P1002 [NOIP2002 普及组] 过河卒 - 洛谷 | 计算机科学教育新生态 (luogu.com.cn)
过河卒,首先科普一下象棋里面的马的跳跃一步的规则吧(这题真够坑人的,连个规则都不给出,害得我第一次交就全wa)。一张图解释
大家看所有标记为p的点的坐标就是马跳一步和原来的位置。解决这个问题之后,我们大家来想怎么做,从起点怎么到达终点,一个好想的思路是我们使用深搜dfs从起点来出发不断搜索,遇到封锁点换到另一个路线,最终到达终点。我们开个变量记录到达终点的总路线。本题由于只是向右和向下并不需要标记位置。但是显然会超时。给一下深搜的代码:
java
import java.awt.FontFormatException;
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.lang.reflect.AnnotatedWildcardType;
import java.math.BigInteger;
import java.net.DatagramPacket;
import java.sql.SQLIntegrityConstraintViolationException;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collections;
import java.util.Comparator;
import java.util.HashMap;
import java.util.HashSet;
import java.util.Iterator;
import java.util.LinkedHashSet;
import java.util.LinkedList;
import java.util.Map;
import java.util.Objects;
import java.util.PriorityQueue;
import java.util.Scanner;
import java.util.Spliterator.OfPrimitive;
import java.util.function.IntToDoubleFunction;
import java.util.function.LongBinaryOperator;
import java.util.TreeMap;
import java.util.TreeSet;
import javax.management.relation.InvalidRelationTypeException;
import javax.print.attribute.standard.JobMessageFromOperator;
import javax.print.attribute.standard.JobPriority;
import javax.swing.plaf.ColorChooserUI;
import javax.swing.table.TableModel;
import javax.swing.text.TabSet;
import javax.xml.crypto.dsig.spec.DigestMethodParameterSpec;
public class Main {
public static void main(String[] args) throws IOException {
Scanner sc=new Scanner(System.in);
BufferedReader br1=new BufferedReader(new InputStreamReader(System.in));
PrintWriter pw1=new PrintWriter(System.out);
String[] aStrings=br1.readLine().split(" ");
xx=Integer.parseInt(aStrings[0]);
yy=Integer.parseInt(aStrings[1]);
visited=new int[xx+1][yy+1];
int a=Integer.parseInt(aStrings[2]);
int b=Integer.parseInt(aStrings[3]);
visited[a][b]=-1;
int c,d;
int e=a-2;
int f=b-1;
if(e>=0&&e<=xx&&f>=0&&f<=yy) {
visited[e][f]=-1;
}
int e1=a-1;
int f1=b-2;
if(e1>=0&&e1<=xx&&f1>=0&&f1<=yy) {
visited[e1][f1]=-1;
}
int e2=a+1;
int f2=b-2;
if(e2>=0&&e2<=xx&&f2>=0&&f2<=yy) {
visited[e2][f2]=-1;
}
int e3=a+2;
int f3=b-1;
if(e3>=0&&e3<=xx&&f3>=0&&f3<=yy) {
visited[e3][f3]=-1;
}
int e4=a-2;
int f4=b+1;
if(e4>=0&&e4<=xx&&f4>=0&&f4<=yy) {
visited[e4][f4]=-1;
}
int e5=a-1;
int f5=b+2;
if(e5>=0&&e5<=xx&&f5>=0&&f5<=yy) {
visited[e5][f5]=-1;
}
int e6=a+1;
int f6=b+2;
if(e6>=0&&e6<=xx&&f6>=0&&f6<=yy) {
visited[e6][f6]=-1;
}
int e7=a+2;
int f7=b+1;
if(e7>=0&&e7<=xx&&f7>=0&&f7<=yy) {
visited[e7][f7]=-1;
}
dfs(0, 0);
System.out.println(answer);
}
public static int xx;
public static int yy;
public static int[] xx1= {0,-1,0,1};
public static int[] yy1= {-1,0,1,0};
public static int[][] visited;
public static long answer=0;
public static void dfs(int a,int b) {
if(a==xx&&b==yy) {
answer++;
return;
}
int c;
for(c=2;c<=3;c++) {
int x=a+xx1[c];
int y=b+yy1[c];
if(x>=0&&x<=xx&&y>=0&&y<=yy) {
if(visited[x][y]!=-1) {
dfs(x, y);
}
}
}
}
}
超时了:
这提示我们比赛时实在不会dp可以写dfs来骗分吗。(蓝桥专属)
(ps:蒟蒻只会dfs骗分,我哪会一开始就写dp还不是看了正解代码才会的)
