結果

問題 No.8014 多項式ハッシュに関する教育的な問題
ユーザー uwi
提出日時 2018-09-28 16:48:44
言語 Java
(openjdk 23)
結果
AC  
実行時間 1,153 ms / 5,000 ms
コード長 9,814 bytes
コンパイル時間 4,437 ms
コンパイル使用メモリ 97,972 KB
実行使用メモリ 68,716 KB
最終ジャッジ日時 2024-10-12 04:53:58
合計ジャッジ時間 7,252 ms
ジャッジサーバーID
（参考情報） 		judge4 / judge2
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ファイルパターン 結果
other AC * 1
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ソースコード

diff #
プレゼンテーションモードにする

package etc;
import java.io.ByteArrayInputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.math.BigDecimal;
import java.math.BigInteger;
import java.math.MathContext;
import java.math.RoundingMode;
import java.util.Arrays;
import java.util.InputMismatchException;
public class N3014 {
    InputStream is;
    PrintWriter out;
    String INPUT = "";
    
    void solve()
    {
        BigInteger mod = new BigInteger(ns());
        BigInteger mul = new BigInteger(ns());
        BigInteger[] res = hack(mul, mod, 25, 1);
        char[][] ret = restore(res, 'a');
        for(char[] x : ret) {
            out.println(new String(x));
        }
    }
    
        static MathContext mc = new MathContext(30);
//      static final BigInteger big = new BigInteger("999999999999999999999");
//      static final BigInteger big = new BigInteger("999999");
        static final BigInteger big = new BigInteger("9999999999");
        
        /**
         * ハッシュ値が0となる列から、ハッシュ値が等しくなる2個の文字列を作成する
         * @param a
         * @param base 文字の最小値。'a'とか'A'とか。
         * @return
         */
        public static char[][] restore(BigInteger[] a, char base)
        {
            int n = a.length;
            char[][] ret = new char[2][n];
            for(int i = 0;i < n;i++) {
                int v = a[i].intValue();
                if(v < 0) {
                    ret[1][i] = base;
                    ret[0][i] = (char)(base - v);
                }else {
                    ret[1][i] = (char)(base + v);
                    ret[0][i] = base;
                }
            }
            return ret;
        }
                
        /**
         * 複数のmuls,modsを1個のmul,modにまとめる
         * @param muls
         * @param mods
         * @return
         */
        public static BigInteger[] collect(long[] muls, long[] mods)
        {
            BigInteger cr = crtx(mods, muls);
            BigInteger mod = BigInteger.valueOf(1);
            for(long m : mods)mod = mod.multiply(BigInteger.valueOf(m));
            return new BigInteger[] {cr, mod};
        }
        
        public static BigInteger b(long x){ return BigInteger.valueOf(x); }
        
        public static BigInteger[] exGCD(BigInteger a, BigInteger b)
        {
            BigInteger p = BigInteger.ONE, q = BigInteger.ZERO, r = BigInteger.ZERO, s = BigInteger.ONE;
            while(b.signum() > 0){
                BigInteger c = a.divide(b);
                BigInteger d;
                d = a; a = b; b = d.mod(b);
                d = p; p = q; q = d.subtract(c.multiply(q));
                d = r; r = s; s = d.subtract(c.multiply(s));
            }
            return new BigInteger[]{a, p, r};
        }
        
        
        public static BigInteger crtx(long[] divs, long[] mods)
        {
            BigInteger div = b(divs[0]), mod = b(mods[0]);
            for(int i = 1;i < divs.length;i++){
                BigInteger[] apr = exGCD(div, b(divs[i]));
                BigInteger mm = b(mods[i]).subtract(mod);
                if(mm.mod(apr[0]).signum() != 0)return b(-1);
                BigInteger da = div.divide(apr[0]);
                BigInteger ndiv = b(divs[i]).multiply(da);
                BigInteger nmod = apr[1].multiply(mm).multiply(da).add(mod).mod(ndiv);
                if(nmod.signum() < 0)nmod = nmod.add(ndiv);
                div = ndiv;
                mod = nmod;
            }
            return mod;
        }
        
