Pots

Description

You are given two pots, having the volume of A and B liters respectively. The following operations can be performed:

1. FILL(i)        fill the pot i (1 ≤ i ≤ 2) from the tap;
2. DROP(i)      empty the pot i to the drain;
3. POUR(i,j)    pour from pot i to pot j; after this operation either the pot j is full (and there may be some water left in the pot i), or the pot i is empty (and all its contents have been moved to the pot j).

Write a program to find the shortest possible sequence of these operations that will yield exactly C liters of water in one of the pots.

Input

On the first and only line are the numbers A, B, and C. These are all integers in the range from 1 to 100 and C≤max(A,B).

Output

The first line of the output must contain the length of the sequence of operations K. The following K lines must each describe one operation. If there are several sequences of minimal length, output any one of them. If the desired result can’t be achieved, the first and only line of the file must contain the word ‘impossible’.

Sample Input

3 5 4

Sample Output

6
FILL(2)
POUR(2,1)
DROP(1)
POUR(2,1)
FILL(2)
POUR(2,1)

Source

Northeastern Europe 2002, Western Subregion

问题链接：POJ3414 Pots。

题意简述：

　　给出两个壶的容量A和B, 一个目标水量C，对A、B可以有３种操作，求最少经过几步操作能够在某个壶中得到目标水量C。输入A、B和C，输入最少操作数和操作过程。

1. FILL(i)     fill the pot i (1 ≤ i ≤ 2) from the tap;
2. DROP(i)     empty the pot i to the drain;
3. POUR(i,j)    pour from pot i to pot j; after this operation either the potj is full (and there may be some water left in the poti), or the poti is empty (and all its contents have been moved to the potj).

问题分析：

　　把A和B壶中水量作为状态，初始状态为<0,0>，每个操作都是状态变化的过程。因为有２个壶，所以总共有６种操作。使用ＢＦＳ搜索来找到最少的操作步数。同时需要考虑操作的条件，以减少操作来加快程序运行速度。

程序说明：

　　搜索过的状态就不需要再搜索了，用数组notvist[][]来标记搜索过的状态。操作的前提条件已经写在程序中。

AC的C++语言程序如下：

/* POJ3414 Pots */

#include <iostream>
#include <queue>
#include <cstring>

using namespace std;

const int MAXN = 100;

int a, b, c;
bool notvist[MAXN+1][MAXN+1];

struct node {
    int a, b, level;
    char path[MAXN+1];
    int plen;
};

string path[] = {
     "FILL(1)"
    ,"FILL(2)"
    ,"DROP(1)"
    ,"DROP(2)"
    ,"POUR(1,2)"
    ,"POUR(2,1)"
};

void output_result(int lvl, char p[], int n)
{
    cout << lvl << endl;
    for(int i=0; i<n; i++)
        cout << path[(int)p[i]] << endl;
}

void bfs()
{
    queue<node> q;

    memset(notvist, true, sizeof(notvist));

    node f;
    f.a = 0;
    f.b = 0;
    f.level = 0;
    memset(f.path, 0, sizeof(f.path));
    f.plen = 0;
    q.push(f);

    notvist[f.a][f.b] = false;

    while(!q.empty()) {
        f = q.front();
        q.pop();

        if(f.a == c || f.b == c) {
            output_result(f.level, f.path, f.plen);
            return;
        }

        node v;

        v = f;
        v.level++;
        v.plen++;
        // FILL(a)
        if(a - f.a > 0) {
            v.a = a;
            v.b = f.b;
            if(notvist[v.a][v.b]) {
                v.path[f.plen] = 0;
                q.push(v);
                notvist[v.a][v.b] = false;
            }
        }
        // FILL(b)
        if(b - f.b > 0) {
            v.a = f.a;
            v.b = b;
            if(notvist[v.a][v.b]) {
                v.path[f.plen] = 1;
                q.push(v);
                notvist[v.a][v.b] = false;
            }
        }
        // DROP(a)
        if(f.a) {
            v.a = 0;
            v.b = f.b;
            if(notvist[v.a][v.b]) {
                v.path[f.plen] = 2;
                q.push(v);
                notvist[v.a][v.b] = false;
            }
        }
        // DROP(b)
        if(f.b) {
            v.a = f.a;
            v.b = 0;
            if(notvist[v.a][v.b]) {
                v.path[f.plen] = 3;
                q.push(v);
                notvist[v.a][v.b] = false;
            }
        }
        // POUR(a,b)
        if(f.a && (f.b < b)) {
            if(f.a > (b - f.b)) {
                v.a = f.a -(b - f.b);
                v.b = b;
            } else {
                v.a = 0;
                v.b = f.b + f.a;
            }
            if(notvist[v.a][v.b]) {
                v.path[f.plen] = 4;
                q.push(v);
                notvist[v.a][v.b] = false;
            }
        }
        // POUR(b,a)
        if(f.b && (f.a < a)) {
            if(f.b > (a - f.a)) {
                v.a = a;
                v.b = f.b -(a - f.a);
            } else {
                v.a = f.a + f.b;
                v.b = 0;
            }
            if(notvist[v.a][v.b]) {
                v.path[f.plen] = 5;
                q.push(v);
                notvist[v.a][v.b] = false;
            }
        }
    }

    cout << "impossible" << endl;
}

int main()
{
    cin >> a >> b >> c;

    bfs();

    return 0;
}