Problem:
Score : 400 points

Problem Statement

Your friend gave you a dequeue D as a birthday present.

D is a horizontal cylinder that contains a row of N jewels.

The values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.

In the beginning, you have no jewel in your hands.

You can perform at most K operations on D, chosen from the following, at most K times (possibly zero):

Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.

Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.

Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.

Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.

Find the maximum possible sum of the values of jewels in your hands after the operations.

Constraints

All values in input are integers.

1 \leq N \leq 50

1 \leq K \leq 100

-10^7 \leq V_i \leq 10^7

Input

Input is given from Standard Input in the following format:

N K
V_1 V_2 ... V_N

Output

Print the maximum possible sum of the values of jewels in your hands after the operations.

Sample Input 1

6 4
-10 8 2 1 2 6

Sample Output 1

14

After the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.

Do operation A. You take out the jewel of value -10 from the left end of D.

Do operation B. You take out the jewel of value 6 from the right end of D.

Do operation A. You take out the jewel of value 8 from the left end of D.

Do operation D. You insert the jewel of value -10 to the right end of D.

Sample Input 2

6 4
-6 -100 50 -2 -5 -3

Sample Output 2

44

Sample Input 3

6 3
-6 -100 50 -2 -5 -3

Sample Output 3

0

It is optimal to do no operation.

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.Collections;
import java.util.Deque;
import java.util.HashMap;
import java.util.LinkedList;
import java.util.List;
import java.util.Map;
import java.util.PriorityQueue;
import java.util.Scanner;
import java.util.function.Supplier;

public class Main {

    static void solve() {
        int n = nextInt();
        int k = nextInt();
        List<Integer> list = new ArrayList<>();
        for (int i = 0; i < n; i++) {
            int v = nextInt();
            list.add(v);
        }

        Deque<Integer> deque;
        PriorityQueue<Integer> pque = new PriorityQueue<>();
        int ans = Integer.MIN_VALUE;

        // 左右から取得する場合
        for (int num = k / 2; num <= k; num++) { // 取得する回数
            for (int left = 0; left <= num; left++) {
                deque = new LinkedList<>(list);
                pque.clear();
                int right = num - left;
                for (int i = 0; i < left; i++) { // 取得処理(左から)
                    Integer v = deque.pollFirst();
                    if (v == null)
                        break;
                    pque.add(v);
                }
                for (int i = 0; i < right; i++) { // 取得処理(右から)
                    Integer v = deque.pollLast();
                    if (v == null)
                        break;
                    pque.add(v);
                }
                for (int i = 0; i < k - num; i++) { // 戻す回数
                    Integer v = pque.peek();
                    if (v != null && v < 0)
                        pque.remove();
                }
                int candidate = pque.stream().mapToInt(e -> e).sum();
                if (candidate > ans)
                    ans = candidate;
            }
        }

        out.println(ans);
    }

    static final int MOD = 1_000_000_007;
    static long[] fac, finv, inv;

    // 階乗(n!)
    static long factorial(long n) {
        long ans = 1;
        for (long i = n; i > 0; i--) {
            ans = ans * i % MOD;
        }
        return ans;
    }

    // nCkの初期化
    static void comInit(int max) {
        fac = new long[max];
        finv = new long[max];
        inv = new long[max];
        fac[0] = fac[1] = 1;
        finv[0] = finv[1] = 1;
        inv[1] = 1;

        for (int i = 2; i < max; i++) {
            fac[i] = fac[i - 1] * i % MOD;
            inv[i] = MOD - inv[MOD % i] * (MOD / i) % MOD;
            finv[i] = finv[i - 1] * inv[i] % MOD;
        }
    }

    // nCkの計算
    static long com(int n, int k) {
        if (n < k)
            return 0;
        if (n < 0 || k < 0)
            return 0;
        return fac[n] * (finv[k] * finv[n - k] % MOD) % MOD;
    }

    static Map<Integer, List<Integer>> primeFactors = new HashMap<>();

    // 素因数分解
    static List<Integer> primeFactorize(int n) {
        if (primeFactors.containsKey(n))
            return primeFactors.get(n);

        List<Integer> list = new ArrayList<>();
        if (n % 2 == 0) {
            list.addAll(primeFactorize(n / 2));
            list.add(2);
        } else {
            boolean ok = true;
            for (int j = 3; j * j <= n; j += 2) {
                if (n % j == 0) {
                    list.addAll(primeFactorize(n / j));
                    list.addAll(primeFactorize(j));
                    ok = false;
                    break;
                }
            }
            if (ok) {
                if (n != 1)
                    list.add(n);
            }
        }

        primeFactors.put(n, Collections.unmodifiableList(list));
        return list;
    }

    // 2〜nまでの数値を素因数分解したmapを返す.
    // map: 数値 -> (素因数)
    // ex)
    // 2 -> (2)
    // 4 -> (2,2)
    // 12 -> (2,2,3)
    @Deprecated // primeFactorizeを使うべき
    static Map<Integer, List<Integer>> primeFactorizeTo(int n) {
        Map<Integer, List<Integer>> map = new HashMap<>();
        for (int i = 2; i <= n; i++) {
            List<Integer> list = new ArrayList<>();
            int tmp = i;
            for (int j = i - 1; j > 1; j--) {
                if (tmp % j == 0) {
                    if (map.containsKey(j)) {
                        list.addAll(map.get(j));
                    } else {
                        list.add(j);
                    }
                    tmp /= j;
                }
            }
            if (tmp != 1)
                list.add(tmp);
            map.put(i, list);
        }
        return map;
    }

    static PrintWriter out;
    static Scanner sc;
    static boolean debugEnabled;
    static long start;
    static long end;

