Problem:
Score : 200 points

Problem Statement

There are K pieces of cakes.
Mr. Takahashi would like to eat one cake per day, taking K days to eat them all.

There are T types of cake, and the number of the cakes of type i (1 ≤ i ≤ T) is a_i.

Eating the same type of cake two days in a row would be no fun,
so Mr. Takahashi would like to decide the order for eating cakes that minimizes the number of days on which he has to eat the same type of cake as the day before.

Compute the minimum number of days on which the same type of cake as the previous day will be eaten.

Constraints

1 ≤ K ≤ 10000

1 ≤ T ≤ 100

1 ≤ a_i ≤ 100

a_1 + a_2 + ... + a_T = K

Input

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

K T
a_1 a_2 ... a_T

Output

Print the minimum number of days on which the same type of cake as the previous day will be eaten.

Sample Input 1

7 3
3 2 2

Sample Output 1

0

For example, if Mr. Takahashi eats cakes in the order of 2, 1, 2, 3, 1, 3, 1, he can avoid eating the same type of cake as the previous day.

Sample Input 2

6 3
1 4 1

Sample Output 2

1

There are 6 cakes.
For example, if Mr. Takahashi eats cakes in the order of 2, 3, 2, 2, 1, 2, he has to eat the same type of cake (i.e., type 2) as the previous day only on the fourth day.
Since this is the minimum number, the answer is 1.

Sample Input 3

100 1
100

Sample Output 3

99

Since Mr. Takahashi has only one type of cake, he has no choice but to eat the same type of cake as the previous day from the second day and after.

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

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Numerics;

namespace CompetitionPrograming2
{
    public sealed class Program : BaseProgram
    {
        public static void Main(string[] _) { using (new SetConsole()) { Solve(); } }
        public static void Solve()
        {
            var kt = GetArray<int>();
            var k = kt[0];
            var t = kt[1];
            var array = GetArray<int>();
            var total = array.Sum();

            if (array.Length == 1)
            {
                Write(total - 1); return;
            }

            var dic = new Dictionary<int, int>();
            for (int i = 0; i < array.Length; i++)
            {
                dic.Add(i + 1, array[i]);
            }
            var num = 1;
            while (total > 0)
            {
                if (total - dic[num] <= 0) { break; }

                var dec = dic.Where(x => x.Key != num && x.Value > 0).First();
                num = dec.Key;
                dic[dec.Key]--;
                total--;
            }

            Write(dic[num]);
        }
    }
    public abstract class BaseProgram
    {
        protected readonly static long divisor = 1000000007;
        protected static void Write(object obj) => Console.WriteLine(obj);
        protected static string GetString()
        {
            var str = Console.ReadLine();
            if (str == null) { throw new NullReferenceException("標準入力がnullです Solveメソッド内の解答コードが間違っています"); }
            return str;
        }
        protected static T GetNumber<T>() where T : IComparable, IComparable<T>, IConvertible, IEquatable<T>, IFormattable
        {
            var str = GetString();
            var t = typeof(T);
            try
            {
                if (t == typeof(int))
                {
                    return (T)(object)str.ToInt();
                }
                else if (t == typeof(long))
                {
                    return (T)(object)str.ToLong();
                }
                else if (t == typeof(double))
                {
                    return (T)(object)str.ToDouble();
                }
                else if (t == typeof(decimal))
                {
                    return (T)(object)str.ToDecimal();
                }
                else if (t == typeof(BigInteger))
                {
                    return (T)(object)str.ToBigInteger();
                }
            }
            catch (OverflowException)
            {
                throw new OverflowException("より大きい数値型を指定してください");
            }
            throw new NotSupportedException();
        }
        protected static T[] GetArray<T>() where T : IComparable, IComparable<T>, IConvertible, IEquatable<T> => ToArray<T>(GetString());
        protected static List<T> GetList<T>() where T : IComparable, IComparable<T>, IConvertible, IEquatable<T> => ToList<T>(GetString());
        protected static void Swap<T>(ref T item1, ref T item2)
        {
            var tmp = item1;
            item1 = item2;
            item2 = tmp;
        }
        protected static IEnumerable<char> AtoZ()
        {
            for (char i = 'a'; i <= 'z'; i++)
            {
                yield return i;
            }
        }
        private static T[] ToArray<T>(string str) where T : IComparable, IComparable<T>, IConvertible, IEquatable<T>
        {
            return ConvertEnumerator<T>(str).ToArray();
        }
        private static List<T> ToList<T>(string str) where T : IComparable, IComparable<T>, IConvertible, IEquatable<T>
        {
            if (typeof(T) == typeof(string)) { return (List<T>)(object)str.Split().ToList(); }

