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3180. Maximum Total Reward Using Operations I

Description

You are given an integer array rewardValues of length n, representing the values of rewards.

Initially, your total reward x is 0, and all indices are unmarked. You are allowed to perform the following operation any number of times:

  • Choose an unmarked index i from the range [0, n - 1].
  • If rewardValues[i] is greater than your current total reward x, then add rewardValues[i] to x (i.e., x = x + rewardValues[i]), and mark the index i.

Return an integer denoting the maximum total reward you can collect by performing the operations optimally.

 

Example 1:

Input: rewardValues = [1,1,3,3]

Output: 4

Explanation:

During the operations, we can choose to mark the indices 0 and 2 in order, and the total reward will be 4, which is the maximum.

Example 2:

Input: rewardValues = [1,6,4,3,2]

Output: 11

Explanation:

Mark the indices 0, 2, and 1 in order. The total reward will then be 11, which is the maximum.

 

Constraints:

  • 1 <= rewardValues.length <= 2000
  • 1 <= rewardValues[i] <= 2000

Solutions

We can sort the rewardValues array and then use memoization to solve for the maximum total reward.

We define a function $\textit{dfs}(x)$, representing the maximum total reward that can be obtained when the current total reward is $x$. Thus, the answer is $\textit{dfs}(0)$.

The execution process of the function $\textit{dfs}(x)$ is as follows:

  1. Perform a binary search in the rewardValues array for the index $i$ of the first element greater than $x$;
  2. Iterate over the elements in the rewardValues array starting from index $i$, and for each element $v$, calculate the maximum value of $v + \textit{dfs}(x + v)$.
  3. Return the result.

To avoid repeated calculations, we use a memoization array f to record the results that have already been computed.

The time complexity is $O(n \times (\log n + M))$, and the space complexity is $O(M)$. Where $n$ is the length of the rewardValues array, and $M$ is twice the maximum value in the rewardValues array.

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class Solution:
    def maxTotalReward(self, rewardValues: List[int]) -> int:
        @cache
        def dfs(x: int) -> int:
            i = bisect_right(rewardValues, x)
            ans = 0
            for v in rewardValues[i:]:
                ans = max(ans, v + dfs(x + v))
            return ans

        rewardValues.sort()
        return dfs(0)

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class Solution {
    private int[] nums;
    private Integer[] f;

    public int maxTotalReward(int[] rewardValues) {
        nums = rewardValues;
        Arrays.sort(nums);
        int n = nums.length;
        f = new Integer[nums[n - 1] << 1];
        return dfs(0);
    }

    private int dfs(int x) {
        if (f[x] != null) {
            return f[x];
        }
        int i = Arrays.binarySearch(nums, x + 1);
        i = i < 0 ? -i - 1 : i;
        int ans = 0;
        for (; i < nums.length; ++i) {
            ans = Math.max(ans, nums[i] + dfs(x + nums[i]));
        }
        return f[x] = ans;
    }
}

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class Solution {
public:
    int maxTotalReward(vector<int>& rewardValues) {
        sort(rewardValues.begin(), rewardValues.end());
        int n = rewardValues.size();
        int f[rewardValues.back() << 1];
        memset(f, -1, sizeof(f));
        function<int(int)> dfs = [&](int x) {
            if (f[x] != -1) {
                return f[x];
            }
            auto it = upper_bound(rewardValues.begin(), rewardValues.end(), x);
            int ans = 0;
            for (; it != rewardValues.end(); ++it) {
                ans = max(ans, rewardValues[it - rewardValues.begin()] + dfs(x + *it));
            }
            return f[x] = ans;
        };
        return dfs(0);
    }
};

