Leetcode 207 Course Schedule 課程表

原題地址

https://leetcode.com/problems/course-schedule/

題目描述

There are a total of n courses you have to take, labeled from 0 to n - 1.
你需要上n個課程，標記爲0~n-1。

Some courses may have prerequisites, for example to take course 0 you have to first take course 1, which is expressed as a pair: [0,1]
有些課程有前導課，比如你要上課程0必須先上課程1，我們用一個整數對[0,1]來表示。

Given the total number of courses and a list of prerequisite pairs, is it possible for you to finish all courses?
給出課程的總數和一系列的課程先導關係對，判斷是否可能上完所有的課程。

For example:
例如：

2, [[1,0]]

There are a total of 2 courses to take. To take course 1 you should have finished course 0. So it is possible.
共有2個課程，你需要先上課程0再上課程1，因此是可能的。

2, [[1,0],[0,1]]

There are a total of 2 courses to take. To take course 1 you should have finished course 0, and to take course 0 you should also have finished course 1. So it is impossible.
共有2個課程，你需要先上課程0再上課程1，而且上課程1之前你必須先上課程0，因此這是不可能的。

Note:
注意：

The input prerequisites is a graph represented by a list of edges, not adjacency matrices.
輸入的先序關係代表着一個圖的邊，不是鄰接矩陣。

Hints:
提示：

1. This problem is equivalent to finding if a cycle exists in a directed graph. If a cycle exists, no topological ordering exists and therefore it will be impossible to take all courses.
   這個問題等價於判定一個有向圖中是否存在環。如果存在環，就不存在拓撲序，也就是不可能上完所有的課程。
2. Topological Sort via DFS.
   拓撲排序可以用深度優先搜索來實現。
3. Topological sort could also be done via BFS.
   拓撲排序可以用廣度優先搜索來實現。

Tags Depth-first Search Breadth-first Search Graph Topological Sort

解題思路

使用廣度優先搜索來實現拓撲排序的解題思路類似於這個題 hihocoder 47 拓撲排序 · 一，所以此處就不做描述了。代碼和測試數據後面給出。

使用深度優先搜索的解題思路跟廣度優先搜索類似，差別在於我們不是根據入度來確定搜索次序，而是根據出度來確定搜索次序，即維護出度爲0的結點的集合，依次移除這些結點，並更新未其相應邊的出度，直至所有結點的出度不再變化，如果最終所有結點的出度都爲0則不存在環。

在判斷有向圖是否有環的問題上，bfs和dfs的區別僅在是按照入度還是出度來考慮問題上。

另外，關於課程的具體順序，詳見Leetcode 210 Course Schedule II

代碼

dfs

class Solution {
public:
    /**
     * 能否完成所有的課程
     * dfs拓撲排序,看有向圖是否存在環
     * @param numCourses 課程數量
     * @param prerequisites 課程先序關係
     * @return 能完成返回true,否則false
     */
    bool canFinish(int numCourses, vector<pair<int, int> >& prerequisites) {
        vector<int> heads(numCourses, -1), degree(numCourses, 0), points, args;
        pair<int, int> p;
        int from, to, count = 0, len = prerequisites.size();

        /* 構造有向圖,鄰接表 */
        for (int i = 0; i < len; ++i) {
            p = prerequisites[i];
            from = p.second;
            to = p.first;
            ++degree[from];
            args.push_back(heads[to]);
            points.push_back(from);
            heads[to] = count++;
        }

        /* dfs拓撲排序,依次移除出度爲0的點 */
        queue<int> q;
        for (int i = 0; i < numCourses; ++i) 
            if (degree[i] == 0) q.push(i);
        while (!q.empty()) {
            to = q.front();
            q.pop();
            from = heads[to];
            while (from != -1) {
                if(--degree[points[from]] == 0) q.push(points[from]);
                from = args[from];
            }
        }

        /* 判定是否所有的點入度都爲0,若是則不存在環,否則存在環 */
        for (int i = 0; i < numCourses; ++i)
            if (degree[i] > 0) 
                 return false;

        return true;
    }
};

完整代碼 https://github.com/Orange1991/leetcode/blob/master/207/cpp/dfs.cpp

bfs

class Solution {
public:
    /**
     * 能否完成所有的課程
     * bfs拓撲排序,看有向圖是否存在環
     * @param numCourses 課程數量
     * @param prerequisites 課程先序關係
     * @return 能完成返回true,否則false
     */
    bool canFinish(int numCourses, vector<pair<int, int> >& prerequisites) {
        vector<int> heads(numCourses, -1), degree(numCourses, 0), points, args;
        pair<int, int> p;
        int from, to, count = 0, len = prerequisites.size();

