Problem:
Score : 1500 points

Problem Statement

Iroha has a sequence of N strings s_1, s_2, ..., s_N.

She will choose some (possibly all) strings from the sequence, then concatenate those strings retaining the relative order, to produce a long string.

Among all strings of length K that she can produce in this way, find the lexicographically smallest one.

Constraints

1 ≦ N ≦ 2000

1 ≦ K ≦ 10^4

For each i, 1 ≦ |s_i| ≦ K.

|s_1| + |s_2| + ... + |s_N| ≦ 10^6

For each i, s_i consists of lowercase letters.

There exists at least one string of length K that Iroha can produce.

Input

The input is given from Standard Input in the following format:

N K
s_1
s_2
:
s_N

Output

Print the lexicographically smallest string of length K that Iroha can produce.

Sample Input 1

3 7
at
coder
codar

Sample Output 1

atcodar

at and codar should be chosen.

Sample Input 2

3 7
coder
codar
at

Sample Output 2

codarat

codar and at should be chosen.

Sample Input 3

4 13
kyuri
namida
zzzzzzz
aaaaaa

Sample Output 3

namidazzzzzzz

namida and zzzzzzz should be chosen.

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

#define A
using System;
using System.Linq;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using StringBuilder = System.Text.StringBuilder;
using System.Numerics;
using Point = System.Numerics.Complex;
using Number = System.Int32;
namespace Program
{
    public class Solver
    {
        public void Solve()
        {

            var n = sc.Integer();
            var k = sc.Integer();
            var back = new int[k + 1];
            for (int i = 0; i < k; i++)
                back[i] = -1;
            back[k] = n;
            var s = sc.Scan(n);
            for (int i = n - 1; i >= 0; i--)
            {
                for (int j = 0; j <= k; j++)
                    if (back[j] != -1 && j - s[i].Length >= 0) back[j - s[i].Length] = Math.Max(back[j - s[i].Length], i);
            }
            SuffixArray sa;
            {
                var sb = new StringBuilder();
                for (int i = 0; i < n; i++)
                {
                    sb.Append(s[i]);
                    sb.Append((char)('z' + 1));
                }
                sa = new SuffixArray(sb.ToString());
                sa.Init();
                sa.BuildLCP();
            }
            var dp = new bool[k + 1];
            dp[0] = true;
            var best = Enumerate(k, x => '|');

            var sum = 0;
            for (int i = 0; i < n; i++)
            {
                Debug.WriteLine(best.AsString());
                var len = s[i].Length;
                var zs = StringEx.ZAlgorithm(string.Concat(s[i], "$", new string(best)));
                var npos = -1;
                var next = Enumerate(k + 1, x => -100);
                for (int j = k - 1; j >= 0; j--)
                {
                    if (j + len > k) continue;
                    if (!dp[j]) continue;
                    if (back[j + len] <= i) continue;
                    var lcp = zs[len + 1 + j];
                    if (j + lcp != k && lcp != s[i].Length && best[j + lcp] > s[i][lcp])
                    {
                        npos = j;
                        next[j + len] = npos;
                        continue;
                    }
                    if (j + lcp != k && lcp != s[i].Length && best[j + lcp] < s[i][lcp]) continue;
                    if (npos != -1)
                    {
                        if (j + len <= npos)
                        {
                            npos = j;
                            next[j + len] = npos;
                        }
                        else
                        {
                            var lcp2 = sa.GetLCP(sum, sum + npos - j);
                            if (npos - j + lcp2 == len) continue;
                            else if (s[i][lcp2] > s[i][npos - j + lcp2])
                            {
                                npos = j;
                                next[j + len] = npos;
                            }
                        }
                    }
                }
                if (npos != -1)
                {
                    for (int j = 0; j < len; j++)
                        best[npos + j] = s[i][j];
                    for (int j = 0; j <= k; j++)
                        if (next[j] == npos) dp[j] |= true;
                }
                sum += len + 1;
            }
            IO.Printer.Out.WriteLine(best);
        }
        public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());
        static T[] Enumerate<T>(int n, Func<int, T> f) { var a = new T[n]; for (int i = 0; i < n; ++i) a[i] = f(i); return a; }
        static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
    }
}

#region main
static class Ex
{
    static public string AsString(this IEnumerable<char> ie) { return new string(System.Linq.Enumerable.ToArray(ie)); }
    static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") { return string.Join(st, ie); }
    static public void Main()
    {
        var solver = new Program.Solver();
        solver.Solve();
        Program.IO.Printer.Out.Flush();
    }
}
#endregion
#region Ex
namespace Program.IO
{
    using System.IO;
    using System.Text;
    using System.Globalization;
    public class Printer: StreamWriter
    {
        static Printer() { Out = new Printer(Console.OpenStandardOutput()) { AutoFlush = false }; }
        public static Printer Out { get; set; }
        public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
        public Printer(System.IO.Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
        public Printer(System.IO.Stream stream, Encoding encoding) : base(stream, encoding) { }
        public void Write<T>(string format, T[] source) { base.Write(format, source.OfType<object>().ToArray()); }
        public void WriteLine<T>(string format, T[] source) { base.WriteLine(format, source.OfType<object>().ToArray()); }
    }
    public class StreamScanner
    {
        public StreamScanner(Stream stream) { str = stream; }
        public readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private int len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream { get { return isEof; } }
        private byte read()
        {
            if (isEof) return 0;
            if (ptr >= len) { ptr = 0; if ((len = str.Read(buf, 0, 1024)) <= 0) { isEof = true; return 0; } }
            return buf[ptr++];
        }
        public char Char() { byte b = 0; do b = read(); while ((b < 33 || 126 < b) && !isEof); return (char)b; }

