接结点的时间 `finishTime[u]` ,其中 `u` 代表一个结点。对于结点 `u`,DFS 算法遍历完其所有子结点之后才会判断其遍历结束,更新时间戳 `finishTime[u]` 。可以证明如果图中存在一条从 `u` 指向 `v` 的边,那么 `finishTime[v] < f[u]` 。所以,按照 `finishTime[u]` 的逆序输出结点就可以得到拓扑排序之后的序列。该算法有点类似回溯算法,我们需要在递归之前和递归之后更新每个结点的状态。每个结点有三种的可能的状态,分别是: +一句话总结就是**拓扑排序就是图后序遍历的倒序输出**。我们现在将这句话展开来理解。根据《算法导论》第 22 章,我们可以在 DFS 的过程中给每个结点盖两个时间戳,分别是发现这个结点的时间 `discoverTime[u]` 和遍历完该结点所有邻接结点的时间 `finishTime[u]` ,其中 `u` 代表一个结点。对于结点 `u`,DFS 算法遍历完其所有子结点之后才会判断其遍历结束,更新时间戳 `finishTime[u]` 。可以证明如果图中存在一条从 `u` 指向 `v` 的边,那么 `finishTime[v] < finishTime[u]` 。所以,按照 `finishTime` 的逆序输出结点就可以得到拓扑排序之后的序列。该算法有点类似回溯算法,我们需要在递归之前和递归之后更新每个结点的状态。每个结点有三种的可能的状态,分别是: * `UNVISITED`:未被发现; * `VISITING`:已发现,但是邻接结点还未遍历完; * `VISITED`:已发现,邻接结点已经遍历完。 @@ -146,9 +142,11 @@ while 存在状态为 UNVISITED 的结点 do function visit(node n) if n 的状态为 VISITING then - stop (not a DAG) + # 存在环,无法进行拓扑排序 + stop if n 的状态为 VISITED then + # 不要重复遍历 return 将 n 的状态标记为 VISITING @@ -160,9 +158,7 @@ function visit(node n) 将 n 加入 topoOrder 的首位 ``` -**时间复杂度**:在 DFS 过程中,每一个结点和每一条边都会被访问一次,将一个结点按照完成时间 `finishTime[u]` 的顺序放入输出数组需要 $O(1)$ 的时间,所以总的时间复杂度是 $O(|V|+|E|)$。 - -**空间复杂度**:我们需要 `states` 数组或者字典来保存每个结点的状态,所以空间复杂度是 $O(|V|)$ 。 +#### 代码 ```python from typing import * @@ -182,18 +178,20 @@ class State(Enum): class Solution(object): def __init__(self): + # 全局时间 self.time = 0 - self.finishTime = collections.defaultdict(lambda: 0) + # 发现时间 self.discoverTime = collections.defaultdict(lambda: 0) + # 完成时间 + self.finishTime = collections.defaultdict(lambda: 0) def topSort(self, graph: List[DirectedGraphNode]) -> List[int]: - numNodes = len(graph) states = {node: State.UNVISITED for node in graph} res = [] - for i in range(numNodes): + for node in graph: # 如果存在环,返回空列表 - if (self.hasCycle(graph, graph[i], states, res)): + if (self.hasCycle(graph, node, states, res)): return [] # 倒序输出后序遍历结果 @@ -233,6 +231,12 @@ class Solution(object): return False ``` +**复杂度分析** + +* 时间复杂度:在 DFS 过程中,每一个结点和每一条边都会被访问一次,将一个结点按照完成时间 `finishTime[u]` 的顺序放入输出数组需要 $O(1)$ 的时间,所以总的时间复杂度是 $O(|V|+|E|)$。 + +* 空间复杂度:我们需要 `states` 数组或者字典来保存每个结点的状态,所以空间复杂度是 $O(|V|)$ 。 + 为了形象的理解 DFS 的过程,我们将上述代码的执行过程以及每个结点的发现时间 `discoverTime` 和 完成时间 `finishTime` 的值都表示在下面的图中。我们用白色代表 `UNVISITED` ,灰色代表 `VISITING`,黑色代表 `VISITED`。 ![DFS_B](/Users/billywu/Coding/AlgoWiki/AlgoWiki/TopologicalSort/TopoSort_DFS_B.png) @@ -246,15 +250,17 @@ class Solution(object): 例题 2: 「力扣」 第 210 题:[课程表 II(Course Schedule II)](https://leetcode-cn.com/problems/course-schedule-ii/) > 现在你总共有 `numCourses` 门课需要选,记为 `0`到 `numCourses - 1`。 -> 给你一个数组 `prerequisites` ,其中 `prerequisites[i] = [a$_i$, b$_i$]` ,表示在选修课程 `a_i` 前**必须**先选修 `b_i` 。 +> 给你一个数组 `prerequisites` ,其中 `prerequisites[i] = [a, b]` ,表示在选修课程 `a` 前**必须**先选修 `b` 。 +> > * 返回你为了学完所有课程所安排的学习顺序。 > * 可能会有多个正确的顺序,你只要返回**任意一种**就可以了。 > * 如果不可能完成所有课程,返回**一个空数组**。 **分析** + * 这两道题例题就是最经典的拓扑排序应用,我们直接套用上述模板即可。 -BFS 答案 +BFS 代码: ```python from typing import * @@ -264,10 +270,10 @@ import collections class Solution(object): def findOrder(self, numCourses: int, prerequisites: List[List[int]]) -> List[int]: + # 通过 prerequisites 构造图 graph = self.buildGraph(numCourses, prerequisites) - + # 计算入度 inDegree = {i: 0 for i in range(numCourses)} - for prerequisite in prerequisites: inDegree[prerequisite[0]] += 1 @@ -276,6 +282,7 @@ class Solution(object): visited = set(startNodeList) topoOrder = [] count = numCourses + # BFS while queue: node = queue.popleft() topoOrder.append(node) @@ -290,7 +297,8 @@ class Solution(object): if inDegree[child] == 0: queue.append(child) visited.add(child) - + + # 判断是否存在环 if count != 0: return [] @@ -305,13 +313,13 @@ class Solution(object): return graph ``` -DFS 答案 +DFS 代码: ```python from typing import * from enum import Enum - +# 结点的三种状态 class State(Enum): UNVISITED = 0 VISITING = 1 @@ -321,15 +329,18 @@ class State(Enum): class Solution(object): def findOrder(self, numCourses: int, prerequisites: List[List[int]]) -> List[int]: + # 通过 prerequisites 构造图 graph = self.buildGraph(numCourses, prerequisites) - + # 初始化结点状态 states = {node: State.UNVISITED for node in graph} res = [] - + + # DFS for node in graph: if (self.hasCycle(graph, node, states, res)): return [] - + + # 逆序输出后序遍历结果 return res[::-1] def buildGraph(self, N: int, relations: List[List[int]]) -> Dict[int, List[int]]: @@ -360,21 +371,22 @@ class Solution(object): ``` **复杂度分析**: -* 时间复杂度:$O(|V|+|E|)$,其中 $V$ 是所有字符的集合,$E$ 是所有字典序线索的集合。 -* 空间复杂度:$O(|V|+|E|)$ + +* 时间复杂度:$O(|V|+|E|)$,其中 $V$ 是所有课程的集合,$E$ 是所有先修课程要求的集合。 +* 空间复杂度:$O(|V|+|E|)$。 ### 题型二:最大/最小字典序拓扑排序 -例题 3: Lintcode 第892题,[外星人字典( Alien Dictionary)](https://www.lintcode.com/problem/892/) +例题 3: LintCode 第892题,[外星人字典( Alien Dictionary)](https://www.lintcode.com/problem/892/) > 有一种新的使用拉丁字母的外来语言。但是,你不知道字母之间的顺序。你会从词典中收到一个非空的单词列表,其中的单词在这种新语言的规则下按字典顺序排序。请推导出这种语言的字母顺序。 > * 假设所有字母都是小写。 -> * 如果单词 a 是单词 b 的前缀且 b 出现在 a 之前,那么这个顺序是无效的。 +> * 如果单词 `a` 是单词 `b` 的前缀且 `b` 出现在 `a` 之前,那么这个顺序是无效的。 > * 如果顺序是无效的,则返回空字符串。 > * 这里可能有多个有效的字母顺序,返回以正常字典顺序看来最小的。 **分析**: -* 字典序可以看成一幅图,结点是所有出现过的拉丁字母,有向边 $(u, v)$ 代表在字典序当中字母 $u$ 排在字母 $v$ 前面。 +* 字典序可以看成一幅图,结点是所有出现过的拉丁字母,有向边 `(a, b)` 代表在字典序当中字母 `a` 排在字母 `b` 前面。 * 通过比较相邻的单词,可以得到关于该字典序的线索,将其作为边加入图。 * 为了返回用正常英文字典顺序看来最小的结果,用 BFS 遍历的时候,需要保证每次出列的字符都是队列中最小的。 @@ -400,7 +412,6 @@ class Solution(object): return ''.join(topoOrder) def buildGraph(self, words: List[str]): - graph = {char: [] for word in words for char in word} for i in range(1, len(words)): @@ -411,15 +422,15 @@ class Solution(object): graph[w1[i]].append(w2[i]) break - # 如果w2是w1的前缀且w1出现在w2之前,则不存在有效的字典序 + # 如果 w2 是 w1 的前缀且 w1 出现在 w2 之前,则不存在有效的字典序 if i == min(len(w1), len(w2))-1 and len(w1) > len(w2): return None return graph def topoSort_BFS_PriorityQueue(self, graph: Dict[str, List[str]]): + # 计算入度 inDegree = {char: 0 for char in graph} - for char, neighbors in graph.items(): for neighbor in neighbors: inDegree[neighbor] += 1 From 312757b41ad1cde14c13fd83fbb75e5b54ed6938 Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Sun, 31 Oct 2021 22:37:46 -0700 Subject: [PATCH 13/24] Add Java and C++ solution for LC 210. --- TopologicalSort/README.md | 112 +++++++++++++++++++++++++++++++++++++- 1 file changed, 111 insertions(+), 1 deletion(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index 6e7c8e0..6b35d0f 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -262,6 +262,10 @@ class Solution(object): BFS 代码: + + +#### **Python3** + ```python from typing import * import collections @@ -312,11 +316,117 @@ class Solution(object): return graph ``` +#### **C++** +```c++ +class Solution { + public: + vector findOrder(int numCourses, vector>& prerequisites) { + vector> graph(numCourses); + vector inDegree(numCourses, 0); + + for (auto& prerequisite : prerequisites) { + const int in = prerequisite[0]; + const int out = prerequisite[1]; + + graph[out].push_back(in); + inDegree[in]++; + } + + vector topoOrder; + set visited; + queue queue; + + for (int i = 0; i < numCourses; i++) { + if (inDegree[i] == 0) { + queue.push(i); + visited.insert(i); + } + } + + while (!queue.empty()) { + const int node = queue.front(); + queue.pop(); + topoOrder.push_back(node); + + for (const int child : graph[node]) { + if (visited.find(child) != visited.end()) { + continue; + } + + inDegree[child]--; + if (inDegree[child] == 0) { + queue.push(child); + visited.insert(child); + } + } + } + + if (topoOrder.size() == numCourses) { + return topoOrder; + } + return vector(); + } +}; +``` + +#### **Java** + +```java +class Solution { + public static int[] findOrder(int numCourses, int[][] prerequisites) { + List[] graph = new List[numCourses]; + int[] inDegree = new int[numCourses]; + + for (int i = 0; i < graph.length; i++) { + graph[i] = new ArrayList<>(); + } + + for (int[] prerequisite : prerequisites) { + final int out = prerequisite[1]; + final int in = prerequisite[0]; + graph[out].add(in); + inDegree[in]++; + } + + List topoOrder = new ArrayList<>(); + HashSet visited = new HashSet(); + Queue queue = new LinkedList(); + + for (int i = 0; i < numCourses; i++) { + if (inDegree[i] == 0) { + queue.offer(i); + visited.add(i); + } + } + + while (!queue.isEmpty()) { + final int node = queue.poll(); + topoOrder.add(node); + + for (final int child : graph[node]) { + if (visited.contains(child)) + continue; + inDegree[child]--; + if (inDegree[child] == 0) { + queue.offer(child); + visited.add(child); + } + } + } + + if (topoOrder.size() == numCourses) { + return topoOrder.stream().mapToInt(i -> i).toArray(); + } + return new int[] {}; + } +} +``` + + DFS 代码: ```python -from typing import * from enum import Enum # 结点的三种状态 From be784d30918b3cb1a755527fba8ececc62d5774c Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Sun, 31 Oct 2021 22:43:57 -0700 Subject: [PATCH 14/24] Use default path for figures. --- TopologicalSort/README.