为什么可以修改 Dijkstra 算法以找到 K 条最短路径？

问：

我试图找到一个直观的解释，解释为什么我们可以推广 Dijkstra 算法，以在没有负边的有向加权图中从单个源中找到 K 条最短（简单）路径。根据维基百科，修改后的 Dijkstra 的伪代码如下：

Definitions:
  G(V, E): weighted directed graph, with set of vertices V and set of directed edges E,
  w(u, v): cost of directed edge from node u to node v (costs are non-negative).
  Links that do not satisfy constraints on the shortest path are removed from the graph
  s: the source node
  t: the destination node
  K: the number of shortest paths to find
  P[u]: a path from s to u
  B: a heap data structure containing paths
  P: set of shortest paths from s to t
  count[u]: number of shortest paths found to node u

Algorithm:
  P = empty
  for all u in V:
    count[u] = 0
  insert path P[s] = {s} into B with cost 0

  while B is not empty:
    let P[u] be the shortest cost path in B with cost C
    remove P[u] from B
    count[u] = count[u] + 1
    
    if count[u] <= K then:
      for each vertex v adjacent to u:
        let P[v] be a new path with cost C + w(u, v) formed by concatenating edge (u, v) to path P[u]
        insert P[v] into B
  return P

我知道，对于原始的 Dijkstra 算法，可以通过归纳法证明，当一个节点被添加到封闭集（或者如果它以 BFS + 堆的形式实现，则从堆中弹出）时，该节点的成本必须从源头开始最小。

这里的算法似乎基于这样一个事实，即当一个节点从堆中弹出第 i 次时，我们从源中获得^{第 i 个}最小的成本。为什么会这样？