Using Dijkstra's algorithm only

Let G = (V, E) be a directed graph. Suppose every node v has a color c[v], which is either black or white. Let k be a positive integer. A path in G is called k-alternating if it changes color at least k times. Note that such a path need not be simple (that is, it may contain repeated nodes)

You are to determine if the graph contains a k-alternating path, and if so, to find a k-alternating path with a minimum number of black nodes. Your algorithm should run in time O(k (|V | + |E|)).