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index 6b35d0f..ef9f9ef 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -28,7 +28,7 @@ graph[3].neighbors = [graph[4], graph[5]] 其拓扑排序可以是: `0->1->2->3->4->5`,也可以是 `0->3->2->5->1->4`。 -![TopoSort](/Users/billywu/Coding/AlgoWiki/AlgoWiki/TopologicalSort/TopoSort.png) +![TopoSort](TopoSort.png) ## 拓扑排序的应用 @@ -115,7 +115,7 @@ class Solution(object): 为了形象的理解 BFS 的过程,我们将代码的执行过程以及对应的 `queue` 和 `topoOrder` 的值都表示在下图中。 -![BFS](/Users/billywu/Coding/AlgoWiki/AlgoWiki/TopologicalSort/TopoSort_BFS.png) +![BFS](TopoSort_BFS.png) ### 深度优先搜索( Depth-first search, DFS) @@ -130,7 +130,7 @@ class Solution(object): 沿用上面的例子,DFS 算法得到的拓扑排序序列和每个结点的发现/完成时间如下图所示。 -![DFS_A](/Users/billywu/Coding/AlgoWiki/AlgoWiki/TopologicalSort/TopoSort_DFS_A.png) +![DFS_A](TopoSort_DFS_A.png) DFS 算法的伪代码如下: @@ -239,7 +239,7 @@ class Solution(object): 为了形象的理解 DFS 的过程,我们将上述代码的执行过程以及每个结点的发现时间 `discoverTime` 和 完成时间 `finishTime` 的值都表示在下面的图中。我们用白色代表 `UNVISITED` ,灰色代表 `VISITING`,黑色代表 `VISITED`。 -![DFS_B](/Users/billywu/Coding/AlgoWiki/AlgoWiki/TopologicalSort/TopoSort_DFS_B.png) +![DFS_B](TopoSort_DFS_B.png) ## 精选例题 From 01c471e7b9aed23d5cdcb76daa2e96c5e47bba68 Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Wed, 3 Nov 2021 16:34:49 -0700 Subject: [PATCH 15/24] Add Java and C++ solutions for LintCode 892 Alien Dictionary. --- TopologicalSort/README.md | 280 ++++++++++++++++++++++++++++++++------ 1 file changed, 236 insertions(+), 44 deletions(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index ef9f9ef..449caa2 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -426,6 +426,10 @@ class Solution { DFS 代码: + + +#### **Python** + ```python from enum import Enum @@ -480,6 +484,8 @@ class Solution(object): return False ``` + + **复杂度分析**: * 时间复杂度:$O(|V|+|E|)$,其中 $V$ 是所有课程的集合,$E$ 是所有先修课程要求的集合。 @@ -502,77 +508,263 @@ class Solution(object): 以下是 BFS + 最小堆实现的代码。 + + +#### **Python** + ```python from typing import * -import heapq +import heapq class Solution(object): - def alienOrder(self, words: List[str]): - # 比较相邻单词构造图 - graph = self.buildGraph(words) + def alienOrder(self, words: List[str]) -> str: + graph, inDegree = self.buildGraph(words) + if not graph: return "" + + # BFS + minheap = [char for char in graph if inDegree[char] == 0] + heapq.heapify(minheap) + visited = set(minheap) + topoOrder = "" + count = len(inDegree) + + while minheap: + node = heapq.heappop(minheap) + topoOrder += node + count -= 1 - # 如果存在环, 则不存在有效的字典序 - topoOrder = self.topoSort_BFS_PriorityQueue(graph) - if not topoOrder: + for child in graph[node]: + if child in visited: + continue + inDegree[child] -= 1 + if inDegree[child] == 0: + heapq.heappush(minheap, child) + visited.add(child) + + if count != 0: return "" - return ''.join(topoOrder) + return topoOrder - def buildGraph(self, words: List[str]): - graph = {char: [] for word in words for char in word} + def buildGraph(self, words: List[str]) -> Tuple[Dict[str, Set[str]], Dict[str, int]]: + graph = {char: set() for word in words for char in word} + inDegree = {char: 0 for char in graph} + # Build graph for i in range(1, len(words)): w1 = words[i-1] w2 = words[i] - for i in range(min(len(w1), len(w2))): - if w1[i] != w2[i]: - graph[w1[i]].append(w2[i]) + for j in range(min(len(w1), len(w2))): + if w1[j] != w2[j]: + graph[w1[j]].add(w2[j]) break - # 如果 w2 是 w1 的前缀且 w1 出现在 w2 之前,则不存在有效的字典序 - if i == min(len(w1), len(w2))-1 and len(w1) > len(w2): - return None + # Check invalid input order: e.g., w1="abc", w2="ab" + if j == min(len(w1), len(w2)) - 1 and len(w1) > len(w2): + graph.clear() + inDegree.clear() + return {}, {} - return graph + # Build inDegree + for char in graph: + for child in graph[char]: + inDegree[child] += 1 - def topoSort_BFS_PriorityQueue(self, graph: Dict[str, List[str]]): - # 计算入度 - inDegree = {char: 0 for char in graph} - for char, neighbors in graph.items(): - for neighbor in neighbors: - inDegree[neighbor] += 1 + return graph, inDegree +``` - pq = [char for char in graph.keys() if inDegree[char] == 0] +#### *Java* - # 最小堆实现的优先队列 - heapq.heapify(pq) - visited = set(pq) - topoOrder = [] - count = len(inDegree) +```java +public class Solution { + public static String alienOrder(String[] words) { + Map> graph = new HashMap<>(); + Map inDegree = new HashMap<>(); + buildGraph(words, graph, inDegree); + + if (graph.isEmpty()) + return ""; + + // BFS + StringBuilder topoOrder = new StringBuilder(); + PriorityQueue minHeap = new PriorityQueue<>(); + Set visited = new HashSet<>(); + + for (final char c : inDegree.keySet()) { + if (inDegree.get(c) == 0) { + minHeap.offer(c); + visited.add(c); + } + } - while pq: - node = heapq.heappop(pq) - topoOrder.append(node) - count -= 1 + int count = inDegree.size(); - for child in graph[node]: - if child in visited: - continue - inDegree[child] -= 1 - if inDegree[child] == 0: - heapq.heappush(pq, child) - visited.add(child) + while (!minHeap.isEmpty()) { + final char node = minHeap.poll(); + topoOrder.append(node); + count--; + for (final char child : graph.get(node)) { + if (visited.contains(child)) + continue; - # 存在环,不存在有效的字典序 - if count != 0: - return [] + inDegree.put(child, inDegree.get(child) - 1); - return topoOrder + if (inDegree.get(child) == 0) { + minHeap.offer(child); + visited.add(child); + } + } + } + + if (count != 0) + return ""; + + return topoOrder.toString(); + } + + private static void buildGraph(String[] words, Map> graph, + Map inDegree) { + // Initialize graph and InDegree + for (final String word : words) { + for (final char c : word.toCharArray()) { + graph.putIfAbsent(c, new HashSet()); + inDegree.putIfAbsent(c, 0); + } + } + + // Build graph + for (int i = 1; i < words.length; i++) { + final String w1 = words[i - 1]; + final String w2 = words[i]; + final int minLen = Math.min(w1.length(), w2.length()); + for (int j = 0; j < minLen; j++) { + final char u = w1.charAt(j); + final char v = w2.charAt(j); + if (u != v) { + graph.get(u).add(v); + break; + } + + // Check invalid input order: e.g., w1="abc", w2="ab" + if (j == minLen - 1 && w1.length() > w2.length()) { + graph.clear(); + inDegree.clear(); + return; + } + } + } + + // Build inDegree + for (final Map.Entry> entry : graph.entrySet()) { + final char c = entry.getKey(); + final Set children = entry.getValue(); + for (final char child : children) { + inDegree.put(child, inDegree.get(child) + 1); + } + } + } +} +``` + +#### **C++** + +```c++ +class Solution { + public: + string alienOrder(vector& words) { + unordered_map> graph; + unordered_map