# Prime Path (POJ 3126)

For prime numbers between 1000 and 9999, you are asked how many step it will cost to transfrom one to another one. During each step you can only change one place in the integer and no step can produce a number that is not prime.

The solution to this problems is that we first find all prime numbers in required range and build a bidirectional graph between them with each prime number as vertex. Then we perform basic graph algorithm to find the shortest path between to given prime. Here we can use dijkstra algorithm.

import java.io.*;
import java.util.*;

class Edge implements Comparable<Edge> {
int u;
int v;
int cost;

public Edge(int u, int v, int cost) {
this.u = u;
this.v = v;
this.cost = cost;
}

public int compareTo(Edge b) {
return this.cost - b.cost;
}
}

public class PrimePath {
private static int[] getPrimes() {
boolean[] a = new boolean[10000];
for (int i = 3; i < a.length; i += 2) {
if (a[i]) continue;

for (int j = i; j * i < a.length; j += 2) {
a[i * j] = true;
}
}

int count = 0;
for (int i = 1001; i < a.length; i += 2) {
if (!a[i]) count++;
}

int[] res = new int[count];

int h = 0;

for (int i = 1001; i < a.length; i += 2) {
if (!a[i]) res[h++] = i;
}

return res;
}

private static boolean connected(int a, int b) {
int count = 0;

for (int i = 0; i < 4; i++) {
if ((a % 10) != (b % 10)) count++;
a /= 10;
b /= 10;
}

return count == 1;
}

private static List<List<Edge>> getGraph(int[]primes) {
int N = primes.length;
ArrayList<List<Edge>> graph = new ArrayList<List<Edge>>(N);
for (int i = 0; i < N; i++) {
}

for (int i = 0; i < N; i++) {
for (int j = i + 1; j < N; j++) {
if (connected(primes[i], primes[j])) {
}
}
}

return graph;
}

private static int[] dijkstra(int N, int s, List<List<Edge>> graph) {
int[] dist = new int[N];
Arrays.fill(dist, Integer.MAX_VALUE);
dist[s] = 0;

PriorityQueue<Edge> que = new PriorityQueue<Edge>();
que.offer(new Edge(s, s, 0));

while (!que.isEmpty()) {
Edge p = que.poll();
if (dist[p.u] < p.cost) continue;

for (Edge e : graph.get(p.u)) {
if (dist[e.v] > p.cost + e.cost) {
dist[e.v] = p.cost + e.cost;
que.offer(new Edge(e.v, s, dist[e.v]));
}
}
}

return dist;
}

public static void main(String[] args) throws IOException {
String[] str;

int[] primes = getPrimes();
List<List<Edge>> graph = getGraph(primes);

for (int i = 0; i < N; i++) {
int a = Integer.parseInt(str[0]), b = Integer.parseInt(str[1]);

a = Arrays.binarySearch(primes, a);
b = Arrays.binarySearch(primes, b);

int[] dist = dijkstra(primes.length, a, graph);

if (a < 0 || b < 0 ||  dist[b] > 1000) {
System.out.println("Impossible");
} else {
System.out.println(dist[b]);
}
}
}
}