(Analysis by Dhruv Rohatgi )

There's a fairly simple quadratic-time solution: let every cow be the vertex of a graph, and say that there is a directed edge of weight $|i-j|$ from cow $i$ to cow $j$ if cow $i$ is willing to talk to cow $j$. Then the cost of the shortest path from cow $1$ to cow $N$ is precisely the answer, and we can find it with e.g. Dijkstra's algorithm. However, since the graph has $O(N^2)$ edges, this is too slow.

That approach uses none of the structure of the original problem: that the edge weights are linear, or that the number of types of cows is very small. Let's try to use this structure to make a different graph with fewer edges that has the same shortest-path costs.

The physical intuition for a cost of $|i-j|$ to transmit a message from cow $i$ to $j$ is that the message travels $1$ cow per unit time. Leveraging this intuition, let's make a graph where each vertex encodes the location of the message. Then in one timestep, the message can either move left by one, or right by one. So we no longer have a quadratic number of edges.

Of course, the location of the message is not quite enough information to determine where it could go next. If we knew the cow who most recently sent the message, that would be enough information: in one unit time, a message sent by cow $i$ can either move one unit away from cow $i$; or it can be "received" by the cow at its current location if cow $i$ is happy to talk to this cow. The former edge updates the message's location, and the latter edge updates the message's sender. Thus, this graph has $O(N^2)$ vertices (for every location/sender pair) and $O(N^2)$ edges ($2$ per vertex).

Despite appearances, that is progress: we used the specifically chosen edge weights to lower the degree of the graph to $O(1)$. It remains to decrease the number of vertices, using the other symmetry of the problem: the small number of cow breeds. The key is that we don't need to remember the sender, just the sender's breed. Now we only have $O(NK)$ vertices and $O(NK)$ edges. Every edge has weight either $0$ or $1$ (cost $1$ to move left or right; cost $0$ to be received by the cow at the current location, if the breed matches the sending breed). The shortest path from (cow $1$, breed of cow $1$) to (cow $N$, breed of cow $N$) in this graph is (almost) the answer we need, and can be found by 0-1 BFS or Dijkstra's algorithm.

There is one more catch: if the first and last cows are the same breed, this graph will always output $N-1$ as the shortest path, which may be incorrect if this breed doesn't talk to itself. There are a number of ways to resolve this, e.g. by changing the breed of cow $N$ to a fake breed $0$, and remembering which breeds of cows are willing to talk to cow $N$.

Here is Danny Mittal's code:

import java.io.BufferedReader;
import java.io.IOException;
import java.util.Arrays;
import java.util.StringTokenizer;

public class TelephoneCorrect {

public static void main(String[] args) throws IOException {
int n = Integer.parseInt(tokenizer.nextToken());
int k = Integer.parseInt(tokenizer.nextToken());
int[] breeds = new int[n + 1];
for (int j = 1; j <= n; j++) {
breeds[j] = Integer.parseInt(tokenizer.nextToken());
}
boolean[][] adj = new boolean[k + 1][k + 1];
for (int b = 1; b <= k; b++) {
String line = " " + in.readLine();
for (int c = 1; c <= k; c++) {
}
}
breeds[n] = 0;
int[][] dist = new int[k + 1][n + 1];
for (int b = 0; b <= k; b++) {
Arrays.fill(dist[b], -1);
}
dist[breeds[1]][1] = 0;
while (!q.isEmpty()) {
int b = q.remove();
int j = q.remove();
if (j > 1 && dist[b][j - 1] == -1) {
dist[b][j - 1] = dist[b][j] + 1;
}
if (j < n && dist[b][j + 1] == -1) {
dist[b][j + 1] = dist[b][j] + 1;
}
if (adj[b][breeds[j]] && dist[breeds[j]][j] == -1) {
dist[breeds[j]][j] = dist[b][j];

Additional note (thanks to Justin Wu for suggesting that this be included): As an alternative means of simplifying the graph, we can note that if the solution includes a transition from a cow at index $i$ to another cow (say, of breed $b$), then without loss of generality we can assume this other cow is one of the two cows of breed $b$ closest to index $i$ --- either the one closest on the left or closest on the right (the only exception here is the transition to the final cow in the last position).
One can argue this by taking an optimal path and noting that if any of its transitions (except the last) do not fit this pattern, they can be modified without penalty to fit the pattern. E.g., if the optimal solution goes from index $1$ to $i$ to $j$ to $N$ and there is another cow of the same breed as the one at position $i$ between 1 and $i$ (say at $i'$), we loose nothing by changing the first leg of the path so it moves from 1 to $i'$ (so now we have an optimal solution that moves from $1$ to $i'$ to $j$ to $n$); we then adjust the next leg of the path the same way, and so on.