inDegree; + buildGraph(words, graph, inDegree); + + if (graph.empty()) return ""; + + // BFS + priority_queue, greater> minHeap; + unordered_set visited; + string topoOrder; + + for (const auto& [c, count] : inDegree) { + if (count == 0) { + minHeap.push(c); + visited.insert(c); + } + } + + int count = inDegree.size(); + + while (!minHeap.empty()) { + const auto node = minHeap.top(); + + minHeap.pop(); + topoOrder += node; + count--; + + for (const auto child : graph[node]) { + if (visited.find(child) != visited.end()) continue; + inDegree[child]--; + if (inDegree[child] == 0) { + minHeap.push(child); + visited.insert(child); + } + } + } + + if (count != 0) return ""; + + return topoOrder; + } + + private: + // Initialize graph and inDegree + void buildGraph(vector& words, + unordered_map>& graph, + unordered_map& inDegree) { + + // Initialize graph and inDegree + for (const auto& word : words) { + for (const auto& c : word) { + graph[c] = unordered_set(); + inDegree[c] = 0; + } + } + + // Build graph + for (int i = 1; i < words.size(); i++) { + const auto w1 = words[i - 1]; + const auto w2 = words[i]; + const int minLen = min(w1.length(), w2.length()); + for (int j = 0; j < minLen; j++) { + if (w1[j] != w2[j]) { + graph[w1[j]].insert(w2[j]); + break; + } + + // Check invalid input order: e.g., w1="abc", w2="ab" + if (j == minLen - 1 && w1.length() > w2.length()) { + graph.clear(); + inDegree.clear(); + return; + } + } + } + + // Build inDegree + for (const auto& [c, children] : graph) { + for (const auto& child : children) { + inDegree[child]++; + } + } + + return; + } +}; ``` + + **复杂度分析**: * 时间复杂度:$O(|V|+|E|)$,其中 $V$ 是所有字符的集合,$E$ 是所有字典序线索的集合。 From ca001a473f10ccc8155ac1cb444b63511dcbd7d5 Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Wed, 3 Nov 2021 22:31:11 -0700 Subject: [PATCH 16/24] Add Java and Python DFS solutions to LeetCode 210 Course Schedule II. --- TopologicalSort/README.md | 375 ++++++++++++++++++++++++-------------- 1 file changed, 243 insertions(+), 132 deletions(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index 449caa2..4f585e9 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -316,7 +316,9 @@ class Solution(object): return graph ``` + #### **C++** + ```c++ class Solution { public: @@ -324,7 +326,7 @@ class Solution { vector> graph(numCourses); vector inDegree(numCourses, 0); - for (auto& prerequisite : prerequisites) { + for (const auto& prerequisite : prerequisites) { const int in = prerequisite[0]; const int out = prerequisite[1]; @@ -373,52 +375,52 @@ class Solution { ```java class Solution { - public static int[] findOrder(int numCourses, int[][] prerequisites) { - List[] graph = new List[numCourses]; - int[] inDegree = new int[numCourses]; + public static int[] findOrder(int numCourses, int[][] prerequisites) { + Map> graph = new HashMap>(); + int[] inDegree = new int[numCourses]; - for (int i = 0; i < graph.length; i++) { - graph[i] = new ArrayList<>(); - } + for (int i = 0; i < numCourses; i++) + graph.put(i, new ArrayList()); - for (int[] prerequisite : prerequisites) { - final int out = prerequisite[1]; - final int in = prerequisite[0]; - graph[out].add(in); - inDegree[in]++; - } + for (final int[] prerequisite : prerequisites) { + final int out = prerequisite[1]; + final int in = prerequisite[0]; + graph.get(out).add(in); + inDegree[in]++; + } - List topoOrder = new ArrayList<>(); - HashSet visited = new HashSet(); - Queue queue = new LinkedList(); + List topoOrder = new ArrayList<>(); + HashSet visited = new HashSet(); + Queue queue = new LinkedList(); - for (int i = 0; i < numCourses; i++) { - if (inDegree[i] == 0) { - queue.offer(i); - visited.add(i); - } - } + for (int i = 0; i < numCourses; i++) { + if (inDegree[i] == 0) { + queue.offer(i); + visited.add(i); + } + } - while (!queue.isEmpty()) { - final int node = queue.poll(); - topoOrder.add(node); - - for (final int child : graph[node]) { - if (visited.contains(child)) - continue; - inDegree[child]--; - if (inDegree[child] == 0) { - queue.offer(child); - visited.add(child); - } - } - } + while (!queue.isEmpty()) { + final int node = queue.poll(); + topoOrder.add(node); - if (topoOrder.size() == numCourses) { - return topoOrder.stream().mapToInt(i -> i).toArray(); + for (final int child : graph.get(node)) { + if (visited.contains(child)) + continue; + inDegree[child]--; + if (inDegree[child] == 0) { + queue.offer(child); + visited.add(child); } - return new int[] {}; + } + } + + if (topoOrder.size() == numCourses) { + return topoOrder.stream().mapToInt(i -> i).toArray(); } + + return new int[] {}; + } } ``` @@ -484,6 +486,115 @@ class Solution(object): return False ``` +#### **C++** + +```c++ +enum State { UNVISITED, VISITING, VISITED }; + +class Solution { + public: + vector findOrder(int numCourses, vector>& prerequisites) { + vector> graph(numCourses); + + // Build graph and inDegree + for (const auto& prerequisite : prerequisites) { + const int in = prerequisite[0]; + const int out = prerequisite[1]; + + graph[out].push_back(in); + } + + vector states(numCourses); + vector res; + + // DFS + for (int node = 0; node < graph.size(); node++) { + if (hasCycle(graph, node, states, res)) { + return {}; + } + } + + reverse(res.begin(), res.end()); + return res; + } + + private: + bool hasCycle(const vector>& graph, const int node, + vector& states, vector& res) { + if (states[node] == VISITING) return true; + if (states[node] == VISITED) return false; + + states[node] = VISITING; + + for (const auto& child : graph[node]) { + if (hasCycle(graph, child, states, res)) return true; + } + + res.push_back(node); + states[node] = VISITED; + return false; + } +}; +``` + +#### **Java** + +```java +class Solution { + public enum States { + UNVISITED, VISITING, VISITED + } + + public static int[] findOrder(int numCourses, int[][] prerequisites) { + Map> graph = new HashMap>(); + States[] states = new States[numCourses]; + ArrayList res = new ArrayList(); + + Arrays.fill(states, States.UNVISITED); + + for (int i = 0; i < numCourses; i++) + graph.put(i, new ArrayList()); + + for (final int[] prerequisite : prerequisites) { + final int in = prerequisite[0]; + final int out = prerequisite[1]; + graph.get(out).add(in); + } + + for (int i = 0; i < numCourses; i++) { + if (hasCycle(graph, i, states, res)) { + return new int[] {}; + } + } + + Collections.reverse(res); + + return res.stream().mapToInt(i -> i).toArray(); + } + + private static boolean hasCycle(Map> graph, int node, States[] states, + ArrayList res) { + if (states[node] == States.VISITING) + return true; + if (states[node] == States.VISITED) + return false; + + states[node] = States.VISITING; + + for (final int child : graph.get(node)) { + if (hasCycle(graph, child, states, res)) { + return true; + } + } + + res.add(node); + states[node] = States.VISITED; + + return false; + } +} +``` + **复杂度分析**: @@ -576,99 +687,6 @@ class Solution(object): return graph, inDegree ``` -#### *Java* - -```java -public class Solution { - public static String alienOrder(String[] words) { - Map> graph = new HashMap<>(); - Map inDegree = new HashMap<>(); - buildGraph(words, graph, inDegree); - - if (graph.isEmpty()) - return ""; - - // BFS - StringBuilder topoOrder = new StringBuilder(); - PriorityQueue minHeap = new PriorityQueue<>(); - Set visited = new HashSet<>(); - - for (final char c : inDegree.keySet()) { - if (inDegree.get(c) == 0) { - minHeap.offer(c); - visited.add(c); - } - } - - int count = inDegree.size(); - - while (!minHeap.isEmpty()) { - final char node = minHeap.poll(); - topoOrder.append(node); - count--; - for (final char child : graph.get(node)) { - if (visited.contains(child)) - continue; - - inDegree.put(child, inDegree.get(child) - 1); - - if (inDegree.get(child) == 0) { - minHeap.offer(child); - visited.add(child); - } - } - } - - if (count != 0) - return ""; - - return topoOrder.toString(); - } - - private static void buildGraph(String[] words, Map> graph, - Map inDegree) { - // Initialize graph and InDegree - for (final String word : words) { - for (final char c : word.toCharArray()) { - graph.putIfAbsent(c, new HashSet()); - inDegree.putIfAbsent(c, 0); - } - } - - // Build graph - for (int i = 1; i < words.length; i++) { - final String w1 = words[i - 1]; - final String w2 = words[i]; - final int minLen = Math.min(w1.length(), w2.length()); - for (int j = 0; j < minLen; j++) { - final char u = w1.charAt(j); - final char v = w2.charAt(j); - if (u != v) { - graph.get(u).add(v); - break; - } - - // Check invalid input order: e.g., w1="abc", w2="ab" - if (j == minLen - 1 && w1.length() > w2.length()) { - graph.clear(); - inDegree.clear(); - return; - } - } - } - - // Build inDegree - for (final Map.Entry> entry : graph.entrySet()) { - final char c = entry.getKey(); - final Set children = entry.getValue(); - for (final char child : children) { - inDegree.put(child, inDegree.get(child) + 1); - } - } - } -} -``` - #### **C++** ```c++ @@ -763,6 +781,99 @@ class Solution { }; ``` +#### **Java** + +```java +public class Solution { + public static String alienOrder(String[] words) { + Map> graph = new HashMap<>(); + Map inDegree = new HashMap<>(); + buildGraph(words, graph, inDegree); + + if (graph.isEmpty()) + return ""; + + // BFS + StringBuilder topoOrder = new StringBuilder(); + PriorityQueue minHeap = new PriorityQueue<>(); + Set visited = new HashSet<>(); + + for (final char c : inDegree.keySet()) { + if (inDegree.get(c) == 0) { + minHeap.offer(c); + visited.add(c); + } + } + + int count = inDegree.size(); + + while (!minHeap.isEmpty()) { + final char node = minHeap.poll(); + topoOrder.append(node); + count--; + for (final char child : graph.get(node)) { + if (visited.contains(child)) + continue; + + inDegree.put(child, inDegree.get(child) - 1); + + if (inDegree.get(child) == 0) { + minHeap.offer(child); + visited.add(child); + } + } + } + + if (count != 0) + return ""; + + return topoOrder.toString(); + } + + private static void buildGraph(String[] words, Map> graph, + Map inDegree) { + // Initialize graph and InDegree + for (final String word : words) { + for (final char c : word.toCharArray()) { + graph.putIfAbsent(c, new HashSet()); + inDegree.putIfAbsent(c, 0); + } + } + + // Build graph + for (int i = 1; i < words.length; i++) { + final String w1 = words[i - 1]; + final String w2 = words[i]; + final int minLen = Math.min(w1.length(), w2.length()); + for (int j = 0; j < minLen; j++) { + final char u = w1.charAt(j); + final char v = w2.charAt(j); + if (u != v) { + graph.get(u).add(v); + break; + } + + // Check invalid input order: e.g., w1="abc", w2="ab" + if (j == minLen - 1 && w1.length() > w2.length()) { + graph.clear(); + inDegree.clear(); + return; + } + } + } + + // Build inDegree + for (final Map.Entry> entry : graph.entrySet()) { + final char c = entry.getKey(); + final Set children = entry.getValue(); + for (final char child : children) { + inDegree.put(child, inDegree.get(child) + 1); + } + } + } +} +``` + **复杂度分析**: From bd2e33035bd96ce5af1ef480092ee43e907b3a13 Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Thu, 4 Nov 2021 22:42:21 -0700 Subject: [PATCH 17/24] Added LeetCode 310 as the 3nd example of topological sort. 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zpIqnOyY<=AJ}GyUysRHZi}IFQsz9(oQ{*;!Pu$$JqhxROeFf94k?t>)3rsw{14y09 zqY|rP5k!J8MEKXhZ>056j;E_0s(`|5TAD1c+6|})zf)V2cH|clWlQ$PhV)1Q@7ja1 zfY0ke7EUqIZBSgMzE0>#=>;sfzLbiF>HQP-&jCIt(2G~<8tfzVXG>?_RAHgA#Po18 z>eQ4}P`8acl~mY4XLg~$?J!xx@V_1s1U1~+W?*6D5wOB)RNA*-d3LsUO+QxXBP6p~ z5tQR|q<``ubw+h#udrwXbuSg}!MXJ=McWdmE*Q<^qB+>!?roG5L|NJYFle7&krsx$ zQ#MAC#UJkXH~Rd{1giKyGlL;T;WL=!BFud!vVX&U!nlUeYNaV)_r<=8kh9F2{5tUY zEdVlPJi075o#KlQH?M{!x4{jbP#rBxu%ybX{4s@aAR$bd+yOXY3a|0(tmCP{&l2In)unvvIEWHnE&~ke;u_ 给你一棵包含 `n` 个节点的树,标记为 `0` 到 `n-1` 。给定数字 `n` 和一个有 `n-1` 条无向边的 `edges` 列表,其中 `edges[i] = [ai, bi]` 表示树中节点 `ai` 和 `bi` 之间存在一条无向边。 +> 可选择树中任何一个节点作为根。当选择节点 `x` 作为根节点时,设结果树的高度为 `h` 。在所有可能的树中,具有最小高度的树(即 `min(h)` )被称为最小高度树 。 +> 请你找到所有的 最小高度树 并按**任意顺序**返回它们的根节点标签列表。 +> 树的**高度**是指根节点和叶子节点之间最长向下路径上边的数量。 +> +> * `1<= n <= 2 * 10^4`。 +> * 给定的输入保证是一棵树,不会有重复的边。 + +**分析**: +* 这道题也可以从每个节点出发用BFS得到以当前节点为根的树的高度,然后取对应高度最小的那些根节点。但是这样效率显然很低,有大量的重复计算。采用拓扑排序的方法,只需要一次遍历就可以得到最小高度树对应的根节点列表。 +* 首先,一个重要的结论是:这样的根节点最多只有 2 个。假设有 3 个及以上这样的根节点,根据提议,这些根节点也必须是相连的,那么就可以以靠中间的节点为根的树就会比以靠边节点为根的树高,从而可以反证得到最小高度树的根节点数量只能小于等于 2 。 +* 通过层级遍历的方法,我们可以从所有叶子节点出发,每次往上走一步,直到最后只省一个或者两个节点,那么这些节点就是我们要找的最小高度树的根节点。 +* 注意,这里给的 `edges` 对应的是无向图,所以 `[1, 2]` 实际代表了两条有向边,在构造邻接表的过程中要增加 `1->2` 和 `2->1` 两条边。与前几道题不同的是,这里的叶子节点判断条件是入度为 1 。 +* 针对一个例子( `n=10`, `edges = {{0, 1}, {0, 2}, {0, 3}, {2, 4}, {0, 5}, {5, 6}, {6, 7}, {2, 8}, {7, 9}}` ),下图展示了从叶子节点每一步 BFS 层级遍历时队列 queue 里面存放的节点。白色的节点代表还没被遍历,黑色的节点已经被遍历。 + +![Minimum_Height_Trees](Minimum_Height_Trees.png) + +以下是 BFS 实现的拓扑排序代码。 + + + +#### **Python** + +```python +class Solution: + def findMinHeightTrees(self, n: int, edges: List[List[int]]) -> List[int]: + if n <= 1: + return [i for i in range(n)] + + res = [] + # Build directed graph + graph = {i: [] for i in range(n)} + inDegree = {i: 0 for i in range(n)} + for edge in edges: + graph[edge[0]].append(edge[1]) + inDegree[edge[1]] += 1 + graph[edge[1]].append(edge[0]) + inDegree[edge[0]] += 1 + + # BFS + startNodes = [i for i in range(n) if inDegree[i] == 1] + queue = collections.deque(startNodes) + visited = set(startNodes) + + count = n + while queue: + if count <= 2: + break + # Level-order traversal + size = len(queue) + for _ in range(size): + node = queue.popleft() + count -= 1 + + for child in graph[node]: + if child in visited: + continue + + inDegree[child] -= 1 + if inDegree[child] == 1: + queue.append(child) + visited.add(child) + + # Convert queue to list + return list(queue) +``` + +#### **C++** + +```c++ +class Solution { + public: + /** + * @param n: n nodes labeled from 0 to n - 1 + * @param edges: a undirected graph + * @return: a list of all the MHTs root labels + */ + vector findMinHeightTrees(int n, vector> &edges) { + if (n == 1) return vector(1, 0); + vector res; + + // Build graph and inDegree + vector> graph(n); + vector inDegree(n); + + for (const auto &edge : edges) { + const auto from = edge[0]; + const auto to = edge[1]; + graph[from].push_back(to); + inDegree[to]++; + graph[to].push_back(from); + inDegree[from]++; + } + + // BFS + queue q; + unordered_set visited; + for (int i = 0; i < n; i++) { + if (inDegree[i] == 1) { + q.push(i); + visited.insert(i); + } + } + + int count = n; + while (!q.empty()) { + if (count <= 2) { + break; + } + // Level-order traversal + int size = q.size(); + for (int i = 0; i < size; i++) { + int node = q.front(); + q.pop(); + count--; + + for (const auto &child : graph[node]) { + if (visited.find(child) != visited.end()) continue; + inDegree[child]--; + if (inDegree[child] == 1) { + q.push(child); + visited.insert(child); + } + } + } + } + + // Convert queue to vector + while (!q.empty()) { + res.emplace_back(move(q.front())); + q.pop(); + } + + return res; + } +}; +``` + +#### **Java** + +```java +class Solution { + public List findMinHeightTrees(int n, int[][] edges) { + List res = new ArrayList<>(); + if (n == 1) { + res.add(0); + return res; + } + + // Build graph and inDegree + Map> graph = new HashMap<>(); + int[] inDegree = new int[n]; + + for (int i = 0; i < n; i++) { + graph.put(i, new ArrayList()); + } + + for (final int[] edge : edges) { + final int n0 = edge[0]; + final int n1 = edge[1]; + graph.get(n0).add(n1); + inDegree[n1]++; + graph.get(n1).add(n0); + inDegree[n0]++; + } + + // BFS + Queue q = new LinkedList(); + Set visited = new HashSet(); + int count = n; + + for (int i = 0; i < n; i++) { + if (inDegree[i] == 1) { + q.offer(i); + visited.add(i); + } + } + + while (!q.isEmpty()) { + if (count <= 2) { + break; + } + // Level-order traversal + final int size = q.size(); + for (int i = 0; i < size; i++) { + final int node = q.poll(); + count--; + + for (final int child : graph.get(node)) { + if (visited.contains(child)) + continue; + inDegree[child]--; + if (inDegree[child] == 1) { + q.offer(child); + visited.add(child); + } + } + } + } + + // Convert Queue into List + while (!q.isEmpty()) { + int node = q.poll(); + res.add(node); + } + + return res; + } +} +``` + + + +**复杂度分析**: + +* 时间复杂度:$O(n)$,`n` 为节点个数。 +* 空间复杂度:$O(n)$。 + ## 精选练习 | 题目 | 类型 | | ------------------------------------------------------------ | ---------------------------- | -| [LC 1136. 平行课程](https://leetcode-cn.com/problems/parallel-courses/) | 中等,必做 | -| [LC 310. 最小高度树](https://leetcode-cn.com/problems/minimum-height-trees/) | 中等, 必做 | -| [LC 329. 矩阵中的最长递增路径](https://leetcode-cn.com/problems/longest-increasing-path-in-a-matrix/) | 困难,必做 | -| [CF 1385E. Directing Edges](https://codeforces.com/problemset/problem/1385/E) | | - +| [LeetCode 1136. 平行课程](https://leetcode-cn.com/problems/parallel-courses/) | 中等,必做 | +| [LeetCode 1462. 课程表 IV](https://leetcode.com/problems/course-schedule-iv/) | 中等, 必做 | +| [LeetCode 329. 矩阵中的最长递增路径](https://leetcode-cn.com/problems/longest-increasing-path-in-a-matrix/) | 困难,必做 | +| [LintCode 605. 序列重构](https://www.lintcode.com/problem/605/) | 中等, 必做| ## 参考 From 379935dbf70d317bce6509fcc7c2e1ba522eabbd Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Mon, 3 Jan 2022 16:44:13 -0800 Subject: [PATCH 18/24] 1. Capitalized `Directed Acyclic Graph`. 2. Used Baidu Baike's definition of topological sorting. 3. Splited the first plot into two such that the linear series are more understandable. 4. Added more description of BFS algorithm before presenting the code template. 5. Added discussion that DFS is not required in most cases. --- TopologicalSort/README.md | 20 ++++++++++++-------- TopologicalSort/TopoSort.png | Bin 261573 -> 0 bytes TopologicalSort/TopoSort_DAG.png | Bin 0 -> 91746 bytes TopologicalSort/TopoSort_not_unique.png | Bin 0 -> 151080 bytes 4 files changed, 12 insertions(+), 8 deletions(-) delete mode 100644 TopologicalSort/TopoSort.png create mode 100644 TopologicalSort/TopoSort_DAG.png create mode 100644 TopologicalSort/TopoSort_not_unique.png diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index 6966378..52bfa84 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -2,14 +2,16 @@ 作者:wu2meng3 -拓扑排序是一个有向无环图( directed acyclic graph ,简称 DAG )所有结点的线性序列,满足以下两个条件: +对一个有向无环图( Directed Acyclic Graph ,简称 DAG )进行拓扑排序,是指将图中所有结点排成一个线性序列,并满足以下两个条件: 1. 每个结点只出现一次。 2. 若存在一条从结点 A 指向结点 B 的路径,则在序列当中 A 出现在 B 之前。 若一幅图存在环路,则无法进行拓扑排序。另外,对于一幅图,其拓扑排序结果并不唯一。我们采用《算法导论》第 22 章的规范来定义图:$G=(V,E)$,其中 $V$ 是代表结点的集合,$E$ 代表边的集合,$|V|$ 和 $|E|$ 分别代表结点和边的个数。$(u,v) \in E$ 代表存在一个从结点 $u$ 指向结点 $v$ 的有向边。因为我们本章中主要研究稀疏图 ( sparse graph ),所以我们采用邻接表( adjacency list )的方式来表示图,其空间开销是 $O(|V|+|E|)$。 -例子:如下图所示,我们定义了一幅有向无环图, +例子:下图中,我们定义了一幅有向无环图, + +![TopoSort_DAG](TopoSort_DAG.png) ```python class DirectedGraphNode(object): @@ -26,9 +28,9 @@ graph[2].neighbors = [graph[4], graph[5]] graph[3].neighbors = [graph[4], graph[5]] ``` -其拓扑排序可以是: `0->1->2->3->4->5`,也可以是 `0->3->2->5->1->4`。 +如图所示,其拓扑排序的结果可以是: (a) `0->1->2->3->4->5` 或者 (b) `0->3->2->5->1->4`。 -![TopoSort](TopoSort.png) +![TopoSort_not_unique](TopoSort_not_unique.png) ## 拓扑排序的应用 @@ -46,6 +48,10 @@ graph[3].neighbors = [graph[4], graph[5]] 该算法也叫 [Kahn 算法](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm)。首先,我们需要介绍**入度**( indegree )的概念。在有向图当中,一个结点的入度表示有多少条边指向它。对应的还有一个概念叫**出度** ( outdegree ),代表从一个结点出发有多少条边,但是在拓扑排序当中我们只会用到入度。入度为 0 的点的集合将作为 BFS 的起点。如果没有入度为 0 的点,则无法进行拓扑排序。BFS 由先进先出 (First In, First Out, FIFO)的队列 ( queue )数据结构来实现。BFS 算法实现拓扑排序的伪代码如下: +BFS 算法理解起来非常直观,我们先拿出不被任何结点指向的节点(入度为 0 )的结点,然后把它从图中删去(同时更新余下所有结点的入度)并加入到输出序列。我们用一个队列 `queue` 来维护当前图中所有结点,用 `topoOrder` 保存输出序列。具体执行过程以及对应的 `queue` 和 `topoOrder` 的值都展示在下图中。 + +![BFS](TopoSort_BFS.png) + ```python topoOrder ← 输出列表,初始为空 queue ← 包含所有入度为 0 的结点的队列 @@ -113,12 +119,10 @@ class Solution(object): * 空间复杂度:我们需要一个数组或者字典来存储每个结点的入度,还需要一个队列来存储入度为 0 的结点,总的空间复杂度为 $O(|V|)$ 。 -为了形象的理解 BFS 的过程,我们将代码的执行过程以及对应的 `queue` 和 `topoOrder` 的值都表示在下图中。 - -![BFS](TopoSort_BFS.png) - ### 深度优先搜索( Depth-first search, DFS) +对于拓扑排序而言,读者需要熟练掌握 BFS 解法,因为其更常见,且思路相对自然。 DFS 解法可以作为补充,了解即可。 + #### 思路 一句话总结就是**拓扑排序就是图后序遍历的倒序输出**。我们现在将这句话展开来理解。根据《算法导论》第 22 章,我们可以在 DFS 的过程中给每个结点盖两个时间戳,分别是发现这个结点的时间 `discoverTime[u]` 和遍历完该结点所有邻接结点的时间 `finishTime[u]` ,其中 `u` 代表一个结点。对于结点 `u`,DFS 算法遍历完其所有子结点之后才会判断其遍历结束,更新时间戳 `finishTime[u]` 。可以证明如果图中存在一条从 `u` 指向 `v` 的边,那么 `finishTime[v] < finishTime[u]` 。所以,按照 `finishTime` 的逆序输出结点就可以得到拓扑排序之后的序列。该算法有点类似回溯算法,我们需要在递归之前和递归之后更新每个结点的状态。每个结点有三种的可能的状态,分别是: diff --git a/TopologicalSort/TopoSort.png 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z*e?D0mXQQ3XZ$O#Gnsy6)}9&Srl#-2VqEygI-`@u|1}Q)4kz(VR8$}~e)U&f8hHE& z9eaCi=&bAX%7}U5O^_|<^QbSuUP!*$?w~^Hud5^)2+RS2u3JWhVXL|XbFwalT__vk zmzuel>h#!hL- zZv2~j-)Fa>{%OgjizhQCQm;Q9`Xx6Ob^+)vks4-fFm*zPHr?3Gs#TnkG2Lx`b8>>o z3gh}N5!E{K9%R>Cx6XUsMXY^sP3Gst@Sj?pGwq8eIiGJ$E#e0cF6HA*!^Z|fPWC1T zhuckm9w=XKKOGUkp)X*h7+iaBJk&Pg`N&V-t_e}o)#=YMgYGh)G(RYMKeznuId@%g zetw=iQ~*u*ZJUdS3E=W7V`&p%q=9R&Zp7yr|6e((;O)PEoIXK++5jRRfnB= Date: Mon, 3 Jan 2022 16:50:32 -0800 Subject: [PATCH 19/24] `queue` is for all nodes with indegree = 0. --- TopologicalSort/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index 52bfa84..f3cdcd9 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -48,7 +48,7 @@ graph[3].neighbors = [graph[4], graph[5]] 该算法也叫 [Kahn 算法](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm)。首先,我们需要介绍**入度**( indegree )的概念。在有向图当中,一个结点的入度表示有多少条边指向它。对应的还有一个概念叫**出度** ( outdegree ),代表从一个结点出发有多少条边,但是在拓扑排序当中我们只会用到入度。入度为 0 的点的集合将作为 BFS 的起点。如果没有入度为 0 的点,则无法进行拓扑排序。BFS 由先进先出 (First In, First Out, FIFO)的队列 ( queue )数据结构来实现。BFS 算法实现拓扑排序的伪代码如下: -BFS 算法理解起来非常直观,我们先拿出不被任何结点指向的节点(入度为 0 )的结点,然后把它从图中删去(同时更新余下所有结点的入度)并加入到输出序列。我们用一个队列 `queue` 来维护当前图中所有结点,用 `topoOrder` 保存输出序列。具体执行过程以及对应的 `queue` 和 `topoOrder` 的值都展示在下图中。 +BFS 算法理解起来非常直观,我们先拿出不被任何结点指向的节点(入度为 0 )的结点,然后把它从图中删去(同时更新余下所有结点的入度)并加入到输出序列。我们用一个队列 `queue` 来维护当前图中所有入度为 0 的结点,用 `topoOrder` 保存输出序列。具体执行过程以及对应的 `queue` 和 `topoOrder` 的值都展示在下图中。 ![BFS](TopoSort_BFS.png) From 9dfb2923e41b34ff8b4330d8ec65b99ba9224cef Mon Sep 17 00:00:00 2001 From: Meng Wu Date: Mon, 3 Jan 2022 16:53:04 -0800 Subject: [PATCH 20/24] Cosmetic change 1. Add space before and after `BFS`. 2. queue should be `queue`. --- TopologicalSort/README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index f3cdcd9..cde4c4d 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -898,11 +898,11 @@ public class Solution { > * 给定的输入保证是一棵树,不会有重复的边。 **分析**: -* 这道题也可以从每个节点出发用BFS得到以当前节点为根的树的高度,然后取对应高度最小的那些根节点。但是这样效率显然很低,有大量的重复计算。采用拓扑排序的方法,只需要一次遍历就可以得到最小高度树对应的根节点列表。 +* 这道题也可以从每个节点出发用 BFS 得到以当前节点为根的树的高度,然后取对应高度最小的那些根节点。但是这样效率显然很低,有大量的重复计算。采用拓扑排序的方法,只需要一次遍历就可以得到最小高度树对应的根节点列表。 * 首先,一个重要的结论是:这样的根节点最多只有 2 个。假设有 3 个及以上这样的根节点,根据提议,这些根节点也必须是相连的,那么就可以以靠中间的节点为根的树就会比以靠边节点为根的树高,从而可以反证得到最小高度树的根节点数量只能小于等于 2 。 * 通过层级遍历的方法,我们可以从所有叶子节点出发,每次往上走一步,直到最后只省一个或者两个节点,那么这些节点就是我们要找的最小高度树的根节点。 * 注意,这里给的 `edges` 对应的是无向图,所以 `[1, 2]` 实际代表了两条有向边,在构造邻接表的过程中要增加 `1->2` 和 `2->1` 两条边。与前几道题不同的是,这里的叶子节点判断条件是入度为 1 。 -* 针对一个例子( `n=10`, `edges = {{0, 1}, {0, 2}, {0, 3}, {2, 4}, {0, 5}, {5, 6}, {6, 7}, {2, 8}, {7, 9}}` ),下图展示了从叶子节点每一步 BFS 层级遍历时队列 queue 里面存放的节点。白色的节点代表还没被遍历,黑色的节点已经被遍历。 +* 针对一个例子( `n=10`, `edges = {{0, 1}, {0, 2}, {0, 3}, {2, 4}, {0, 5}, {5, 6}, {6, 7}, {2, 8}, {7, 9}}` ),下图展示了从叶子节点每一步 BFS 层级遍历时队列 `queue` 里面存放的节点。白色的节点代表还没被遍历,黑色的节点已经被遍历。 ![Minimum_Height_Trees](Minimum_Height_Trees.png) From 4dc524f73221f7f2bfd3d56f63efd12320359a85 Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Mon, 3 Jan 2022 16:54:28 -0800 Subject: [PATCH 21/24] Shrinked the first plot. --- TopologicalSort/TopoSort_DAG.png | Bin 91746 -> 72155 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/TopologicalSort/TopoSort_DAG.png b/TopologicalSort/TopoSort_DAG.png index b2042827c6d9263b7436a2f9a7be475491e0d1db..4fdc339f6c3c9f11e0e6b82e8d58a3396d2e80c8 100644 GIT binary patch literal 72155 zcmeFZWk6J27d8wy;D8_uA*G~rNl6Ge;LzQTN_V#?-60?iQj*e*geU@nNOy^JNlOay z?E$@?c%S$8_vgKT+#Y7m*=Oy&*S^-;*ShwEDk(_fV3A>=prGJLOCeNHP#`oYC}_%1 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zPYFv*2?wu{1i=Zmy24eL6>XzE&#h)H;Cqz6jM6!oP4puc>N6Tz6{D>M^7CZ-O}aIy znmi`_=~Eu}8N!}WoN6(X^UYTma;H(D!z*-m4DuTU3^mAoNu2fu^az`%0EM3Zy>{bnX5km^!$ySVOA{`qFfv!)>2iAI>_WA|Nnnm;lLIT zk3&Fz7#)9nT`~a(U`xZ6-%*19S~ED6jV=F0BmawU+3CBIB?L+9oIHBy!p4pNNy!^T z{^w~>MbHLc|KH*34PO3}dN+9a&(ql8<-bn@7*79-b@T==|9M{y3BSP|5xQT5eDD+nlZ^IP%TlsE;2>w#O8*as4 z%6B8C{YN1E-@yKcDY9XTY*=3#*4Ku?xnXc_*w}yR0`!KBy8*ash rTe0CuZFo`}7_h(ME&rpiHwvowg1d-d-KRbX{5yF}?`Y~_^Lzgf;^CEk From 63248d4be6f1bbafc938b0a6ea105a302f534c41 Mon Sep 17 00:00:00 2001 From: Meng Wu Date: Mon, 3 Jan 2022 17:01:17 -0800 Subject: [PATCH 22/24] Reorganize the pseudocode of BFS --- TopologicalSort/README.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index cde4c4d..f63f651 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -46,12 +46,14 @@ graph[3].neighbors = [graph[4], graph[5]] #### 思路 -该算法也叫 [Kahn 算法](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm)。首先,我们需要介绍**入度**( indegree )的概念。在有向图当中,一个结点的入度表示有多少条边指向它。对应的还有一个概念叫**出度** ( outdegree ),代表从一个结点出发有多少条边,但是在拓扑排序当中我们只会用到入度。入度为 0 的点的集合将作为 BFS 的起点。如果没有入度为 0 的点,则无法进行拓扑排序。BFS 由先进先出 (First In, First Out, FIFO)的队列 ( queue )数据结构来实现。BFS 算法实现拓扑排序的伪代码如下: +该算法也叫 [Kahn 算法](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm)。首先,我们需要介绍**入度**( indegree )的概念。在有向图当中,一个结点的入度表示有多少条边指向它。对应的还有一个概念叫**出度** ( outdegree ),代表从一个结点出发有多少条边,但是在拓扑排序当中我们只会用到入度。入度为 0 的点的集合将作为 BFS 的起点。如果没有入度为 0 的点,则无法进行拓扑排序。BFS 由先进先出 (First In, First Out, FIFO)的队列 ( queue )数据结构来实现。 BFS 算法理解起来非常直观,我们先拿出不被任何结点指向的节点(入度为 0 )的结点,然后把它从图中删去(同时更新余下所有结点的入度)并加入到输出序列。我们用一个队列 `queue` 来维护当前图中所有入度为 0 的结点,用 `topoOrder` 保存输出序列。具体执行过程以及对应的 `queue` 和 `topoOrder` 的值都展示在下图中。 ![BFS](TopoSort_BFS.png) +BFS 算法实现拓扑排序的伪代码如下: + ```python topoOrder ← 输出列表,初始为空 queue ← 包含所有入度为 0 的结点的队列 From e4096b072d58c45b60af9fdea7b5f4cb898ddb16 Mon Sep 17 00:00:00 2001 From: Meng Wu Date: Mon, 3 Jan 2022 17:01:52 -0800 Subject: [PATCH 23/24] Cosmetic change --- TopologicalSort/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index f63f651..a0cd260 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -48,7 +48,7 @@ graph[3].neighbors = [graph[4], graph[5]] 该算法也叫 [Kahn 算法](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm)。首先,我们需要介绍**入度**( indegree )的概念。在有向图当中,一个结点的入度表示有多少条边指向它。对应的还有一个概念叫**出度** ( outdegree ),代表从一个结点出发有多少条边,但是在拓扑排序当中我们只会用到入度。入度为 0 的点的集合将作为 BFS 的起点。如果没有入度为 0 的点,则无法进行拓扑排序。BFS 由先进先出 (First In, First Out, FIFO)的队列 ( queue )数据结构来实现。 -BFS 算法理解起来非常直观,我们先拿出不被任何结点指向的节点(入度为 0 )的结点,然后把它从图中删去(同时更新余下所有结点的入度)并加入到输出序列。我们用一个队列 `queue` 来维护当前图中所有入度为 0 的结点,用 `topoOrder` 保存输出序列。具体执行过程以及对应的 `queue` 和 `topoOrder` 的值都展示在下图中。 +BFS 算法理解起来非常直观,我们先拿出不被任何结点指向的结点(入度为 0 )的结点,然后把它从图中删去(同时更新余下所有结点的入度)并加入到输出序列。我们用一个队列 `queue` 来维护当前图中所有入度为 0 的结点,用 `topoOrder` 保存输出序列。具体执行过程以及对应的 `queue` 和 `topoOrder` 的值都展示在下图中。 ![BFS](TopoSort_BFS.png) From b7329dfc16aa982c14f5120535eebf92e7d33386 Mon Sep 17 00:00:00 2001 From: Meng Wu Date: Mon, 3 Jan 2022 17:02:36 -0800 Subject: [PATCH 24/24] Cosmetic change --- TopologicalSort/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index a0cd260..32a082c 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -48,7 +48,7 @@ graph[3].neighbors = [graph[4], graph[5]] 该算法也叫 [Kahn 算法](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm)。首先,我们需要介绍**入度**( indegree )的概念。在有向图当中,一个结点的入度表示有多少条边指向它。对应的还有一个概念叫**出度** ( outdegree ),代表从一个结点出发有多少条边,但是在拓扑排序当中我们只会用到入度。入度为 0 的点的集合将作为 BFS 的起点。如果没有入度为 0 的点,则无法进行拓扑排序。BFS 由先进先出 (First In, First Out, FIFO)的队列 ( queue )数据结构来实现。 -BFS 算法理解起来非常直观,我们先拿出不被任何结点指向的结点(入度为 0 )的结点,然后把它从图中删去(同时更新余下所有结点的入度)并加入到输出序列。我们用一个队列 `queue` 来维护当前图中所有入度为 0 的结点,用 `topoOrder` 保存输出序列。具体执行过程以及对应的 `queue` 和 `topoOrder` 的值都展示在下图中。 +BFS 算法理解起来非常直观,我们先拿出不被任何结点指向(入度为 0 )的结点,然后把它从图中删去(同时更新余下所有结点的入度)并加入到输出序列。我们用一个队列 `queue` 来维护当前图中所有入度为 0 的结点,用 `topoOrder` 保存输出序列。具体执行过程以及对应的 `queue` 和 `topoOrder` 的值都展示在下图中。 ![BFS](TopoSort_BFS.png)

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+例子:如下图所示,我们定义了一幅有向无环图, + +```python +class DirectedGraphNode: + + def __init__(self, x): + self.label = x + self.neighbors = [] + +graph = [DirectedGraphNode(i) for i in range(0, 6)] +graph[0].neighbors = [graph[1], graph[2], graph[3]] +graph[1].neighbors = [graph[4]] +graph[2].neighbors = [graph[4], graph[5]] +graph[3].neighbors = [graph[4], graph[5]] +``` + +其拓扑排序可以是: `0->1->2->3->4->5`,也可以是 `0->3->2->5->1->4`。 + +![BFS](/Users/billywu/Coding/AlgoWiki/AlgoWiki/TopologicalSort/BFS.png) + +![DFS](/Users/billywu/Coding/AlgoWiki/AlgoWiki/TopologicalSort/DFS.png) + ## 拓扑排序的应用 + + ## 拓扑排序的两个模板 -* BFS +* 宽度优先搜索: + + 0. 构造图的邻接表。 + 1. 遍历所有结点,计算每个结点的入度。 + 2. 从入度为0的结点出发进行宽度优先搜索,将入度为0的结点放入输出列表。遍历入度为0结点的邻接结点时将其入度减一,如果入度为0,放入队列。 + + * 时间复杂度:$O(|V|+|E|)$ + * 空间复杂度:$O(|V|)$ + ```python +import collections + class Solution(object): def topSort(self, graph: Dict[int, List[int]]) -> List[int]: @@ -15,10 +56,9 @@ class Solution(object): @type: 邻接表表示的有向图 @rtype: 图的结点列表 """ - import collections # 计算入度 inDegree = self.getInDegree(graph) - + # 将入度为0的结点放入队列 queue 和哈希集合 visited 防止重复遍历 startNodeList = [ node for node in graph if inDegree[node] == 0] # 用 deque 实现队列 @@ -26,6 +66,8 @@ class Solution(object): visited = set(startNodeList) # 保存结果的数组 topoOrder = [] + # 判断是否存在环 + count = len(graph) # 广度优先搜索 while queue: @@ -40,13 +82,67 @@ class Solution(object): queue.append(neighbor) visited.add(neighbor) + if count != 0: + return [] return topoOrder ``` -* DFS +* 深度优先搜索:一句话总结就是**拓扑排序就是后序遍历的倒序输出**。 + * 时间复杂度:$O(|V|+|E|)$ + * 空间复杂度:$O(|V|)$ + +```python +from enum import Enum + +class State(Enum): + UNVISITED = 0 + VISITING = 1 + VISITED = 2 + +class Solution: + + def topSort(self, graph: List[DirectedGraphNode]) -> List[int]: + numNodes = len(graph) + states = {node: State.UNVISITED for node in graph} + res = [] + + for i in range(numNodes): + # 如果存在环,返回空列表 + if (self.hasCycle(graph, graph[i], states, res)): + return [] + + # 倒序输出后序遍历结果 + return res[::-1] + + # 判断是否存在环,存在则返回 True + def hasCycle(self, graph, node, states, res): + if states[node] == State.VISITING: + return True + + if states[node] == State.VISITED: + return False + + # 回溯 + states[node] = State.VISITING + + # 递归处理子结点,将子结点加入 res 列表 + for child in node.neighbors: + if (self.hasCycle(graph, child, states, res)): + return True + + # 后序遍历 + res.append(node) + states[node] = State.VISITED + + return False +``` ## 精选例题 -### 题型一:有向无环图中的最长路径 (课程表) +### 题型一:图中是否存在环 (课程表) + +### 题型二:有向无环图中的最长路径 (课程表) + +### 题型三:有向无环图的拓扑排序 (课程表) ## 精选练习 From aa2b274a57036bed16315560b2136acd9490b66f Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Sun, 3 Oct 2021 17:12:00 -0700 Subject: [PATCH 04/24] Add a simple application of topological sort. --- TopologicalSort/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index 4033605..41e9764 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -33,7 +33,7 @@ graph[3].neighbors = [graph[4], graph[5]] ## 拓扑排序的应用 - +举个穿衣服的例子,穿鞋子之前要穿袜子,以及先穿衬衫再穿外套。