Problem:
Score : 600 points

Problem Statement

AtCoDeer the deer has N cards with positive integers written on them. The number on the i-th card (1≤i≤N) is a_i.
Because he loves big numbers, he calls a subset of the cards good when the sum of the numbers written on the cards in the subset, is K or greater.

Then, for each card i, he judges whether it is unnecessary or not, as follows:

If, for any good subset of the cards containing card i, the set that can be obtained by eliminating card i from the subset is also good, card i is unnecessary.

Otherwise, card i is NOT unnecessary.

Find the number of the unnecessary cards. Here, he judges each card independently, and he does not throw away cards that turn out to be unnecessary.

Constraints

All input values are integers.

1≤N≤5000

1≤K≤5000

1≤a_i≤10^9 (1≤i≤N)

Partial Score

300 points will be awarded for passing the test set satisfying N,K≤400.

Input

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

N K
a_1 a_2 ... a_N

Output

Print the number of the unnecessary cards.

Sample Input 1

3 6
1 4 3

Sample Output 1

1

There are two good sets: {2,3} and {1,2,3}.

Card 1 is only contained in {1,2,3}, and this set without card 1, {2,3}, is also good. Thus, card 1 is unnecessary.

For card 2, a good set {2,3} without card 2, {3}, is not good. Thus, card 2 is NOT unnecessary.

Neither is card 3 for a similar reason, hence the answer is 1.

Sample Input 2

5 400
3 1 4 1 5

Sample Output 2

5

In this case, there is no good set. Therefore, all the cards are unnecessary.

Sample Input 3

6 20
10 4 3 10 25 2

Sample Output 3

3

Solution in C (GCC 5.4.1) (judge verdict: Accepted)

#include<stdio.h>
#include<stdlib.h>
#include<stdint.h>
#include<inttypes.h>
#include<assert.h>

typedef uint64_t bitset_val;

typedef struct bitSetBy64bit{
  uint64_t *array;
  size_t array_len;
  size_t num;
} bitset;

bitset* new_bitset (const size_t num) {
  bitset *b = (bitset *) calloc (1, sizeof (bitset));
  b->num = num;
  const size_t bit = sizeof (uint64_t) * 8;
  b->array_len = (num + bit - 1) / bit;
  b->array = (uint64_t *) calloc (b->array_len, sizeof (uint64_t));
  return b;
}

void free_bitset (bitset * const b) {
  free (b->array);
  free (b);
}

static inline void fix_bitset (bitset * const b) {
  if (b->num % (8 * sizeof (uint64_t)) != 0) {
    b->array[b->array_len - 1] &= ((uint64_t)1 << (b->num % (8 * sizeof (uint64_t)))) - 1;
  }
}

void set_bitset (bitset * const b) {
  const uint64_t t = ~(uint64_t)0;
  for (size_t i = 0; i < b->array_len; ++i) {
    b->array[i] = t;
  }
  fix_bitset (b);
}

void clear_bitset (bitset * const b) {
  const uint64_t t = 0;
  for (size_t i = 0; i < b->array_len; ++i) {
    b->array[i] = t;
  }
}

void flip_bitset (bitset * const b) {
  for (size_t i = 0; i < b->array_len; ++i) {
    b->array[i] = ~b->array[i];
  }
  fix_bitset (b);
}

inline int getAt_bitset (const bitset *b, const size_t index) {
  assert (0 <= index && index < b->num);
  return (b->array[index / (8 * sizeof (uint64_t))] >> (index % (8 * sizeof (uint64_t)))) & 1;
}

inline void setAt_bitset (bitset *const b, const size_t index) {
  assert (0 <= index && index < b->num);
  b->array[index / (8 * sizeof (uint64_t))] |= (uint64_t)1 << (index % (8 * sizeof (uint64_t)));
}

inline void clearAt_bitset (bitset * const b, const size_t index) {
  assert (0 <= index && index < b->num);
  b->array[index / (8 * sizeof (uint64_t))] &= ~((uint64_t)1 << (index % (8 * sizeof (uint64_t))));
}

inline void flipAt_bitset (bitset * const b, const size_t index) {
  assert (0 <= index && index < b->num);
  b->array[index / (8 * sizeof (uint64_t))] ^= (uint64_t)1 << (index % (8 * sizeof (uint64_t)));
}

void assign_bitset (bitset * const x, const bitset * const a) {
  assert (x->num >= a->num);
  size_t i;
  for (i = 0; i < a->array_len; ++i) {
    x->array[i] = a->array[i];
  }
  for (; i < x->array_len; ++i) {
    x->array[i] = 0;
  }
}

void left_shift_bitset (bitset * const x, const bitset * const a, const size_t shift) {
  clear_bitset (x);
  size_t p = shift / (8 * sizeof (uint64_t));
  size_t q = shift % (8 * sizeof (uint64_t));
  if (q == 0) {
    for (size_t i = 0; i + p < x->array_len && i < a->array_len; ++i) {
      x->array[i + p] = a->array[i];
    }
  } else {
    size_t i;
    for (i = 0; i + p + 1 < x->array_len && i + p < a->array_len; ++i) {
      uint64_t v = a->array[i];
      x->array[i + p] |= v << q;
      x->array[i + p + 1] |= v >> ((8 * sizeof (uint64_t)) - q);
    }
    if (i < a->array_len && i + p < x->array_len) {
      x->array[i + p] |= a->array[i] << q;
    }
  }
  fix_bitset (x);
}

typedef uint32_t u32;

void trans (bitset * const b, const u32 a) {
  static bitset *tmp = NULL;
  if (tmp == NULL) tmp = new_bitset (b->num);
  left_shift_bitset (tmp, b, a);
  for (size_t i = 0; i < b->array_len; ++i) {
    b->array[i] |= tmp->array[i];
  }
}

u32 calc (const u32 *a, const u32 n, const u32 k, bitset * const b) {
  if (n == 1) {
    if (a[0] >= k) return 0;
    for (u32 i = k - 1; i >= k - a[0]; --i) {
      if (getAt_bitset (b, i)) {
	return 0;
      }
    }
    return 1;
  }
  bitset *save = new_bitset (k);
  assign_bitset (save, b);
  u32 m = n / 2;
  for (u32 i = 0; i < m; ++i) {
    trans (b, a[i]);
  }
  u32 ans = calc (a + m, n - m, k, b);
  assign_bitset (b, save);
  free_bitset (save);
  for (u32 i = m; i < n; ++i) {
    trans (b, a[i]);
  }
  return ans + calc (a, m, k, b);
}

void run (void) {
  u32 n, k;
  scanf ("%" SCNu32 "%" SCNu32, &n, &k);
  u32 *a = (u32 *) calloc (n, sizeof (u32));
  for (u32 i = 0; i < n; ++i) {
    scanf ("%" SCNu32, a + i);
  }
  bitset *b = new_bitset (k);
  setAt_bitset (b, 0);
  u32 ans = calc (a, n, k, b);
  printf ("%" PRIu32 "\n", ans);
}

int main (void) {
  run ();
  return 0;
}