        /* 構造有向圖,鄰接表 */
        for (int i = 0; i < len; ++i) {
            p = prerequisites[i];
            from = p.second;
            to = p.first;
            ++degree[to];
            args.push_back(heads[from]);
            points.push_back(to);
            heads[from] = count++;
        }

        /* bfs拓撲排序,依次移除入度爲0的點 */
        queue<int> q;
        for (int i = 0; i < numCourses; ++i) 
            if (degree[i] == 0) q.push(i);
        while (!q.empty()) {
            from = q.front();
            q.pop();
            to = heads[from];
            while (to != -1) {
                if(--degree[points[to]] == 0) q.push(points[to]);
                to = args[to];
            }
        }

        /* 判定是否所有的點入度都爲0,若是則不存在環,否則存在環 */
        for (int i = 0; i < numCourses; ++i)
            if (degree[i] > 0)
                 return false;

        return true;
    }
};

完整代碼 https://github.com/Orange1991/leetcode/blob/master/207/cpp/bfs.cpp

測試數據

(1) : 2, [[0,1]] : You can finish all the classes.
(2) : 2, [[0,1],[1,0]] : You can not finish all the classes.
(3) : 6, [[1,0],[2,0],[2,1],[3,2],[4,1],[5,3],[5,0],[5,3],[5,4]] : You can finish all the classes.
(4) : 800, [[695,229],[199,149],[443,397],[258,247],[781,667],[350,160],[678,629],[467,166],[500,450],[477,107],[483,151],[792,785],[752,368],[659,623],[316,224],[487,268],[743,206],[552,211],[314,20],[720,196],[421,103],[493,288],[762,24],[528,318],[472,32],[684,502],[641,354],[586,480],[629,54],[611,412],[719,680],[733,42],[549,519],[697,316],[791,634],[546,70],[676,587],[460,58],[605,530],[617,579],[484,89],[571,482],[767,200],[555,547],[771,695],[624,542],[708,551],[432,266],[656,468],[724,317],[423,248],[621,593],[781,399],[535,528],[578,12],[770,549],[576,295],[318,247],[400,372],[465,363],[786,482],[441,398],[577,411],[524,30],[741,540],[459,59],[758,96],[550,89],[402,295],[476,336],[645,346],[750,116],[551,207],[343,226],[568,498],[530,228],[525,84],[507,128],[526,210],[535,381],[635,330],[654,535],[710,275],[397,213],[412,44],[131,70],[508,49],[679,223],[519,11],[626,286],[242,160],[778,199],[606,281],[226,16],[340,46],[578,127],[212,208],[674,343],[778,108],[749,451],[735,105],[544,131],[600,229],[691,314],[608,74],[613,491],[754,500],[722,449],[486,11],[786,70],[212,23],[717,11],[692,410],[503,157],[783,177],[220,215],[419,363],[182,17],[321,54],[711,78],[312,106],[560,101],[501,178],[583,403],[577,9],[595,227],[601,386],[792,619],[550,167],[589,431],[793,243],[395,76],[197,3],[357,6],[763,7],[599,48],[178,92],[325,307],[620,10],[334,117],[556,296],[454,394],[485,236],[140,80],[404,301],[651,58],[504,455],[101,93],[712,42],[559,421],[594,230],[505,98],[719,654],[672,283],[109,73],[556,183],[617,94],[133,100],[771,515],[613,587],[285,50],[579,432],[282,244],[669,527],[783,494],[628,560],[716,661],[177,127],[430,166],[383,159],[746,19],[653,284],[495,243],[376,57],[560,143],[679,198],[751,355],[339,157],[409,140],[729,389],[518,315],[623,352],[651,133],[761,269],[442,44],[379,245],[313,180],[773,583],[291,221],[271,54],[799,44],[200,102],[568,67],[695,167],[327,36],[431,73],[782,167],[611,129],[630,122],[563,497],[697,93],[596,436],[611,131],[627,256],[658,559],[591,419],[193,156],[302,52],[409,33],[405,249],[384,151],[214,142],[558,164],[565,557],[492,445],[681,271],[797,396],[251,195],[784,266],[607,179],[671,30],[752,179],[787,390],[749,532],[618,220],[659,298],[567,134],[229,208],[298,147],[787,459],[572,359],[794,351],[53,14],[646,422],[234,66],[274,255],[744,626],[730,462],[498,428],[573,288],[688,355],[603,25],[191,16],[793,544],[750,682],[415,156],[460,209],[749,85],[269,186],[441,338],[319,278],[505,18],[672,260],[420,233],[493,