{"id":26640,"date":"2017-12-22T19:27:01","date_gmt":"2017-12-22T13:57:01","guid":{"rendered":"https:\/\/www.wikitechy.com\/technology\/?p=26640"},"modified":"2017-12-22T19:27:01","modified_gmt":"2017-12-22T13:57:01","slug":"c-algorithm-maximum-bipartite-matching","status":"publish","type":"post","link":"https:\/\/www.wikitechy.com\/technology\/c-algorithm-maximum-bipartite-matching\/","title":{"rendered":"C++ Algorithm – Maximum Bipartite Matching"},"content":{"rendered":"

A matching in a Bipartite Graph is a set of the edges chosen in such a way that no two edges share an endpoint. A maximum matching is a matching of maximum size (maximum number of edges). <\/span>In a maximum matching, if any edge is added to it, it is no longer a matching. There can be more than one maximum matchings for a given Bipartite Graph.<\/p>\n

Why do we care?<\/strong>
\nThere are many real world problems that can be formed as Bipartite Matching. For example, consider the following problem:
\nThere are M job applicants and N jobs. Each applicant has a subset of jobs that he\/she is interested in. Each job opening can only accept one applicant and a job applicant can be appointed for only one job. Find an assignment of jobs to applicants in such that as many applicants as possible get jobs.<\/em><\/p>\n

 <\/p>\n

We strongly recommend to read the following post first.<\/p>\n

Ford-Fulkerson Algorithm for Maximum Flow Problem<\/p>\n[ad type=”banner”]\n

Maximum Bipartite Matching and Max Flow Problem<\/strong>
\nM<\/strong>aximum B<\/strong>ipartite M<\/strong>atching (MBP<\/strong>) problem can be solved by converting it into a flow network (See this video to know how did we arrive this conclusion). Following are the steps.<\/p>\n

1) Build a Flow Network<\/strong><\/em>
\nThere must be a source and sink in a flow network. So we add a source and add edges from source to all applicants. Similarly, add edges from all jobs to sink. The capacity of every edge is marked as 1 unit.<\/p>\n

2) Find the maximum flow.
\nWe use Ford-Fulkerson algorithm to find the maximum flow in the flow network built in step 1. The maximum flow is actually the MBP we are looking for.
\nHow to implement the above approach?<\/strong>
\nLet us first define input and output forms. Input is in the form of Edmonds matrix which is a 2D array \u2018bpGraph[M][N]\u2019 with M rows (for M job applicants) and N columns (for N jobs). The value bpGraph[i][j] is 1 if i\u2019th applicant is interested in j\u2019th job, otherwise 0.
\nOutput is number maximum number of people that can get jobs.<\/p>\n

A simple way to implement this is to create a matrix that represents adjacency matrix representation of a directed graph with M+N+2 vertices. Call the fordFulkerson() for the matrix. This implementation requires O((M+N)*(M+N)) extra space.<\/p>\n

Extra space can be be reduced and code can be simplified using the fact that the graph is bipartite and capacity of every edge is either 0 or 1. The idea is to use DFS traversal to find a job for an applicant (similar to augmenting path in Ford-Fulkerson). We call bpm() for every applicant, bpm() is the DFS based function that tries all possibilities to assign a job to the applicant.<\/p>\n

In bpm(), we one by one try all jobs that an applicant \u2018u\u2019 is interested in until we find a job, or all jobs are tried without luck. For every job we try, we do following.
\nIf a job is not assigned to anybody, we simply assign it to the applicant and return true. If a job is assigned to somebody else say x, then we recursively check whether x can be assigned some other job. To make sure that x doesn\u2019t get the same job again, we mark the job \u2018v\u2019 as seen before we make recursive call for x. If x can get other job, we change the applicant for job \u2018v\u2019 and return true. We use an array maxR[0..N-1] that stores the applicants assigned to different jobs.<\/p>\n

If bmp() returns true, then it means that there is an augmenting path in flow network and 1 unit of flow is added to the result in maxBPM()<\/p>\n

