Graph theory-Two-point graph matching-Hungarian algorithm

Related concepts:

Binary graphs: the set of points in Figure G can be divided into two disjoint subsets, and the two points of each edge in G belong to these two subsets respectively .

Binary graph matching: The sub-figure m of the binary graph G has only one edge on each node, then M is a match .

Great match: Cannot add edges to a binary graph and match matching criteria

Maximum match: One of the most significant

Augmented path: If p is a path connecting two unmatched nodes on figure g, and the matched and mismatched edges appear alternately on p, then P is a augmented path relative to M

The Hungarian algorithm is used to find the maximal match of the binary graph.

Ideas:

The augmented path p is found in Figure G , and each edge on p is reversed (that is, the matched edge is changed to an unmatched edge, the unmatched edge is changed to a matched edge), and repeated until the augmentation path is not found .

Sample derivation:

Give a picture of a binary image

Starting from X1, search x1-y1, into the augmented road, connected X1-y1

[ATTENTION] from X2, search to X2-y1-x1-y3, into the augmented road

[ATTENTION] Cancel connection x1-y1, connection x1-y3,x2-y1

Starting from the X3, the search x3-y1-x2, does not pass, does not make the change

Search to X3-y2, into augmented road, connect X3-y2

Starting from the X4, the search x4-y3-x1-y1-x2, does not pass, does not make the change

Search to X4-y4, into augmented road, connect X4-y4

To find the great augmented road

adjacency table

1#include <cstdio>2#include <cstring>3 #defineMaxn4 intN,m,cnt,match[maxn],head[maxn],ans;5 BOOLCHECK[MAXN];6 structnode7 {8     intV,next;9}edge[maxn*5];Ten voidAddintXinty) One { Aedge[++cnt].next=Head[x]; -head[x]=CNT; -edge[cnt].v=y; the } - BOOLHungary (intu) - { -      for(intI=head[u];i;i=edge[i].next) +     { -         intv=edge[i].v; +         if(!check[v])//determine if V is in the augmented path A         { atcheck[v]=true; -             if(!match[v]| |Hungary (Match[v])) -             { -Match[v]=u;//v corresponds to u -                 return true; -             } in         } -     } to     return false; + } - intMain () the { *     //Omit input $      for(intI=1; i<=n;i++)Panax Notoginseng     { -         if(!Match[i]) the         { +memset (check,false,sizeof(check)); A             if(Hungary (i)) ans++; the         } +     } -printf"%d\n", ans); $     return 0; $}

Adjacency Matrix

1#include <cstdio>2#include <cstring>3 #defineMaxn4 intN,m,cnt,match[maxn],map[maxn][maxn],ans;5 BOOLCHECK[MAXN];6 BOOLHungary (intu)7 {8      for(intI=1; i<=n;i++)9     {Ten         if(!check[i]&&Map[u][i]) One         { Acheck[i]=true; -             if(!match[i]| |Hungary (Match[i])) -             { thematch[i]=u; -                 return true; -             } -         } +     } -     return false; + } A intMain () at { -     //Omit input -      for(intI=1; i<=n;i++) -     { -         if(!Match[i]) -         { inmemset (check,false,sizeof(check)); -             if(Hungary (i)) ans++; to         } +     } -printf"%d\n", ans); the     return 0; *}

* Reference: https://www.byvoid.com/blog/hungary/

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Graph theory-Two-point graph matching-Hungarian algorithm