两组点之间的最小距离

Question

我在度量空间中有一组 n 个点。这些点都是蓝色的。我在空间中有另一组 n 点。这些点都是红色的。我想以这样一种方式连接这些点，即每个蓝点都连接到一个红点，每个红点都连接到一个蓝点。 （显然有 n! 种方法。）我希望找到最小化连接总长度的连接集。这个问题叫什么？

Answer 1

问题叫Min weight perfect matching on complete bipartite graph，或者Assignment problem，

通常表述如下：

问题实例有许多代理和许多任务。可以分配任何代理来执行任何任务，产生的成本可能会因代理任务分配而异。要求执行尽可能多的任务，每个任务最多分配一个代理，每个代理最多分配一个任务，以使分配的总成本最小化。

问题可以通过Hungarian algorithm解决，复杂度为O(n^3)。

Answer 2

Thr 下面的代码（Edmonds-Karp 算法）解决了一个类似的问题：两组（男孩和女孩）之间的最优婚姻。 O(n^3) 也是。

# each boy would accept marriage with some
# of the girl. What wedding would maximize
# happiness ?
MensN=["Brad","John","Chrid","Tom","Zack","Moe","Yim","Tim","Eric","Don"]
WomN=["Olivia","Jenny","Michelle","Kate","Jude","Sonia","Yin","Zoe","Amy","Nat"]
       # O J M K J S Y Z A N
MensP=[[ 0,0,1,0,1,0,0,1,0,0],  # Brad
       [ 1,1,0,0,0,1,1,0,1,0],  # John
       [ 0,0,0,1,1,0,0,1,0,1],  # Chris
       [ 0,1,1,0,0,0,1,0,0,0],  # Tom
       [ 0,0,1,0,0,1,0,1,1,1],  # Zack
       [ 1,0,1,0,1,0,0,1,0,0],  # Moe
       [ 0,0,1,0,0,0,0,0,1,1],  # Yim
       [ 0,1,1,0,0,1,0,0,1,0],  # Tim
       [ 0,0,1,1,1,0,1,0,0,0],  # Eric
       [ 1,0,0,0,1,0,0,1,0,1]]  # Don 
       
#Edmonds-Karp Algorithm for optimal matching

def max_flow(C, s, t):
 F,path=[[0]*len(C) for c in C],s!=t
 while path:
  [path,flow]=bfs(C,F,s,t)
  for u,v in path:
   F[u][v],F[v][u]=F[u][v]+flow,F[v][u]-flow
 return F,sum(F[s])

#find path by using BFS
def bfs(C,F,s,t,f=999999):
 queue,paths=[s],{s:[]}
 while queue: 
  u=queue.pop(0)
  for v in range(len(C)):
    if C[u][v]>F[u][v] and v not in paths:
     paths[v]=paths[u]+[(u,v)]
     f=min(f,C[u][v]-F[u][v])
     if v==t:  return [paths[v],f]
     queue.append(v)
 return([[],999999])    

# make a capacity graph
C=[[0]+[1]*len(MensN)+[0]*len(WomN)+[0]]+[ # Source leads to men
[0]*(1+len(MensN))+p+[0] for p in MensP]+[ # Men lead to women with respective prefs
[0]*(1+len(MensN)+len(WomN))+[1] for w in WomN]+[ # Women lead to target
[0]*(1+len(MensN))+[0]*len(WomN)+[0]]  # Target leads nowhere
[F,n]=max_flow(C,0,len(C[0])-1)
print("It is possible to do",n,"marriage(s)")
for i in enumerate(MensN):
    print (i[1]," chooses ",",".join(WomN[j] for j in range(len(WomN)) if F[1+i[0]][1+len(MensN)+j] ))