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func maxTotalReward(rewardValues []int) int {
    sort.Ints(rewardValues)
    n := len(rewardValues)
    f := make([]int, rewardValues[n-1]<<1)
    for i := range f {
        f[i] = -1
    }
    var dfs func(int) int
    dfs = func(x int) int {
        if f[x] != -1 {
            return f[x]
        }
        i := sort.SearchInts(rewardValues, x+1)
        f[x] = 0
        for _, v := range rewardValues[i:] {
            f[x] = max(f[x], v+dfs(x+v))
        }
        return f[x]
    }
    return dfs(0)
}

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function maxTotalReward(rewardValues: number[]): number {
    rewardValues.sort((a, b) => a - b);
    const search = (x: number): number => {
        let [l, r] = [0, rewardValues.length];
        while (l < r) {
            const mid = (l + r) >> 1;
            if (rewardValues[mid] > x) {
                r = mid;
            } else {
                l = mid + 1;
            }
        }
        return l;
    };
    const f: number[] = Array(rewardValues.at(-1)! << 1).fill(-1);
    const dfs = (x: number): number => {
        if (f[x] !== -1) {
            return f[x];
        }
        let ans = 0;
        for (let i = search(x); i < rewardValues.length; ++i) {
            ans = Math.max(ans, rewardValues[i] + dfs(x + rewardValues[i]));
        }
        return (f[x] = ans);
    };
    return dfs(0);
}

Solution 2: Dynamic Programming

We define $f[i][j]$ as whether it is possible to obtain a total reward of $j$ using the first $i$ reward values. Initially, $f[0][0] = \textit{True}$, and all other values are $\textit{False}$.

We consider the $i$-th reward value $v$. If we do not choose it, then $f[i][j] = f[i - 1][j]$. If we choose it, then $f[i][j] = f[i - 1][j - v]$, where $0 \leq j - v < v$. Thus, the state transition equation is:

$$ f[i][j] = f[i - 1][j] \vee f[i - 1][j - v] $$

The final answer is $\max{j \mid f[n][j] = \textit{True}}$.

Since $f[i][j]$ only depends on $f[i - 1][j]$ and $f[i - 1][j - v]$, we can optimize away the first dimension and use only a one-dimensional array for state transitions.

The time complexity is $O(n \times M)$, and the space complexity is $O(M)$. Where $n$ is the length of the rewardValues array, and $M$ is twice the maximum value in the rewardValues array.

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class Solution:
    def maxTotalReward(self, rewardValues: List[int]) -> int:
        nums = sorted(set(rewardValues))
        m = nums[-1] << 1
        f = [False] * m
        f[0] = True
        for v in nums:
            for j in range(m):
                if 0 <= j - v < v:
                    f[j] |= f[j - v]
        ans = m - 1
        while not f[ans]:
            ans -= 1
        return ans

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class Solution {
    public int maxTotalReward(int[] rewardValues) {
        int[] nums = Arrays.stream(rewardValues).distinct().sorted().toArray();
        int n = nums.length;
        int m = nums[n - 1] << 1;
        boolean[] f = new boolean[m];
        f[0] = true;
        for (int v : nums) {
            for (int j = 0; j < m; ++j) {
                if (0 <= j - v && j - v < v) {
                    f[j] |= f[j - v];
                }
            }
        }
        int ans = m - 1;
        while (!f[ans]) {
            --ans;
        }
        return ans;
    }
}

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class Solution {
public:
    int maxTotalReward(vector<int>& rewardValues) {
        sort(rewardValues.begin(), rewardValues.end());
        rewardValues.erase(unique(rewardValues.begin(), rewardValues.end()), rewardValues.end());
        int n = rewardValues.size();
        int m = rewardValues.back() << 1;
        bool f[m];
        memset(f, false, sizeof(f));
        f[0] = true;
        for (int v : rewardValues) {
            for (int j = 1; j < m; ++j) {
                if (0 <= j - v && j - v < v) {
                    f[j] = f[j] || f[j - v];
                }
            }
        }
        int ans = m - 1;
        while (!f[ans]) {
            --ans;
        }
        return ans;
    }
};