C++ Programming:<\/strong><\/p>\n

<\/div> 
\/\/ A C++ program to find maximal Bipartite matching.#include <iostream>#include <string.h>using namespace std; \/\/ M is number of applicants and N is number of jobs#define M 6#define N 6 \/\/ A DFS based recursive function that returns true if a\/\/ matching for vertex u is possiblebool bpm(bool bpGraph[M][N], int u, bool seen[], int matchR[]){    \/\/ Try every job one by one    for (int v = 0; v < N; v++)    {        \/\/ If applicant u is interested in job v and v is        \/\/ not visited        if (bpGraph[u][v] && !seen[v])        {            seen[v] = true; \/\/ Mark v as visited             \/\/ If job 'v' is not assigned to an applicant OR            \/\/ previously assigned applicant for job v (which is matchR[v])             \/\/ has an alternate job available.             \/\/ Since v is marked as visited in the above line, matchR[v]             \/\/ in the following recursive call will not get job 'v' again            if (matchR[v] < 0 || bpm(bpGraph, matchR[v], seen, matchR))            {                matchR[v] = u;                return true;            }        }    }    return false;} \/\/ Returns maximum number of matching from M to Nint maxBPM(bool bpGraph[M][N]){    \/\/ An array to keep track of the applicants assigned to    \/\/ jobs. The value of matchR[i] is the applicant number    \/\/ assigned to job i, the value -1 indicates nobody is    \/\/ assigned.    int matchR[N];     \/\/ Initially all jobs are available    memset(matchR, -1, sizeof(matchR));     int result = 0; \/\/ Count of jobs assigned to applicants    for (int u = 0; u < M; u++)    {        \/\/ Mark all jobs as not seen for next applicant.        bool seen[N];        memset(seen, 0, sizeof(seen));         \/\/ Find if the applicant 'u' can get a job        if (bpm(bpGraph, u, seen, matchR))            result++;    }    return result;} \/\/ Driver program to test above functionsint main(){    \/\/ Let us create a bpGraph shown in the above example    bool bpGraph[M][N] = {  {0, 1, 1, 0, 0, 0},                        {1, 0, 0, 1, 0, 0},                        {0, 0, 1, 0, 0, 0},                        {0, 0, 1, 1, 0, 0},                        {0, 0, 0, 0, 0, 0},                        {0, 0, 0, 0, 0, 1}                      };     cout << "Maximum number of applicants that can get job is "         << maxBPM(bpGraph);     return 0;}<\/code><\/pre> <\/div>\n
\n
\n
\n
Output:<\/strong><\/p>\n
Maximum number of applicants that can get job is 5<\/pre>\n<\/div>\n<\/div>\n<\/div>\n[ad type=”banner”]\n","protected":false},"excerpt":{"rendered":"
C++ Algorithm – Maximum Bipartite Matching – Graph Algorithm – A matching in a Bipartite Graph is a set of the edges chosen in such a way <\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[83515,1,73906,78429],"tags":[79482,79488,79271,79483,79486,79481,79480,79485,79489,79487,79484],"class_list":["post-26640","post","type-post","status-publish","format-standard","hentry","category-c-programming-3","category-coding","category-graph-algorithms","category-maximum-flow","tag-bipartite-matching-network-flow","tag-ford-fulkerson-bipartite-matching","tag-hopcroft-karp-algorithm","tag-hopcroft-karp-algorithm-c","tag-maximal-matching","tag-maximum-bipartite-matching-ford-fulkerson","tag-maximum-matching-algorithm-for-bipartite-graphs","tag-maximum-matching-in-bipartite-graphs-video","tag-perfect-matching-algorithm","tag-perfect-matching-bipartite-graph","tag-weighted-bipartite-matching"],"_links":{"self":[{"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/posts\/26640","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/comments?post=26640"}],"version-history":[{"count":0,"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/posts\/26640\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/media?parent=26640"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/categories?post=26640"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wikitechy.com\/technology\/wp-json\/wp\/v2\/tags?post=26640"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}