        public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        public string ScanLine()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b != '\n'; b = (char)read())
                if (b == 0) break;
                else if (b != '\r') sb.Append(b);
            return sb.ToString();
        }
        public long Long()
        {
            if (isEof) return long.MinValue;
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != 0 && b != '-' && (b < '0' || '9' < b));
            if (b == 0) return long.MinValue;
            if (b == '-') { ng = true; b = read(); }
            for (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else ret = ret * 10 + b - '0';
            }
        }
        public int Integer() { return (isEof) ? int.MinValue : (int)Long(); }
        public double Double() { var s = Scan(); return s != "" ? double.Parse(s, CultureInfo.InvariantCulture) : double.NaN; }
        private T[] enumerate<T>(int n, Func<T> f)
        {
            var a = new T[n];
            for (int i = 0; i < n; ++i) a[i] = f();
            return a;
        }

        public char[] Char(int n) { return enumerate(n, Char); }
        public string[] Scan(int n) { return enumerate(n, Scan); }
        public double[] Double(int n) { return enumerate(n, Double); }
        public int[] Integer(int n) { return enumerate(n, Integer); }
        public long[] Long(int n) { return enumerate(n, Long); }
    }
}
#endregion

#region SuffixArray
/// <summary>
/// O(N) 構築
/// </summary>
/// 
public class SuffixArray
{
    int[] S;
    int N;
    int[] SA;
    int[] rank;
    sparseTableRMQ rmq;
    const char ZERO = 'a';
    const int K = 28;
    public SuffixArray(string str)
    {
        N = str.Length;
        S = new int[N + 3];
        for (int i = 0; i < N; i++) S[i] = str[i] - ZERO + 1;
        SA = new int[N];
        rank = new int[N + 1];
    }
    public void Init()
    {
        rec(S, SA, N, K);
        for (int i = 0; i < N; i++)
            rank[SA[i]] = i + 1;
        rank[N] = 0;
    }
    public void BuildLCP()
    {
        var k = 0;
        var h = new int[N + 1];
        for (int i = 0; i < N; i++)
        {
            var j = this[rank[i] - 1];
            if (k > 0) k--;
            for (; j + k < N && i + k < N; k++) if (S[j + k] != S[i + k]) break;
            h[rank[i] - 1] = k;
        }
        rmq = new sparseTableRMQ(h);
    }
    public int GetLCP(int i, int j)
    {
        i = rank[i]; j = rank[j];
        return rmq.Query(Math.Min(i, j), Math.Max(i, j));
    }
    void rec(int[] s, int[] sa, int n, int k)
    {
        //Debug.WriteLine("{0} {1}", n, k);
        if (n == 1) { sa[0] = 0; return; }
        int a = (n + 2) / 3, b = (n + 1) / 3, c = n / 3;
        int ac = a + c;

        int[] tbc = new int[ac + 3], tabc = new int[ac + 3];
        int[] ta = new int[a];
        int[] taa = new int[a];

        for (int i = 0, j = 0; i < n + (a - c); i++) if (i % 3 != 0) tbc[j++] = i;
        radixPass(tbc, tabc, s, 2, ac, k);
        radixPass(tabc, tbc, s, 1, ac, k);
        radixPass(tbc, tabc, s, 0, ac, k);

        int name = 0, c0 = -1, c1 = -1, c2 = -1;
        for (int i = 0; i < ac; i++)
        {
            if (s[tabc[i]] != c0 || s[tabc[i] + 1] != c1 || s[tabc[i] + 2] != c2)
            {
                name++; c0 = s[tabc[i]]; c1 = s[tabc[i] + 1]; c2 = s[tabc[i] + 2];
            }
            if (tabc[i] % 3 == 1) tbc[tabc[i] / 3] = name;
            else
            {

