In a d-ary heap each internal node has exactly d children. Again, the heap property then states that a node must have a smaller key than each of its d children. Thus, the case d = 2 is the standard (binary) heap. (a) How long does it take to perform a HEAPIFY-UP operation in a d-ary heap? (b) How long does it take to perform a HEAPIFY-DOWN operation? (c) Applying a) and b), how long does it take to run Dijkstra's algorithm (or Prim's algorithm) using a d-ary heap? (d) Assume that the number of arcs m in the graph G satisfies m = = O(n+), for some constant 1 > > 0. What choice of d gives the quickest imple- mentation of Dijkstra's algorithm?