寻找最小瓶颈路径的线性时间算法

Question

我正在上斯坦福大学的在线算法class，其中一题如下：

Define the bottleneck of a path to be the maximum length of one of its edges. A minimum-bottleneck path between two vertices s and t is a path with bottleneck no larger than that of any other s-t path. Suppose now that the graph is undirected. Give a linear-time (O(m)) algorithm to compute a minimum-bottleneck path between two given vertices.

使用修改后的 Dijkstra 算法解决此问题，运行时间复杂度为 O(mlog(n))，不符合要求。 Wikipedia 声称

exists a linear time algorithm for finding a widest s-t path in an undirected graph, that does not use the maximum spanning tree. The main idea of the algorithm is to apply the linear-time path-finding algorithm to the median edge weight in the graph, and then either to delete all smaller edges or contract all larger edges according to whether a path does or does not exist, and recurse in the resulting smaller graph.

有几个问题。算法主要是手挥手，我不是在找最宽的路径，而是相反。

This 论文的文字比维基百科多，但它也没有深入细节，尤其是在收缩边缘时。

我写出了以下伪代码：

1: MBP(G, s, t)
2:  if |E| == 1
3:    return the only edge
4:  else
5:    x = median of all edge weights
6:    E' = E - (v, w) where weight(v, w) < x
7:    construct G'(V, E')
8:    exists = is there a path from s to t in G'

9:    if (exists == FALSE)
10:      compute all the connected components Cᵢ of G'
11:      reinsert the edges deleted into G'

12:      G* = G'
13:      for each Cᵢ
14:        G* = SHRINK(G*, Cᵢ)

15:  return MBP(G', s, t)

16: SHRINK(G, C)
17:  leader = leader vertex of C
18:  V* = {V(G) - C} ∪ {leader}

19:  E* = {}
20:  for each edge (v, w) ∈ E(G)
21:    if v, w ∈ V*
22:      E* = E* ∪ {(v, w, weight(v, w))}
23:    else if v ∈ C, w ∈ V*
24:      E* = E* ∪ {(leader, w, max(weight(v, w)))}

25:  return G*(V*, E*)

有几点我不明白：

第 6 行：删除权重高于或低于中值的边有什么关系？
第20行：有3种边，两个顶点都在连通分量外，两个顶点都在连通分量内，一个顶点在连通分量内，一个在连通分量外。第一种类型保留其边缘权重，第二种类型变为自循环并应删除（？）。第三种边的权重应该是多少？

Answer 1

OP在这里。在my blog上找到了详细的解决方案，但伪代码如下：

1: CRITICAL-EDGE(G, s, t)
2:   if |E(G)| == 1
3:     return the only edge
4:   else
5:     x = median of all edge weights
6:     X = E - (v, w) s.t. weight(v, w) > x
7:     G' = G(V, X)
8:     exists = is there a path from s to t in G'

9:     if (exists == FALSE)
10:      C = {C₁, C₂, ..., Cₖ} s.t. Cᵢ is a connected component of G
11:      G' = G(V, E - X)

12:      for i = 1 to |C|
13:        G' = SHRINK(G', C, i)
14:    else if X == E // no edges were deleted
15:      X = {(v, w)} s.t. weight(v, w) = x
16:      G' = G(V, X)

17:  return CRITICAL-EDGE(G', s, t)

18: SHRINK(G, C, i)
19:   leaderᵢ = leader vertex of C[i]
20:   V* = {V(G) - C[i]} ∪ {leaderᵢ}

21:   E* = {}
22:   for each (v, w) ∈ E(G)
23:     if v ∈ C[i], w ∈ C[j]
24:       E* = E* ∪ {(leaderᵢ, leaderⱼ, min(weight(u, w)))} ∀ u ∈ C[i]
25:     else if v, w ∉ C[i]
          E * = E* ∪ {(v, w, weight(v, w))}

26:   return G*(V*, E*)

寻找最小瓶颈路径的线性时间算法

Linear-Time algorithm for finding a mininum-bottleneck path

algorithm

graph-theory

minimum-spanning-tree

graph-algorithm