                tbc[tabc[i] / 3 + a] = name;
            }
        }
        if (name < ac)
        {
            tbc[ac] = 0;
            rec(tbc, tabc, ac, name);
            for (int i = 0; i < ac; i++) tbc[tabc[i]] = i + 1;
        }
        else
        {
            for (int i = 0; i < ac; i++) tabc[tbc[i] - 1] = i;
        }
        for (int i = 0, j = 0; i < ac; i++) if (tabc[i] < a) ta[j++] = 3 * tabc[i];
        radixPass(ta, taa, s, 0, a, k);
        for (int p = 0, t = a - b, ptr = 0; ptr < n; ptr++)
        {
            int i = (tabc[t] < a) ? (tabc[t] * 3 + 1) : ((tabc[t] - a) * 3 + 2);
            var j = taa[p];
            if ((tabc[t] < a) ?
                s[i] < s[j] || (s[i] == s[j] && tbc[tabc[t] + a] < tbc[j / 3]) :
                s[i] < s[j] ||
                (s[i] == s[j] && s[i + 1] < s[j + 1]) ||
                (s[i] == s[j] && s[i + 1] == s[j + 1] && tbc[tabc[t] - a + 1] < tbc[j / 3 + a]))
            {
                sa[ptr] = i; t++;
                if (t == ac) for (ptr++; p < a; p++, ptr++) sa[ptr] = taa[p];
            }
            else
            {
                sa[ptr] = j; p++;
                if (p == a)
                    for (ptr++; t < ac; t++, ptr++) sa[ptr] = (tabc[t] < a) ? (tabc[t] * 3 + 1) : ((tabc[t] - a) * 3 + 2);

            }
        }
    }
    void radixPass(int[] a, int[] b, int[] s, int offset, int n, int k)
    {
        var c = new int[k + 1];
        for (int i = 0; i < n; i++) c[s[a[i] + offset]]++;
        for (int i = 0, sum = 0; i <= k; i++)
        {
            var t = c[i]; c[i] = sum; sum += t;
        }
        var ret = new int[n];
        for (int i = 0; i < n; i++) b[c[s[a[i] + offset]]++] = a[i];
    }
    /// <summary>
    /// rankがiのものを返す
    /// </summary>
    public int this[int index]
    {
        get { return index == 0 ? N : SA[index - 1]; }
    }
    /// <summary>
    /// s[i:]のランクを返す
    /// </summary>
    public int Rank(int index)
    {
        return rank[index];
    }
    #region SparseTableRMQ
    public class sparseTableRMQ
    {
        int n;
        int[] A;
        public sparseTableRMQ(int[] a)
        {
            var k = 1;
            n = a.Length;
            for (int i = 1; i < n; i <<= 1) k++;

            A = new int[n * k];
            for (int i = 0; i < n; i++)
                A[i] = a[i];
            var d = 0;
            for (int i = 1; i < n; i <<= 1)
            {
                for (int j = 0; j < n; j++)
                    A[d + n + j] = A[d + j];
                d += n;
                for (int j = 0; j < n - i; j++)
                    A[d + j] = Math.Min(A[d + j], A[d + j + i]);
            }
        }
        /// <summary>
        /// value of [l,r)
        /// </summary>
        public int Query(int l, int r)
        {
            r--;
            int z = r - l, k = 0, e = 1, s;
            s = ((z & 0xffff0000) != 0 ? 1 : 0) << 4; z >>= s; e <<= s; k |= s;
            s = ((z & 0x0000ff00) != 0 ? 1 : 0) << 3; z >>= s; e <<= s; k |= s;
            s = ((z & 0x000000f0) != 0 ? 1 : 0) << 2; z >>= s; e <<= s; k |= s;
            s = ((z & 0x0000000c) != 0 ? 1 : 0) << 1; z >>= s; e <<= s; k |= s;
            s = ((z & 0x00000002) != 0 ? 1 : 0) << 0; z >>= s; e <<= s; k |= s;
            //var vl = A[l + n * k];
            //var vr = A[r - e + 1 + (n * k)];
            return Math.Min(A[l + (n * k)], A[r + (n * k) - e + 1]);
        }
    }
    #endregion
}
#endregion


#region Z Algorithm
static public partial class StringEx
{
    /// <summary>ある位置からの部分文字列が接頭辞と何文字一致しているか調べる．O(|S|)</summary>
    static public int[] ZAlgorithm(string s)
    {
        var a = new int[s.Length + 1];
        a[0] = s.Length;
        int i = 1, j = 0;
        while (i < s.Length)
        {
            while (i + j < s.Length && s[j] == s[i + j]) ++j;
            a[i] = j;
            if (j == 0) { ++i; continue; }
            int k = 1;
            while (i + k < s.Length && k + a[k] < j) { a[i + k] = a[k]; ++k; }
            i += k; j -= k;
        }
        return a;
    }
    /// <summary>sのある位置からの部分文字列がtの接頭辞と何文字一致しているかを構築する．O(|S|)</summary>
    /// <returns>ret[i]=i番目の位置からの部分文字列とtの接頭辞が最大何文字一致するか</returns>
    static public int[] ZAlgorithm(string s, string t)
    {
        var ns = t + s;
        /*
        var ns = new int[t.Length + s.Length];
        for (int i = 0; i < t.Length; i++)
            ns[i] = t[i];
        for (int i = 0; i < s.Length; i++)
            ns[i + t.Length] = s[i];
        //*/
        var res = ZAlgorithm(ns);
        var ret = new int[s.Length + 1];
        for (int i = t.Length; i < res.Length; i++)
            ret[i - t.Length] = res[i];
        return ret;


    }
}
#endregion