            return ConvertEnumerator<T>(str).ToList();
        }
        private static IEnumerable<T> ConvertEnumerator<T>(string str) where T : IComparable, IComparable<T>, IConvertible, IEquatable<T>
        {
            var t = typeof(T);
            if (t == typeof(byte))
            {
                return (IEnumerable<T>)str.Split().Select(byte.Parse);
            }
            if (t == typeof(int))
            {
                return (IEnumerable<T>)str.Split().Select(int.Parse);
            }
            if (t == typeof(long))
            {
                return (IEnumerable<T>)str.Split().Select(long.Parse);
            }
            if (t == typeof(double))
            {
                return (IEnumerable<T>)str.Split().Select(double.Parse);
            }
            if (t == typeof(decimal))
            {
                return (IEnumerable<T>)str.Split().Select(decimal.Parse);
            }
            if (t == typeof(BigInteger))
            {
                return (IEnumerable<T>)str.Split().Select(BigInteger.Parse);
            }
            throw new NotSupportedException();
        }
        protected sealed class SetConsole : IDisposable
        {
            readonly StreamWriter sw = new StreamWriter(Console.OpenStandardOutput());
            public SetConsole()
            {
                sw.AutoFlush = false;
                Console.SetOut(sw);
            }
            public void Dispose()
            {
                Console.Out.Flush();
                sw.AutoFlush = true;
                Console.SetOut(sw);
            }
        }
    }
    public static class ExtentionsLibrary
    {
        public static long Combination(this long num, long m)
        {
            if (num == m) { return 1; }
            if (m == 0) { return 1; }
            BigInteger ans = 1;
            var tmpM = m;
            while (tmpM > 0)
            {
                ans *= num;
                num--;
                tmpM--;
            }
            while (m > 0)
            {
                ans /= m;
                m--;
            }
            return (long)ans;
        }
        /// <summary>
        /// 階乗を求める
        /// </summary>
        /// <param name="num"></param>
        /// <returns></returns>
        public static long Factorial(this long num)
        {
            var result = 1L;
            for (int i = 1; i <= num; i++)
            {
                result *= i;
            }
            return result;
        }
        public static T[] NextPermutation<T>(this IEnumerable<T> collection) where T : IComparable
        {
            var array = collection.ToArray();
            int? exchangingNumIndex = null;
            for (int i = 1; i < array.Length; i++)
            {
                if (array[i - 1].CompareTo(array[i]) >= 0) { continue; }

                exchangingNumIndex = i - 1;
            }

            if (exchangingNumIndex == null) { return null; }

            for (int j = array.Length - 1; j >= 0; j--)
            {
                if (array[(int)exchangingNumIndex].CompareTo(array[j]) >= 0) { continue; }

                var tmp = array[j];
                array[j] = array[(int)exchangingNumIndex];
                array[(int)exchangingNumIndex] = tmp;
                array = array.Take((int)exchangingNumIndex + 1).Concat(array.Skip((int)exchangingNumIndex + 1).Reverse()).ToArray();
                break;
            }
            return array;
        }
        public static IEnumerable<T[]> Permutations<T>(this IEnumerable<T> collection) where T : IComparable
        {
            var next = collection.NextPermutation();

            while (next != null)
            {
                yield return next;

                next = next.NextPermutation();
            }
        }
        public static long Gcd(this long a, long b)
        {
            if (a < b) { return Gcd(b, a); }
            long tmp;
            do
            {
                tmp = a % b;
                a = b;
                b = tmp;
            } while (tmp != 0);
            return a;
        }
        /// <summary>
        /// 素因数分解を行う
        /// </summary>
        /// <param name="num"></param>
        /// <returns></returns>
        public static IEnumerable<Tuple<long, int>> Factorize(this long n)
        {
            yield return Tuple.Create(1L, 1);
            var rootN = Math.Sqrt(n);
            for (long i = 2; i <= rootN; i++)
            {
                if (n % i != 0) { continue; }