    static int[][] newIntArray(int h, int w, int value) {
        int[][] ret = new int[h][w];
        for (int i = 0; i < h; i++) {
            for (int j = 0; j < w; j++) {
                ret[i][j] = value;
            }
        }
        return ret;
    }

    static int nextInt() {
        return Integer.parseInt(sc.next());
    }

    static long nextLong() {
        return Long.parseLong(sc.next());
    }

    static String nextString() {
        return sc.next();
    }

    static int[] nextIntArray(int n) {
        int[] a = new int[n];
        for (int i = 0; i < n; i++) {
            a[i] = nextInt();
        }
        return a;
    }

    static long[] nextLongArray(int n) {
        long[] a = new long[n];
        for (int i = 0; i < n; i++) {
            a[i] = nextLong();
        }
        return a;
    }

    static List<Integer> nextIntList(int n) {
        List<Integer> list = new ArrayList<>();
        for (int i = 0; i < n; i++) {
            list.add(nextInt());
        }
        return list;
    }

    static List<Double> nextDoubleList(int n) {
        List<Double> list = new ArrayList<>();
        for (int i = 0; i < n; i++) {
            list.add((double) nextInt());
        }
        return list;
    }

    static List<Long> nextLongList(int n) {
        List<Long> list = new ArrayList<>();
        for (int i = 0; i < n; i++) {
            list.add(nextLong());
        }
        return list;
    }

    static char[][] nextCharArray(int h, int w) {
        char[][] c = new char[h][w];
        for (int i = 0; i < h; i++) {
            String str = nextString();
            for (int j = 0; j < w; j++) {
                c[i][j] = str.charAt(j);
            }
        }
        return c;
    }

    // 昇順
    static <T extends Comparable<? super T>> List<T> sort(List<T> list) {
        return sortAsc(list);
    }

    // 昇順
    static <T extends Comparable<? super T>> List<T> sortAsc(List<T> list) {
        Collections.sort(list);
        return list;
    }

    // 降順
    static <T extends Comparable<? super T>> List<T> sortDesc(List<T> list) {
        Collections.sort(list, (e1, e2) -> e2.compareTo(e1));
        return list;
    }

    // greatest common divisor
    // 最大公約数
    static long gcd(long a, long b) {
        if (b == 0) {
            return a;
        }
        return gcd(b, a % b);
    }

    // least common multiple
    // 最小公倍数
    static long lcm(long a, long b) {
        if (a > b) {
            return (a / gcd(a, b)) * b;
        } else {
            return (b / gcd(a, b)) * a;
        }
    }

    // least common multiple
    // 最小公倍数
    // 素因数分解されたListを引数とした場合
    static List<Integer> lcm(List<Integer> a, List<Integer> b) {
        List<Integer> ret = new ArrayList<>(a);
        List<Integer> tmp = new ArrayList<>(b);
        for (Integer i : a) {
            if (tmp.contains(i))
                tmp.remove(i);
        }
        ret.addAll(tmp);
        return ret;
    }

    // baseのn乗を計算を返す
    static int pow(int base, int n) {
        int ret = 1;
        for (int i = 0; i < n; i++) {
            ret *= base;
        }
        return ret;
    }

    // return n^k mod m
    static long powMod(long n, long k, long m) {
        if (k == 0) {
            return 1;
        } else if (k % 2 == 1) {
            return powMod(n, k - 1, m) * n % m;
        } else {
            long tmp = powMod(n, k / 2, m);
            return tmp * tmp % m;
        }
    }

    // intをlength桁のbit文字列に変換
    static String toBitString(int length, int n) {
        StringBuilder sb = new StringBuilder();
        for (int i = length - 1; i >= 0; i--) {
            if ((n >> i) % 2 == 1) {
                sb.append("1");
            } else {
                sb.append("0");
            }
        }
        return sb.toString();
    }

    static void setDebugEnabled() {
        String className = new Object() {
        }.getClass().getEnclosingClass().getSimpleName();

        if (!className.equals("Main"))
            debugEnabled = true;
    }

    static void debug(String msg) {
        if (debugEnabled)
            System.err.println(msg);
    }

    static void debug(Supplier<String> msg) {
        if (debugEnabled)
            System.err.println(msg.get());
    }

    static class UnionFind {
        int[] parent;

        UnionFind(int n) {
            parent = new int[n];
            init();
        }

        void init() {
            for (int i = 0; i < parent.length; i++) {
                parent[i] = i;
            }
        }

        int getRoot(int i) {
            if (parent[i] == i) {
                return i;
            } else {
                return parent[i] = getRoot(parent[i]);
            }
        }

        boolean isSame(int x, int y) {
            return getRoot(x) == getRoot(y);
        }

        void unit(int x, int y) {
            x = getRoot(x);
            y = getRoot(y);
            if (x != y) {
                parent[x] = y;
            }

        }
    }

    public static void main(String[] args) {
        preProcess();
        // System.exit(0)で終了した場合も、解答を出力するためにはshutdown hookの設定が必要
        // しかし、これを有効化すると100[msec]程度処理時間が長くなる
        // Runtime.getRuntime().addShutdownHook(new Thread(() -> postProcess()));
        solve();
        postProcess();
    }

    static void preProcess() {
        start = System.nanoTime();
        setDebugEnabled();
        out = new PrintWriter(System.out);
        sc = new Scanner(System.in);
    }

    static void postProcess() {
        out.flush();
        sc.close();
        end = System.nanoTime();
        debug(() -> "elapsed time: " + (end - start) / 1_000_000 + "[msec]");
    }

}