134],[493,19],[308,302],[582,282],[755,60],[641,626],[669,69],[772,29],[132,111],[666,120],[605,58],[534,252],[636,491],[777,3],[602,368],[533,287],[401,147],[782,669],[517,161],[686,49],[789,639],[776,379],[376,65],[696,545],[423,81],[448,336],[631,605],[501,387],[413,94],[777,563],[661,332],[756,359],[646,36],[650,283],[656,347],[522,7],[383,382],[438,102],[762,305],[650,15],[249,180],[784,467],[763,122],[163,115],[775,734],[166,132],[634,2],[668,584],[767,274],[595,552],[11,7],[693,407],[789,751],[613,556],[715,402],[751,516],[646,199],[625,52],[572,106],[724,332],[617,409],[573,526],[760,18],[382,202],[207,139],[416,392],[672,358],[233,212],[668,22],[765,452],[294,76],[259,47],[593,271],[510,450],[592,132],[770,558],[296,43],[419,86],[752,347],[615,605],[635,554],[794,635],[613,316],[563,61],[770,715],[771,251],[646,582],[423,79],[576,249],[604,97],[767,348],[736,239],[775,56],[619,601],[790,546],[531,384],[507,84],[564,337],[432,310],[600,543],[747,341],[556,392],[661,113],[449,282],[575,288],[637,7],[635,325],[735,574],[574,387],[705,603],[704,15],[684,588],[495,132],[718,223],[517,206],[272,34],[677,416],[788,167],[649,525],[619,427],[541,277],[489,405],[608,259],[603,264],[435,317],[623,26],[544,511],[72,69],[623,17],[600,544],[551,367],[404,52],[324,272],[706,205],[778,446],[341,155],[581,173],[666,192],[588,529],[554,506],[250,39],[772,116],[569,77],[526,132],[563,221],[655,597],[649,224],[57,4],[679,199],[265,157],[380,335],[558,35],[726,388],[763,567],[437,426],[643,103],[773,181],[726,68],[164,50],[717,427],[681,618],[477,172],[697,423],[525,383],[794,132],[149,70],[704,414],[581,139],[678,204],[107,46],[352,68],[645,178],[758,156],[627,365],[331,144],[547,340],[788,36],[633,259],[588,66],[321,102],[528,322],[212,36],[288,179],[434,189],[749,490],[753,508],[784,341],[550,159],[741,206],[758,688],[766,758],[586,70],[657,654],[701,104],[548,184],[613,162],[620,320],[506,430],[517,65],[571,291],[771,517],[796,756],[735,459],[625,367],[759,345],[582,468],[469,73],[790,352],[493,284],[664,567],[342,207],[669,108],[611,182],[764,485],[214,102],[544,202],[713,447],[793,378],[147,129],[407,198],[608,271],[695,667],[680,277],[222,163],[744,527],[280,116],[430,367],[281,228],[688,488],[733,92],[529,190],[750,718],[793,99],[626,169],[486,329],[620,0],[782,460],[329,16],[753,142],[338,172],[518,361],[688,168],[497,490],[484,365],[365,325],[107,98],[622,407],[527,277],[659,74],[552,538],[493,469],[638,147],[304,3],[573,201],[411,169],[719,309],[287,160],[742,175],[573,299],[562,473],[705,328],[261,98],[580,203],[740,26],[418,296],[764,170],[656,89],[724,536],[730,91],[796,290],[735,270],[512,20],[402,246],[46,30],[426,290],[296,57],[725,222],[324,317],[547,0],[661,136],[636,271],[261,56],[750,668],[647,402],[773,390],[677,62],[249,53],[574,4],[393,304],[701,44],[109,66],[275,109],[679,509],[725,21],[409,311],[368,156],[605,514],[538,42],[690,602],[411,343],[424,240],[78,40],[750,273],[367,230],[167,58],[738,200],[634,341],[409,170],[644,373],[741,296],[702,342],[746,233],[411,67],[526,436],[796,438],[647,312],[717,347],[548,54],[725,50],[549,92],[610,294],[668,350],[578,445],[446,93],[727,246],[526,355],[344,246],[145,54],[355,256],[751,46],[454,271],[587,49],[728,79],[627,49],[522,260],[270,250],[491,113],[337,258],[609,470],[387,147],[656,237],[366,357],[160,98],[761,692],[753,627],[718,5],[335,22],[640,78],[687,317],[315,295],[471,93],[481,147],[724,580],[687,177],[409,41],[355,276],[393,366],[770,85],[697,358],[187,115],[671,318],[716,530],[767,140],[566,543],[318,238],[341,336],[648,204],[496,202],[505,191],[360,32],[408,138],[537,489],[668,102],[400,9],[472,3],[727,469],[713,5],[530,292],[465,381