        /**
         * (mul,mod)の多項式ハッシュに対し値が0となる列を見つける。
         * @param mul
         * @param mod
         * @param w
         * @param least さがす最低長
         * @return
         */
        public static BigInteger[] hack(BigInteger mul, BigInteger mod, int w, int least)
        {
            BigInteger[] ret = null;
            BigInteger W = BigInteger.valueOf(w);
            for(int n = least;ret == null;n++) {
                BigInteger[][] B = new BigInteger[n][n+1];
                BigInteger a = BigInteger.ONE;
                for(int i = n-1;i >= 0;i--) {
                    Arrays.fill(B[i], BigInteger.ZERO);
                    B[i][i] = BigInteger.ONE;
                    B[i][n] = a.multiply(big);
                    a = a.multiply(mul).mod(mod);
                }
//              for(int i = 0;i < n;i++) {
//                  Arrays.fill(B[i], BigInteger.ZERO);
//                  B[i][i] = BigInteger.ONE;
//                  B[i][n] = a.multiply(big);
//                  a = a.multiply(mul).mod(mod);
//              }
                BigInteger[][] res = reduce(B);
                
                inner:
                for(int i = 0;i < n;i++) {
                    if(res[i][n].signum() == 0) {
                        for(int j = 0;j < n;j++) {
                            if(res[i][j].abs().compareTo(W) > 0)continue inner;
                        }
                        ret = Arrays.copyOf(res[i], n);
                        break;
                    }
                }
            }
            check(ret, mul, mod);
            return ret;
        }
        
        static void check(BigInteger[] v, BigInteger mul, BigInteger mod)
        {
            BigInteger h = BigInteger.ZERO;
            int n = v.length;
            for(int i = 0;i < n;i++) {
                h = h.multiply(mul).add(v[i]).mod(mod);
            }
//          for(int i = n-1;i >= 0;i--) {
//              h = h.multiply(mul).add(v[i]).mod(mod);
//          }
            assert h.signum() == 0;
        }
        
        /**
         * LLL
         * @param BO
         * @return
         */
        public static BigInteger[][] reduce(BigInteger[][] BO)
        {
            int n = BO.length;
            int d = BO[0].length;
            
            // deepcopy
            BigInteger[][] B = new BigInteger[n][];
            for(int i = 0;i < n;i++) {
                B[i] = Arrays.copyOf(BO[i], d);
            }
            
            BigDecimal[][] BA = new BigDecimal[n][];
            
            BigDecimal delta = new BigDecimal("0.75"); // 0.25 < delta <= 1
            recalculateOrthoginalization(0, BA, B);
            for(int k = 1;k < n;) {
                for(int j = k-1;j >= 0;j--) {
                    BigDecimal mukj = mu(B[k], BA[j]);
                    if(Math.abs(mukj.doubleValue()) > 0.5) {
                        BigInteger rmu = mukj.setScale(0, RoundingMode.HALF_EVEN).toBigInteger();
                        for(int l = 0;l < d;l++) {
                            B[k][l] = B[k][l].subtract(rmu.multiply(B[j][l]));
                        }
                    }
                }
                recalculateOrthoginalization(k, BA, B);
                BigDecimal mu = mu(B[k], BA[k-1]);
                // Lovasz condition
                if(norm(BA[k]).compareTo(delta.subtract(mu.multiply(mu, mc)).multiply(norm(BA[k-1]))) >= 0) {
                    k++;
                }else {
                    // swap
                    BigInteger[] dum = B[k]; B[k] = B[k-1]; B[k-1] = dum;
                    recalculateOrthoginalization(k-1, BA, B);
                    k = Math.max(k-1, 1);
                }
            }
            return B;
        }
        
        /**
         * Gram-Schmidtの直交化のインデックスごと
         * @param ind
         * @param ovs
         * @param vs
         */
        private static void recalculateOrthoginalization(int ind, BigDecimal[][] ovs, BigInteger[][] vs)
        {
            int D = vs[0].length;
            BigDecimal[] h = new BigDecimal[D];
            for(int l = 0;l < D;l++)h[l] = new BigDecimal(vs[ind][l]);
            for(int l = 0;l < ind;l++){
                BigDecimal dot = BigDecimal.ZERO;
                BigDecimal norm = BigDecimal.ZERO;
                for(int k = 0;k < D;k++){
                    dot = dot.add(ovs[l][k].multiply(new BigDecimal(vs[ind][k]), mc));
                    norm = norm.add(ovs[l][k].multiply(ovs[l][k], mc));
                }
                if(norm.signum() == 0)continue;
                BigDecimal t = dot.divide(norm, mc);
                for(int k = 0;k < D;k++){
                    h[k] = h[k].subtract(ovs[l][k].multiply(t, mc));
                }
            }
            ovs[ind] = h;
        }
        