穿袜子和穿鞋子这两个任务之间就存在完成次序的依赖,而穿鞋子和穿衬衫并没有这种依赖。通过拓扑排序算法,我们就可以制定一个按次序完成任务的计划,比如说按照袜子,鞋子,衬衫,外套这样的次序来着装。 ## 拓扑排序的两个模板 From 2f80eb45122a86f15e8cb9f63df900b7f226abc7 Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Sun, 3 Oct 2021 23:09:03 -0700 Subject: [PATCH 05/24] Finish introduction and code templates. Start working on problems. --- TopologicalSort/README.md | 65 +++++++++++++++++++++++++++------------ 1 file changed, 46 insertions(+), 19 deletions(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index 41e9764..550566f 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -2,12 +2,12 @@ 作者:wu2meng3 -拓扑排序是一个有向无环图所有结点的线性序列,满足以下两个条件: +拓扑排序是一个有向无环图( directed acyclic graph ,简称 DAG )所有结点的线性序列,满足以下两个条件: 1. 每个结点只出现一次。 2. 若存在一条从结点 A 指向结点 B 的路径,则在序列当中 A 出现在 B 之前。 -若一幅图存在环路,则无法进行拓扑排序。另外,对于一幅图,其拓扑排序结果并不唯一。 +若一幅图存在环路,则无法进行拓扑排序。另外,对于一幅图,其拓扑排序结果并不唯一。我们采用《算法导论》第22章的规范来定义图:$G=(V,E)$,其中 $V$ 是代表结点的集合,$E$代表边的集合,$|V|$ 和 $|E|$ 分别代表结点和边的个数。$(i,j) \in E$ 代表存在一个从结点 $i$ 指向结点 $j$ 的有向边。因为我们本章中主要研究稀疏图 ( sparse graph ),所以我们采用邻接表( adjacency list )的方式来表示图,其空间开销是 $O(|V|+|E|)$。 例子:如下图所示,我们定义了一幅有向无环图, @@ -35,16 +35,26 @@ graph[3].neighbors = [graph[4], graph[5]] 举个穿衣服的例子,穿鞋子之前要穿袜子,以及先穿衬衫再穿外套。穿袜子和穿鞋子这两个任务之间就存在完成次序的依赖,而穿鞋子和穿衬衫并没有这种依赖。通过拓扑排序算法,我们就可以制定一个按次序完成任务的计划,比如说按照袜子,鞋子,衬衫,外套这样的次序来着装。 +另外,采用 `Makefile` 进行编译的过程中也要用到拓扑排序对于文件和库的依赖关系进行梳理。 + ## 拓扑排序的两个模板 -* 宽度优先搜索: +使用以下两个模板,我们可以判断一幅图是否可以进行拓扑排序,如果无法进行拓扑排序,输出空列表,如果可以进行拓扑排序则输出一种可能的拓扑排序之后的序列。 + +### 宽度优先搜索 ( Breadth-first search, BFS ): + +#### 思路 + +对于 BFS ,我们需要介绍**入度**( indegree )的概念。在有向图当中,一个结点的入度表示有多少条边指向它。对应的还有一个概念叫**出度** ( outdegree ),代表从一个结点出发有多少条边,但是在拓扑排序当中我们只会用到入度。入度为 0 的点的集合将作为 BFS 的起点。如果没有入度为零的点,则无法进行拓扑排序。BFS 由先进先出 (First In, First Out, FIFO)的队列 ( queue )数据结构来实现。BFS 算法实现拓扑排序的文字描述如下: + +1. 遍历邻接表中所有结点和边,计算每个结点的入度。 +2. 从入度为 0 的结点出发进行 BFS ,将入度为 0 的结点放入输出列表。遍历入度为 0 结点的邻接结点时将这些邻接结点的入度减 1,如果其入度是 0 ,则放入队列。 + +**时间复杂度**:计算入度的循环需要遍历每个结点和每条边,BFS 循环也要遍历每一条边并且将每个结点入列和出列一次,入列和出列操作都是 $O(1)$, 所以总的时间复杂度是 $O(|V|+|E|)$ 。 - 0. 构造图的邻接表。 - 1. 遍历所有结点,计算每个结点的入度。 - 2. 从入度为0的结点出发进行宽度优先搜索,将入度为0的结点放入输出列表。遍历入度为0结点的邻接结点时将其入度减一,如果入度为0,放入队列。 +**空间复杂度**:我们需要一个数组或者字典来存储每个结点的入度,还需要一个队列来存储入度为 0 的结点,总的空间复杂度为 $O(|V|)$ 。 - * 时间复杂度:$O(|V|+|E|)$ - * 空间复杂度:$O(|V|)$ +#### 代码 ```python import collections @@ -52,14 +62,14 @@ import collections class Solution(object): def topSort(self, graph: Dict[int, List[int]]) -> List[int]: - """ - @type: 邻接表表示的有向图 - @rtype: 图的结点列表 - """ - # 计算入度 + """ + @type: 邻接表表示的有向图 graph + @rtype: 图的结点列表 topoOrder + """ + # 计算入度 inDegree = self.getInDegree(graph) - # 将入度为0的结点放入队列 queue 和哈希集合 visited 防止重复遍历 + # 将入度为 0 的结点放入队列 queue 和哈希集合 visited 防止重复遍历 startNodeList = [ node for node in graph if inDegree[node] == 0] # 用 deque 实现队列 queue = collections.deque(startNodeList) @@ -87,16 +97,30 @@ class Solution(object): return topoOrder ``` -* 深度优先搜索:一句话总结就是**拓扑排序就是后序遍历的倒序输出**。 - * 时间复杂度:$O(|V|+|E|)$ - * 空间复杂度:$O(|V|)$ +### 深度优先搜索( Depth-first search, DFS) + +#### 思路 + +一句话总结就是**拓扑排序就是图后序遍历的倒序输出**。我们现在将这句话展开来理解。根据《算法导论》第 22 章,我们可以在 DFS 的过程中给每个结点盖两个时间戳,分别是发现这个结点的时间 $d[u]$和遍历完该结点所有邻接结点的时间 $f[u]$,其中 $u \in V$ 。对于一个结点 $u$,DFS 算法遍历完其所有子结点之后才会判断其遍历结束,更新时间戳 $f[u]$。可以证明如果图中存在一条从 $u$ 指向 $v$ 的边,那么 $f[v] < f[u]$ 。所以,按照 $f[u]$ 的逆序输出结点就可以得到拓扑排序之后的序列。该算法有点类似回溯算法,我们需要在递归之前和递归之后更新每个结点的状态。每个结点有三种的可能的状态,分别是: +* `UNVISITED`:未被发现; +* `VISITING`:已发现,但是邻接结点还未遍历完; +* `VISITED`:已发现,邻接结点已经遍历完。 + +所有结点的初始状态都是`UNVISITED`,在遍历某结点的所以邻接结点进入下一层递归之前要将该结点的状态设为 `VISITING`,所有邻接结点都遍历完递归返回之后将其状态更新为 `VISITED`。 + +**时间复杂度**:在 DFS 过程中,每一个结点和每一条边都会被访问一次,将一个结点按照完成时间 $f[u]$的顺序放入输出数组需要 $O(1)$ 的时间,所以总的时间复杂度是 $O(|V|+|E|)$。 + +**空间复杂度**:我们需要 `states` 数组或者字典来保存每个结点的状态,所以空间复杂度是 $O(|V|)$ 。 ```python from enum import Enum - +# 遍历过程中结点的三种状态 class State(Enum): + # 《算法导论》中的 WHITE 状态:未被发现 UNVISITED = 0 + # 《算法导论》中的 GREY 状态:已发现,但是邻接结点还未遍历完 VISITING = 1 + # 《算法导论》中的 BLACK 状态:已发现,邻接结点已经遍历完 VISITED = 2 class Solution: @@ -116,13 +140,15 @@ class Solution: # 判断是否存在环,存在则返回 True def hasCycle(self, graph, node, states, res): + # 如果该结点被发现两次,说明有环 if states[node] == State.VISITING: return True + # 如果该结点已经遍历结束,则该递归分支结束 if states[node] == State.VISITED: return False - # 回溯 + # VISITING 代表该结点已经被发现,但是其邻接结点还未遍历完 states[node] = State.VISITING # 递归处理子结点,将子结点加入 res 列表 @@ -132,6 +158,7 @@ class Solution: # 后序遍历 res.append(node) + # 邻接结点已经遍历结束 states[node] = State.VISITED return False From 9b570665ada3b42685b582cc27e0d1af97fd0d09 Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Mon, 4 Oct 2021 12:20:44 -0700 Subject: [PATCH 06/24] Add description of Kahn's algorithm. Add example of Alien Dictionary (LintCode 892) --- TopologicalSort/README.md | 117 +++++++++++++++++++++++++++++++++++++- 1 file changed, 114 insertions(+), 3 deletions(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index 550566f..5067937 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -45,10 +45,31 @@ graph[3].neighbors = [graph[4], graph[5]] #### 思路 -对于 BFS ,我们需要介绍**入度**( indegree )的概念。在有向图当中,一个结点的入度表示有多少条边指向它。对应的还有一个概念叫**出度** ( outdegree ),代表从一个结点出发有多少条边,但是在拓扑排序当中我们只会用到入度。入度为 0 的点的集合将作为 BFS 的起点。如果没有入度为零的点,则无法进行拓扑排序。BFS 由先进先出 (First In, First Out, FIFO)的队列 ( queue )数据结构来实现。BFS 算法实现拓扑排序的文字描述如下: +该算法也叫 [Kahn 算法](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm)。首先,我们需要介绍**入度**( indegree )的概念。在有向图当中,一个结点的入度表示有多少条边指向它。对应的还有一个概念叫**出度** ( outdegree ),代表从一个结点出发有多少条边,但是在拓扑排序当中我们只会用到入度。入度为 0 的点的集合将作为 BFS 的起点。如果没有入度为零的点,则无法进行拓扑排序。BFS 由先进先出 (First In, First Out, FIFO)的队列 ( queue )数据结构来实现。BFS 算法实现拓扑排序的文字描述如下: 1. 遍历邻接表中所有结点和边,计算每个结点的入度。 -2. 从入度为 0 的结点出发进行 BFS ,将入度为 0 的结点放入输出列表。遍历入度为 0 结点的邻接结点时将这些邻接结点的入度减 1,如果其入度是 0 ,则放入队列。 +2. 从入度为 0 的结点出发进行 BFS ,将入度为 0 的结点放入输出列表。遍历入度为 0 结点的邻接结点时将这些邻接结点的入度减 1(相当于删掉已经遍历过的边),如果其入度是 0 ,则放入队列。 + +伪代码如下 +```python +L ← 输出列表 +Q ← 包含所有入度为 0 结点的队列 +while ( Q 不为空 ) do + n 出列 + 将 n 放入 L + for ( n 的所有邻接结点 m ) do + 删除该边,m 的入度减 1 + if ( m 入度为 0 ) then + m 入列 + +if (图中存在边) then + # 存在环,不可以拓扑排序 + return None +else + return L +``` + +之前说过拓扑排序结果并不唯一,按照字典序 ( lexicographical order )可以将所有可能的输出结果进行排序。如果需要最大或者最小结果,可以将 Kahn 算法中的队列替换成最大堆/最小堆实现的优先队列即可,具体实现参考[精选例题X]。 **时间复杂度**:计算入度的循环需要遍历每个结点和每条边,BFS 循环也要遍历每一条边并且将每个结点入列和出列一次,入列和出列操作都是 $O(1)$, 所以总的时间复杂度是 $O(|V|+|E|)$ 。 @@ -170,6 +191,96 @@ class Solution: ### 题型二:有向无环图中的最长路径 (课程表) -### 题型三:有向无环图的拓扑排序 (课程表) +### 题型三:最大/最小字典序拓扑排序 + +例题 3: Lintcode 第892题,[外星人字典( Alien Dictionary)](https://www.lintcode.com/problem/892/) +> 有一种新的使用拉丁字母的外来语言。但是,你不知道字母之间的顺序。你会从词典中收到一个非空的单词列表,其中的单词在这种新语言的规则下按字典顺序排序。请推导出这种语言的字母顺序。 +> 假设所有字母都是小写。 +> 如果单词 a 是单词 b 的前缀且 b 出现在 a 之前,那么这个顺序是无效的。 +> 如果顺序是无效的,则返回空字符串。 +> 这里可能有多个有效的字母顺序,返回以正常字典顺序看来最小的。 + +**分析**: +* 字典序可以看出一幅图,结点是所有出现过的拉丁字母,有向边 $(u, v)$ 代表在字典序当中字母 $u$ 排在字母 $v$ 前面。 +* 通过比较相邻的单词,可以得到关于该字典序的线索,将其作为边加入图。 +* 为了返回用正常英文字典顺序看来最小的结果,用 BFS 遍历的时候,需要保证每次出列的字符都是队列中最小的。 + + +以下是 BFS + 最小堆实现的代码。 + +```python +class Solution: + def alienOrder(self, words): + # 比较相邻单词构造图 + graph = self.buildGraph(words) + if not graph: + return "" + + # 如果存在环, 则不存在有效的字典序 + topoOrder = self.topoSort_BFS_PriorityQueue(graph) + if not topoOrder: + return "" + + return ''.join(topoOrder) + + def buildGraph(self, words): + + graph = {char: [] for word in words for char in word} + + for i in range(1, len(words)): + w1 = words[i-1] + w2 = words[i] + for i in range(min(len(w1), len(w2))): + if w1[i] != w2[i]: + graph[w1[i]].append(w2[i]) + break + + # 如果w2是w1的前缀且w1出现在w2之前,则不存在有效的字典序 + if i == min(len(w1), len(w2))-1 and len(w1) > len(w2): + return None + + return graph + + def topoSort_BFS_PriorityQueue(self, graph): + inDegree = {char: 0 for char in graph} + + for char, neighbors in graph.items(): + for neighbor in neighbors: + inDegree[neighbor] += 1 + + pq = [char for char in graph.keys() if inDegree[char] == 0] + + # 最小堆实现的优先队列 + heapq.heapify(pq) + visited = set(pq) + topoOrder = [] + count = len(inDegree) + + while pq: + node = heapq.heappop(pq) + topoOrder.append(node) + count -= 1 + + for child in graph[node]: + if child in visited: + continue + inDegree[child] -= 1 + if inDegree[child] == 0: + heapq.heappush(pq, child) + visited.add(child) + + # 存在环,不存在有效的字典序 + if count != 0: + return [] + + return topoOrder +``` + +**复杂度**: +* 时间复杂度:$O(|V|+|E|)$,其中 $V$ 是所有字符的集合,$E$ 是所有字典序线索的集合。 +* 空间复杂度:$O(|V|+|E|)$ + ## 精选练习 + +# 参考 From e59835719c457a6974883804ca4c5956bb64e0b3 Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Mon, 4 Oct 2021 14:16:43 -0700 Subject: [PATCH 07/24] 1. Add solutions of Course Schedule and Course Schedule II. 2. Add exercises. 3. Add references. --- TopologicalSort/README.md | 156 +++++++++++++++++++++++++++++++++----- 1 file changed, 139 insertions(+), 17 deletions(-) diff --git a/TopologicalSort/README.md b/TopologicalSort/README.md index 5067937..e8183e4 100644 --- a/TopologicalSort/README.md +++ b/TopologicalSort/README.md @@ -144,7 +144,7 @@ class State(Enum): # 《算法导论》中的 BLACK 状态:已发现,邻接结点已经遍历完 VISITED = 2 -class Solution: +class Solution(object): def topSort(self, graph: List[DirectedGraphNode]) -> List[int]: numNodes = len(graph) @@ -187,30 +187,141 @@ class Solution: ## 精选例题 -### 题型一:图中是否存在环 (课程表) +### 题型一:课程表 +例题 1: 「力扣」 第 207 题:[课程表(Course Schedule)](https://leetcode-cn.com/problems/course-schedule/) + +例题 2: 「力扣」 第 210 题:[课程表 II(Course Schedule II)](https://leetcode-cn.com/problems/course-schedule-ii/) + +> 现在你总共有 `numCourses` 门课需要选,记为 `0`到 `numCourses - 1`。给你一个数组 `prerequisites` ,其中 `prerequisites[i] = [a$_i$, b$_i$]` ,表示在选修课程 `a_i` 前**必须**先选修 `b_i` 。 +> 返回你为了学完所有课程所安排的学习顺序。可能会有多个正确的顺序,你只要返回**任意一种**就可以了。如果不可能完成所有课程,返回**一个空数组**。 + +**分析** +* 这两道题例题就是最经典的拓扑排序应用,我们直接套用上述模板即可。 + +BFS 答案 + +```python +class Solution(object): + + def findOrder(self, numCourses: int, prerequisites: List[List[int]]) -> List[int]: + graph = self.buildGraph(numCourses, prerequisites) + + inDegree = {i: 0 for i in range(numCourses)} + + for prerequisite in prerequisites: + inDegree[prerequisite[0]] += 1 + + startNodeList = [i for i in range(numCourses) if inDegree[i] == 0] + queue = collections.deque(startNodeList) + visited = set(startNodeList) + topoOrder = [] + count = numCourses + while queue: + node = queue.popleft() + topoOrder.append(node) + count -= 1 + + for child in graph[node]: + if child in visited: + continue + + inDegree[child] -= 1 + + if inDegree[child] == 0: + queue.append(child) + visited.add(child) + + if count != 0: + return [] + + return topoOrder + + def buildGraph(self, N: int, relations: List[List[int]]): + graph = {i: [] for i in range(N)} + + for relation in relations: + graph[relation[1]].append(relation[0]) + + return graph +``` + +DFS 答案 + +```python +from enum import Enum + +class State(Enum): + UNVISITED = 0 + VISITING = 1 + VISITED = 2 + + +class Solution(object): + + def findOrder(self, numCourses: int, prerequisites: List[List[int]]) -> List[int]: + graph = self.buildGraph(numCourses, prerequisites) -### 题型二:有向无环图中的最长路径 (课程表) + states = {node: State.UNVISITED for node in graph} + res = [] + + for node in graph: + if (self.hasCycle(graph, node, states, res)): + return [] + + return res[::-1] -### 题型三:最大/最小字典序拓扑排序 + def buildGraph(self, N: int, relations: List[List[int]]) -> Dict[int, List[int]]: + graph = {i: [] for i in range(N)} + + for relation in relations: + graph[relation[1]].append(relation[0]) + + return graph + + def hasCycle(self, graph: Dict[int, List[int]], node: int, states: List[State], res: List[int]) -> bool: + if states[node] == State.VISITING: + return True + + if states[node] == State.VISITED: + return False + + states[node] = State.VISITING + + for child in graph[node]: + if (self.hasCycle(graph, child, states, res)): + return True + + res.append(node) + states[node] = State.VISITED + + return False +``` + +**复杂度分析**: +* 时间复杂度:$O(|V|+|E|)$,其中 $V$ 是所有字符的集合,$E$ 是所有字典序线索的集合。 +* 空间复杂度:$O(|V|+|E|)$ + +### 题型二:最大/最小字典序拓扑排序 例题 3: Lintcode 第892题,[外星人字典( Alien Dictionary)](https://www.lintcode.com/problem/892/) + > 有一种新的使用拉丁字母的外来语言。但是,你不知道字母之间的顺序。你会从词典中收到一个非空的单词列表,其中的单词在这种新语言的规则下按字典顺序排序。请推导出这种语言的字母顺序。 -> 假设所有字母都是小写。 -> 如果单词 a 是单词 b 的前缀且 b 出现在 a 之前,那么这个顺序是无效的。 -> 如果顺序是无效的,则返回空字符串。 -> 这里可能有多个有效的字母顺序,返回以正常字典顺序看来最小的。 +> * 假设所有字母都是小写。 +> * 如果单词 a 是单词 b 的前缀且 b 出现在 a 之前,那么这个顺序是无效的。 +> * 如果顺序是无效的,则返回空字符串。 +> * 这里可能有多个有效的字母顺序,返回以正常字典顺序看来最小的。 **分析**: -* 字典序可以看出一幅图,结点是所有出现过的拉丁字母,有向边 $(u, v)$ 代表在字典序当中字母 $u$ 排在字母 $v$ 前面。 +* 字典序可以看成一幅图,结点是所有出现过的拉丁字母,有向边 $(u, v)$ 代表在字典序当中字母 $u$ 排在字母 $v$ 前面。 * 通过比较相邻的单词,可以得到关于该字典序的线索,将其作为边加入图。 * 为了返回用正常英文字典顺序看来最小的结果,用 BFS 遍历的时候,需要保证每次出列的字符都是队列中最小的。 - 以下是 BFS + 最小堆实现的代码。 ```python -class Solution: - def alienOrder(self, words): +class Solution(object): + + def alienOrder(self, words: List[str]): # 比较相邻单词构造图 graph = self.buildGraph(words) if not graph: @@ -223,7 +334,7 @@ class Solution: return ''.join(topoOrder) - def buildGraph(self, words): + def buildGraph(self, words: List[str]): graph = {char: [] for word in words for char in word} @@ -241,7 +352,7 @@ class Solution: return graph - def topoSort_BFS_PriorityQueue(self, graph): + def topoSort_BFS_PriorityQueue(self, graph: Dict[str, List[str]]): inDegree = {char: 0 for char in graph} for char, neighbors in graph.items(): @@ -276,11 +387,22 @@ class Solution: return topoOrder ``` -**复杂度**: -* 时间复杂度:$O(|V|+|E|)$,其中 $V$ 是所有字符的集合,$E$ 是所有字典序线索的集合。 -* 空间复杂度:$O(|V|+|E|)$ +**复杂度分析**: +* 时间复杂度:$O(|V|+|E|)$,其中 $V$ 是所有字符的集合,$E$ 是所有字典序线索的集合。 +* 空间复杂度:$O(|V|+|E|)$。 ## 精选练习 +| 题目 | 类型 | +| ------------------------------------------------------------ | ---------------------------- | +| [LC 1136. 平行课程](https://leetcode-cn.com/problems/parallel-courses/) | 中等,必做 | +| [LC 310. 最小高度树](https://leetcode-cn.com/problems/minimum-height-trees/) | 中等, 必做 | +| [LC 329. 矩阵中的最长递增路径](https://leetcode-cn.com/problems/longest-increasing-path-in-a-matrix/) | 困难,必做 | +| [CF 1385E. Directing Edges](https://codeforces.com/problemset/problem/1385/E) | | + + # 参考 +* [Topological sorting @ Wikipedia](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm) +* https://oi-wiki.org/graph/topo/ +* 《算法导论》第 22 章 From 2bd1925ae056ede7dd6a92fad4560e37fbadccf7 Mon Sep 17 00:00:00 2001 From: wu2meng3 Date: Mon, 4 Oct 2021 18:54:56 -0700 Subject: [PATCH 08/24] 1. Reorganize figures, add snapshots of BFS and DFS workflow. --- TopologicalSort/BFS.png | Bin 215836 -> 0 bytes TopologicalSort/DFS.png | Bin 213045 -> 0 bytes TopologicalSort/README.md | 88 ++++++++++++++++++++++++----- TopologicalSort/TopoSort.png | Bin 0 -> 261573 bytes TopologicalSort/TopoSort_BFS.png | Bin 0 -> 244175 bytes TopologicalSort/TopoSort_DFS_A.png | Bin 0 -> 187351 bytes TopologicalSort/TopoSort_DFS_B.png | Bin 0 -> 555744 bytes 7 files changed, 75 insertions(+), 13 deletions(-) delete mode 100644 TopologicalSort/BFS.png delete mode 100644 TopologicalSort/DFS.png create mode 100644 TopologicalSort/TopoSort.png create mode 100644 TopologicalSort/TopoSort_BFS.png create mode 100644 TopologicalSort/TopoSort_DFS_A.png create mode 100644 TopologicalSort/TopoSort_DFS_B.png diff --git a/TopologicalSort/BFS.png b/TopologicalSort/BFS.png deleted file mode 100644 index 0b5b1d9a887250771e12710d5471a16af4b8da79..0000000000000000000000000000000000000000 GIT binary patch literal 0 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