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func maxTotalReward(rewardValues []int) int {
    slices.Sort(rewardValues)
    nums := slices.Compact(rewardValues)
    n := len(nums)
    m := nums[n-1] << 1
    f := make([]bool, m)
    f[0] = true
    for _, v := range nums {
        for j := 1; j < m; j++ {
            if 0 <= j-v && j-v < v {
                f[j] = f[j] || f[j-v]
            }
        }
    }
    ans := m - 1
    for !f[ans] {
        ans--
    }
    return ans
}

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function maxTotalReward(rewardValues: number[]): number {
    const nums = Array.from(new Set(rewardValues)).sort((a, b) => a - b);
    const n = nums.length;
    const m = nums[n - 1] << 1;
    const f: boolean[] = Array(m).fill(false);
    f[0] = true;
    for (const v of nums) {
        for (let j = 1; j < m; ++j) {
            if (0 <= j - v && j - v < v) {
                f[j] = f[j] || f[j - v];
            }
        }
    }
    let ans = m - 1;
    while (!f[ans]) {
        --ans;
    }
    return ans;
}

Solution 3: Dynamic Programming + Bit Manipulation

We can optimize Solution 2 by defining a binary number $f$ to save the current state, where the $i$-th bit of $f$ being $1$ indicates that a total reward of $i$ is reachable.

Observing the state transition equation from Solution 2, $f[j] = f[j] \vee f[j - v]$, this is equivalent to taking the lower $v$ bits of $f$, shifting them left by $v$ bits, and then performing an OR operation with the original $f$.

Thus, the answer is the position of the highest bit in $f$.

The time complexity is $O(n \times M / w)$, and the space complexity is $O(n + M / w)$. Where $n$ is the length of the rewardValues array, $M$ is twice the maximum value in the rewardValues array, and the integer $w = 32$ or $64$.

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class Solution:
    def maxTotalReward(self, rewardValues: List[int]) -> int:
        nums = sorted(set(rewardValues))
        f = 1
        for v in nums:
            f |= (f & ((1 << v) - 1)) << v
        return f.bit_length() - 1

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import java.math.BigInteger;
import java.util.Arrays;

class Solution {
    public int maxTotalReward(int[] rewardValues) {
        int[] nums = Arrays.stream(rewardValues).distinct().sorted().toArray();
        BigInteger f = BigInteger.ONE;
        for (int v : nums) {
            BigInteger mask = BigInteger.ONE.shiftLeft(v).subtract(BigInteger.ONE);
            BigInteger shifted = f.and(mask).shiftLeft(v);
            f = f.or(shifted);
        }
        return f.bitLength() - 1;
    }
}

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class Solution {
public:
    int maxTotalReward(vector<int>& rewardValues) {
        sort(rewardValues.begin(), rewardValues.end());
        rewardValues.erase(unique(rewardValues.begin(), rewardValues.end()), rewardValues.end());
        bitset<100000> f{1};
        for (int v : rewardValues) {
            int shift = f.size() - v;
            f |= f << shift >> (shift - v);
        }
        for (int i = rewardValues.back() * 2 - 1;; i--) {
            if (f.test(i)) {
                return i;
            }
        }
    }
};

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func maxTotalReward(rewardValues []int) int {
    slices.Sort(rewardValues)
    rewardValues = slices.Compact(rewardValues)
    one := big.NewInt(1)
    f := big.NewInt(1)
    p := new(big.Int)
    for _, v := range rewardValues {
        mask := p.Sub(p.Lsh(one, uint(v)), one)
        f.Or(f, p.Lsh(p.And(f, mask), uint(v)))
    }
    return f.BitLen() - 1
}

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function maxTotalReward(rewardValues: number[]): number {
    rewardValues.sort((a, b) => a - b);
    rewardValues = [...new Set(rewardValues)];
    let f = 1n;
    for (const x of rewardValues) {
        const mask = (1n << BigInt(x)) - 1n;
        f = f | ((f & mask) << BigInt(x));
    }
    return f.toString(2).length - 1;
}

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