回归正解,我们考虑dp做法。由于dfs超时了,我们考虑使用dp。因为本题
只能向右走和向下走,所以任何一个点,如果它可以被走到(这个点不是被封锁的点),那么它一定是从其上方的点和其左边的点走到的,那么我们的状态转移方程也就有了。
java
dp[i][j]=dp[i-1][j]+dp[i][j-1];
不过有鉴于我们的dp通常从1,1开始,所以我们默认将所有的位置都加上1,起点从(1,1)开始。顺便确定边界条件,我们直到起点为(1,1)那么我们从起点开始,假设终点就是(1,1)那肯定就只有一条路径。也就是说从起点开始路径就一条。根据我们的状态转移方程来判断,我们只需把(0,1)或(1,0)中的一个位置变为1即可。顺便如果遇到封锁点,我们直接把它赋值为0即可。
上代码:
java
import java.awt.FontFormatException;
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.lang.reflect.AnnotatedWildcardType;
import java.math.BigInteger;
import java.net.DatagramPacket;
import java.sql.SQLIntegrityConstraintViolationException;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collections;
import java.util.Comparator;
import java.util.HashMap;
import java.util.HashSet;
import java.util.Iterator;
import java.util.LinkedHashSet;
import java.util.LinkedList;
import java.util.Map;
import java.util.Objects;
import java.util.PriorityQueue;
import java.util.Scanner;
import java.util.Spliterator.OfPrimitive;
import java.util.function.IntToDoubleFunction;
import java.util.function.LongBinaryOperator;
import java.util.TreeMap;
import java.util.TreeSet;
import javax.management.relation.InvalidRelationTypeException;
import javax.print.attribute.standard.JobMessageFromOperator;
import javax.print.attribute.standard.JobPriority;
import javax.swing.plaf.ColorChooserUI;
import javax.swing.table.TableModel;
import javax.swing.text.TabSet;
import javax.xml.crypto.dsig.spec.DigestMethodParameterSpec;
public class Main {
public static void main(String[] args) throws IOException {
Scanner sc=new Scanner(System.in);
BufferedReader br1=new BufferedReader(new InputStreamReader(System.in));
PrintWriter pw1=new PrintWriter(System.out);
String[] aStrings=br1.readLine().split(" ");
xx=Integer.parseInt(aStrings[0]);
yy=Integer.parseInt(aStrings[1]);
visited=new long[xx+2][yy+2];
int a=Integer.parseInt(aStrings[2]);
int b=Integer.parseInt(aStrings[3]);
a=a+1;
b=b+1;
visited[a][b]=-1;
int c,d;
int e=a-2;
int f=b-1;
if(e>=0&&e<=xx+1&&f>=0&&f<=yy+1) {//这些if都是枚举封锁点,蒟蒻懒得写数组了,大家不要学我
visited[e][f]=-1;
}
int e1=a-1;
int f1=b-2;
if(e1>=0&&e1<=xx+1&&f1>=0&&f1<=yy+1) {
visited[e1][f1]=-1;
}
int e2=a+1;
int f2=b-2;
if(e2>=0&&e2<=xx+1&&f2>=0&&f2<=yy+1) {
visited[e2][f2]=-1;
}
int e3=a+2;
int f3=b-1;
if(e3>=0&&e3<=xx+1&&f3>=0&&f3<=yy+1) {
visited[e3][f3]=-1;
}
int e4=a-2;
int f4=b+1;
if(e4>=0&&e4<=xx+1&&f4>=0&&f4<=yy+1) {
visited[e4][f4]=-1;
}
int e5=a-1;
int f5=b+2;
if(e5>=0&&e5<=xx+1&&f5>=0&&f5<=yy+1) {
visited[e5][f5]=-1;
}
int e6=a+1;
int f6=b+2;
if(e6>=0&&e6<=xx+1&&f6>=0&&f6<=yy+1) {
visited[e6][f6]=-1;
}
int e7=a+2;
int f7=b+1;
if(e7>=0&&e7<=xx+1&&f7>=0&&f7<=yy+1) {
visited[e7][f7]=-1;
}
dp=new long[xx+2][yy+2];
dp[0][1]=1;
for(int i=1;i<=xx+1;i++) {
for(int j=1;j<=yy+1;j++) {
if(visited[i][j]==-1) {//把封锁点标记为-1,遇到封锁点我们赋值为0表示不能走
dp[i][j]=0;
}
else
dp[i][j]=dp[i-1][j]+dp[i][j-1];
}
}
System.out.println(dp[xx+1][yy+1]);
//记得dp开long
}
public static int xx;
public static int yy;
public static long[][] visited;
public static long[][] dp;
public static int answer=0;
}