- 1 -
Be neat and concise, but complete.
1. (15 points) State the general greedy method for finding minimum spanning trees. Show that
Kruskal’s algorithm is a special case of the general greedy method.
The general greedy method applies one of the following two coloring steps until all edges are colored.
On completion, the blue edges form a minimum spanning tree.
Blue rule. Find a cut crossed by no blue edges and at least one uncolored edge. Select a minimum
weight uncolored edge crossing the cut and color it blue.
Red rule. Find a cycle with no red edges and at least one uncolored edge. Select an uncolored
edge of maximum weight from the cycle and color it red.
Kruskal’s algorithm goes through the edges in non-decreasing order of their weight. On finding an
edge that joins two separate blue trees, it colors it blue. This is a proper application of the blue rule,
since the cut defined by placing one of the blue trees on one side of the cut and all other vertices on
the other side, constitutes a cut that we can apply the blue rule to. Since the edge under consideration
is a minimum weight uncolored edge, it is also minimum weight among edges crossing the cut.
When Kruskal’s algorithm finds an edge joining two vertices in the same blue tree, it colors it red.
This is a proper application of the red rule, since the edge and the tree path joining its endpoints form
a cycle that the red rule can be applied to. Since the edge being considered, is the only uncolored edge,
in the cycle, it must be a maximum weight uncolored edge in the cycle.
2. (10 points) The correctness of any data structure operation depends on its maintaining
certain essential properties of the data structure. The data portion of the class declaration
for the C++ implementation of the d-heap data structure is shown below. What properties
of the data must be maintained by programs that operate on it?
class dheap {
int N; // max number of items in heap
int n; // number of items in heap
int d; // base of heap
item *h; // {h[1],...,h[n]} is set of items
int *pos; // pos[i] gives position of i in h
keytyp *kvec; // kvec[i] is key of item i
...
};
d>1, 0
≤
n
≤
N, for 2
≤
i
≤
n, kvec
[
(i-1)/d
]
≤
kvec[i], for 1
≤
i
≤
n, pos[h[i]] = i, 1
≤
h[i]
≤
n,
1
≤
pos[i]
≤
n.
CS 541 – Algorithms and Programs
Exam 1 Solutions
J
onathan Turner
10/4/01