],[551,262],[514,227],[394,315],[551,121],[655,402],[755,83],[153,144],[303,60],[766,578],[668,527],[796,391],[692,571],[617,616],[229,154],[798,690],[706,504],[610,569],[655,624],[408,108],[569,463],[461,151],[507,13],[781,314],[780,469],[506,171],[552,312],[189,164],[336,171],[571,432],[688,224],[160,119],[470,311],[663,114],[665,420],[556,492],[709,358],[202,99],[170,149],[340,154],[666,385],[617,383],[502,132],[220,42],[778,393],[444,68],[526,357],[217,7],[597,76],[586,406],[481,44],[486,240],[513,217],[790,447],[275,245],[396,1],[369,224],[485,159],[680,151],[387,312],[721,70],[733,25],[457,216],[798,297],[329,169],[766,212],[286,160],[703,164],[765,77],[620,142],[510,35],[475,400],[784,8],[768,189],[668,328],[697,2],[389,169],[550,223],[514,268],[579,285],[419,53],[318,96],[335,117],[729,27],[694,281],[349,137],[545,221],[679,100],[382,116],[707,140],[62,48],[664,312],[499,369],[547,350],[509,279],[778,76],[186,17],[741,683],[635,531],[441,391],[493,385],[354,218],[304,128],[651,271],[693,360],[613,112],[798,393],[743,190],[115,62],[725,592],[525,233],[621,517],[327,70],[501,358],[504,346],[787,321],[94,74],[729,339],[50,13],[603,265],[163,29],[781,373],[586,459],[797,741],[624,364],[411,277],[360,161],[690,686],[746,639],[553,325],[631,328],[388,330],[619,210],[573,43],[559,100],[210,152],[378,5],[776,447],[615,181],[365,299],[708,310],[718,690],[268,225],[639,90],[318,5],[196,89],[361,184],[762,690],[772,465],[729,721],[541,331],[567,350],[269,58],[656,78],[579,163],[711,223],[282,268],[760,533],[404,280],[473,384],[94,48],[340,12],[727,364],[264,221],[591,487],[514,466],[305,168],[372,248],[639,499],[560,435],[541,142],[462,83],[594,353],[618,485],[95,33],[602,595],[605,289],[715,207],[448,293],[752,170],[641,203],[532,198],[608,13],[707,114],[744,211],[110,3],[298,228],[622,496],[286,26],[683,178],[706,192],[751,358],[486,461],[561,251],[466,193],[342,62],[221,37],[731,325],[205,132],[518,173],[502,261],[640,49],[541,522],[747,110],[756,591],[124,76],[639,603],[765,482],[388,5],[34,12],[514,344],[495,254],[770,751],[730,597],[708,105],[683,586],[528,288],[386,225],[287,26],[649,262],[753,670],[789,85],[632,439],[570,176],[672,652],[445,399],[400,226],[655,522],[469,249],[557,500],[275,6],[397,296],[725,43],[605,533],[425,220],[637,118],[628,215],[654,431],[697,421],[512,121],[237,36],[151,85],[574,217],[320,233],[492,272],[552,220],[739,81],[712,219],[612,590],[410,66],[548,40],[320,211],[381,95],[633,482],[742,535],[704,131],[682,435],[508,48],[435,337],[534,96],[663,653],[283,205],[715,74],[484,376],[585,366],[635,479],[753,719],[793,548],[396,171],[156,112],[575,380],[717,464],[612,576],[569,319],[736,259],[406,227],[711,709],[793,132],[528,295],[592,48],[731,217],[408,299],[373,137],[786,327],[791,166],[712,285],[772,603],[723,338],[531,121],[572,548],[786,167],[670,401],[724,440],[280,229],[497,453],[265,70],[733,144],[689,434],[504,384],[93,64],[563,397],[550,106],[224,198],[372,177],[249,31],[667,372],[263,78],[783,446],[791,59],[438,64],[630,270],[216,160],[704,261],[674,506],[704,23],[378,4],[784,437],[196,118],[681,314],[698,663],[397,274],[499,440],[737,265],[697,625],[139,84],[440,231],[453,150],[266,55],[377,11],[728,60],[431,202],[268,47],[763,123],[347,339],[470,117],[466,298],[344,142],[584,55],[417,175],[439,392],[548,55],[714,701],[643,71],[357,69],[649,459],[789,541],[626,5],[752,619],[711,267],[639,12],[750,364],[620,249],[769,721],[636,97],[233,15],[171,72],[488,421],[251,139],[750,98],[199,64],[768,344],[759,537],[435,154],[425,185],[336,221],[418,395],[390,136],[618,603]] : You can finish all the classes.

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2015/8/10