(Analysis by Brian Dean)

Imagine forming a graph (a network) of all the cows, with two cows linked together with edges labeled "S" or "D". Our goal is to count the number of connected components, $K$, in this graph, and then two also figure out if each component can be labeled with 1's and 2's in a consistent way that respects all of the "S" and "D" edges within the component. If labeling each component is possible, the answer is $2^K$, since note that we can reverse the labeling of any component (toggle 1s to 2s and vice versa) to produce another valid labeling; so each component can be labeled in exactly 2 ways. Conveniently, $2^K$ in binary is a 1 followed by $K$ zeros.

In my code below, I use a recursive "depth first" search to solve both of the problems above. I scan all the nodes of the graph and upon finding one that isn't labeled yet, I launch a recursive search from that point, which fans out and visits every node within its connected component, labeling nodes with 1s and 2s along the way in a consistent manner. If it ever discovers an inherent inconsistency (two 1s connected with a "D" edge or a 1 and 2 connected by an "S" edge), it flags the solution as being impossible.

If you are interested in related problems, this is quite similar to the problem of testing if a graph is "bipartite" --- colorable with just 2 colors so that no two connected nodes have the same color (in fact, if all the edges are "D" edges, then this problem is exactly the problem of testing bipartiteness). It's also related to the "2SAT" problem --- setting variables in a boolean expression like (x OR y) AND (not(x) OR y) AND (z OR not(y)) to either true or false so the entire expression evaluates to true. The reason it's called "2SAT" is that the expression must consist of "clauses" that are all ANDed together (so we need to satisfy all of them to satisfy the overall expression), where each clause is an OR of two things (a variable or its negation). To solve a problem like this, observe that each clause turns into an "S" or "D" edge between two variables.

Here is my code. The label $L[x]$ of node $x$ is zero if we haven't visited the node yet, or else 1 or 2.

#include <iostream>
#include <fstream>
#include <vector>
using namespace std;

int N, M, answer;
int L;
vector<int> S_nbrs, D_nbrs;
bool impossible;

void visit(int x, int v)
{
L[x] = v;
for (auto n : S_nbrs[x]) {
if (L[n] == 3-v) impossible = true;
if (L[n] == 0) visit(n, v);
}
for (auto n : D_nbrs[x]) {
if (L[n] == v) impossible = true;
if (L[n] == 0) visit(n, 3-v);
}
}

int main(void)
{
ifstream fin ("revegetate.in");
fin >> N >> M;
for (int i=0; i<M; i++) {
int a, b;
string s;
fin >> s >> a >> b;
if (s=="S") { S_nbrs[a].push_back(b); S_nbrs[b].push_back(a); }
if (s=="D") { D_nbrs[a].push_back(b); D_nbrs[b].push_back(a); }
}

for (int i=1; i<=N; i++)
if (!L[i]) { visit(i,1); answer++; }

ofstream fout ("revegetate.out");
if (impossible) fout << "0\n";
else {
fout << "1";
for (int i=0; i<answer; i++) fout << "0";
fout << "\n";
}
return 0;
}