        static BigDecimal mu(BigInteger[] b, BigDecimal[] ba)
        {
            return dot(b, ba).divide(norm(ba), mc);
        }
        
        public static BigDecimal norm(BigDecimal[] v)
        {
            BigDecimal n = BigDecimal.ZERO;
            for(BigDecimal w : v)n = n.add(w.multiply(w, mc));
            return n;
        }
        
        public static BigDecimal dot(BigDecimal[] a, BigDecimal[] b)
        {
            BigDecimal ret = BigDecimal.ZERO;
            for(int i = 0;i < a.length;i++){
                ret = ret.add(a[i].multiply(b[i], mc));
            }
            return ret;
        }
        
        public static BigDecimal dot(BigInteger[] a, BigDecimal[] b)
        {
            BigDecimal ret = BigDecimal.ZERO;
            for(int i = 0;i < a.length;i++){
                ret = ret.add(new BigDecimal(a[i]).multiply(b[i], mc));
            }
            return ret;
        }
    
    void run() throws Exception
    {
        is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
        out = new PrintWriter(System.out);
        
        long s = System.currentTimeMillis();
        solve();
        out.flush();
        if(!INPUT.isEmpty())tr(System.currentTimeMillis()-s+"ms");
//      Thread t = new Thread(null, null, "~", Runtime.getRuntime().maxMemory()){
//          @Override
//          public void run() {
//              long s = System.currentTimeMillis();
//              solve();
//              out.flush();
//              if(!INPUT.isEmpty())tr(System.currentTimeMillis()-s+"ms");
//          }
//      };
//      t.start();
//      t.join();
    }
    
    public static void main(String[] args) throws Exception { new N3014().run(); }
    
    private byte[] inbuf = new byte[1024];
    public int lenbuf = 0, ptrbuf = 0;
    
    private int readByte()
    {
        if(lenbuf == -1)throw new InputMismatchException();
        if(ptrbuf >= lenbuf){
            ptrbuf = 0;
            try { lenbuf = is.read(inbuf); } catch (IOException e) { throw new InputMismatchException(); }
            if(lenbuf <= 0)return -1;
        }
        return inbuf[ptrbuf++];
    }
    
    private boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
    private int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
    
    private double nd() { return Double.parseDouble(ns()); }
    private char nc() { return (char)skip(); }
    
    private String ns()
    {
        int b = skip();
        StringBuilder sb = new StringBuilder();
        while(!(isSpaceChar(b))){ // when nextLine, (isSpaceChar(b) && b != ' ')
            sb.appendCodePoint(b);
            b = readByte();
        }
        return sb.toString();
    }
    
    private char[] ns(int n)
    {
        char[] buf = new char[n];
        int b = skip(), p = 0;
        while(p < n && !(isSpaceChar(b))){
            buf[p++] = (char)b;
            b = readByte();
        }
        return n == p ? buf : Arrays.copyOf(buf, p);
    }
    
    private int[] na(int n)
    {
        int[] a = new int[n];
        for(int i = 0;i < n;i++)a[i] = ni();
        return a;
    }
    
    private long[] nal(int n)
    {
        long[] a = new long[n];
        for(int i = 0;i < n;i++)a[i] = nl();
        return a;
    }
    
    private char[][] nm(int n, int m) {
        char[][] map = new char[n][];
        for(int i = 0;i < n;i++)map[i] = ns(m);
        return map;
    }
    
    private int[][] nmi(int n, int m) {
        int[][] map = new int[n][];
        for(int i = 0;i < n;i++)map[i] = na(m);
        return map;
    }
    
    private int ni() { return (int)nl(); }
    
    private long nl()
    {
        long num = 0;
        int b;
        boolean minus = false;
        while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
        if(b == '-'){
            minus = true;
            b = readByte();
        }
        
        while(true){
            if(b >= '0' && b <= '9'){
                num = num * 10 + (b - '0');
            }else{
                return minus ? -num : num;
            }
            b = readByte();
        }
    }
    
    private static void tr(Object... o) { System.out.println(Arrays.deepToString(o)); }
}
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