Question

Use the following definition of the union operation, assuming the sentinel of x is not the same as the sentinel of y:

union(x,y) = Make the sentinel of x point to the sentinel of y
Imagine we start with an unconnected forest of nodes labeled 0 through 7. Then, we perform the following operations:

union(0,1)
union(2,3)
union(0,2)
union(4,0)
union(5,6)
union(7,5)
union(5,4)
find(5)

Assuming we are not performing any weighting optimizations (i.e., don't use union-by-size or union-by-height) and assuming find does not perform any path compression, what is the array representation of our disjoint set?

Provide the array representation as space-separated numbers. For example, if you wanted to represent the initial forest of nodes, your answer would look like the following:

-1 -1 -1 -1 -1 -1 -1 -1

Answer 1

The array representation of the disjoint set is -1 -1 -1 0 0 7 7 0.

Step-by-step explanation:

The array representation of the disjoint set is as follows:

-1 -1 -1 0 0 7 7 0

Initially, all the nodes are represented as individual sets with a sentinel value of -1. After performing the given union operations, the array is updated accordingly. For example, union(0,1) makes the sentinel of node 0 point to the sentinel of node 1, resulting in -1 1 -1 0 0 7 7 0. Similarly, the other union operations result in the final array representation.

Answer 2

Following the given union operation definition, here's the array representation after the specified operations:

```

-1 -1 0 0 0 7 7 7

```

Step-by-step explanation:

- Initially, all nodes are in their own sets represented by -1.

- After `union(0,1)`: Nodes 0 and 1 are in the same set, and 0 becomes the representative.

- After `union(2,3)`: Nodes 2 and 3 are in the same set, and 2 becomes the representative.

- After `union(0,2)`: The set containing nodes 0 and 1 is combined with the set containing nodes 2 and 3, and 0 becomes the representative.

- After `union(4,0)`: Nodes 0, 1, 2, 3, and 4 are now in the same set, and 0 becomes the representative.

- After `union(5,6)`: Nodes 5 and 6 are in the same set, and 5 becomes the representative.

- After `union(7,5)`: Nodes 5, 6, and 7 are now in the same set, and 5 becomes the representative.

- After `union(5,4)`: The set containing nodes 0, 1, 2, 3, and 4 is combined with the set containing nodes 5, 6, and 7, and 0 becomes the representative.

- `find(5)`: Returns the representative of the set containing node 5, which is 0.