                var count = 1;
                n /= i;
                while (n % i == 0)
                {
                    n /= i;
                    count++;
                }
                yield return Tuple.Create(i, count);
            }
            if (n != 1) { yield return Tuple.Create(n, 1); }
        }
        public static int ToInt(this string str) => int.Parse(str);
        public static int ToInt(this char chr) => int.Parse(chr.ToString());
        public static long ToLong(this string str) => long.Parse(str);
        public static double ToDouble(this string str) => double.Parse(str);
        public static decimal ToDecimal(this string str) => decimal.Parse(str);
        public static BigInteger ToBigInteger(this string str) => BigInteger.Parse(str);
        public static DateTime ToDateTime(this string str) => DateTime.Parse(str);
        public static string StringJoin<T>(this IEnumerable<T> collection, string separator = "") => string.Join(separator, collection.Select(c => c.ToString()));
        public static string StringConcat(this IEnumerable<char> collection) => string.Concat(collection);
        public static int LowerBound<T>(this IReadOnlyList<T> a, T v) => LowerBound(a, v, Comparer<T>.Default);
        public static int LowerBound<T>(this IReadOnlyList<T> a, T v, Comparer<T> cmp)
        {
            var l = 0;
            var r = a.Count - 1;
            while (l <= r)
            {
                var mid = l + (r - l) / 2;
                var result = cmp.Compare(a[mid], v);
                if (result == -1)
                {
                    l = mid + 1;
                }
                else
                {
                    r = mid - 1;
                }
            }
            return l;
        }
        public static int UpperBound<T>(this IReadOnlyList<T> a, T v) => UpperBound(a, v, Comparer<T>.Default);
        public static int UpperBound<T>(this IReadOnlyList<T> a, T v, Comparer<T> cmp)
        {
            var l = 0;
            var r = a.Count - 1;
            while (l <= r)
            {
                var mid = l + (r - l) / 2;
                var res = cmp.Compare(a[mid], v);
                if (res <= 0) l = mid + 1;
                else r = mid - 1;
            }
            return l;
        }
    }

    public struct Mint
    {
        private static readonly long MOD = (long)1e9 + 7;

        /// <summary>
        /// 0以上MOD未満の整数
        /// </summary>
        public long Value;

        public Mint(long val)
        {
            Value = val % MOD;
            if (Value < 0) Value += MOD;
        }

        private static Mint Ctor(long val) => new Mint() { Value = val };
        public static Mint operator +(Mint a, Mint b)
        {
            long res = a.Value + b.Value;
            if (res > MOD) res -= MOD;
            return Ctor(res);
        }
        public static Mint operator -(Mint a, Mint b)
        {
            long res = a.Value - b.Value;
            if (res < 0) res += MOD;
            return Ctor(res);
        }
        public static Mint operator *(Mint a, Mint b)
        {
            long res = a.Value * b.Value;
            if (res > MOD) res %= MOD;
            return Ctor(res);
        }
        public static Mint operator /(Mint a, Mint b) => a * Inv(b);
        public static Mint Pow(Mint a, long n)
        {
            if (n == 0) return new Mint(1);
            Mint b = Pow(a, n >> 1);
            b *= b;
            if ((n & 1) == 1) b *= a;
            return b;
        }
        public static Mint Inv(Mint n)
        {
            long a = n.Value;
            long b = MOD;

            long x = 1;
            long u = 0;
            while (b != 0)
            {
                long k = a / b;
                long _x = u;
                u = x - k * u;
                x = _x;

                long _a = a;
                a = b;
                b = _a - (k * b);
            }

            return new Mint(x);
        }
        public override string ToString() => Value.ToString();
        public static implicit operator Mint(long a) => new Mint(a);
        public static explicit operator long(Mint a) => a.Value;
        public static Mint Choose(long n, long k)
        {
            Mint numerator = 1, denominator = 1;
            for (int i = 0; i < k; i++)
            {
                numerator *= n - i;
                denominator *= i + 1;
            }
            return numerator / denominator;
        }
    }
    public enum Priority : byte
    {
        Smaller = 0,
        Larger = 1
    }
}