Maximal Expression - Google Top Interview Questions


Problem Statement :


Given a list of integers nums, return the maximal value that can be made by adding any binary operators +, -, and * between the numbers in nums as well as insert any valid brackets.

Constraints

1 ≤ n ≤ 200 where n is the length of nums

Example 1

Input

nums = [-5, -3, -8]

Output

64

Explanation

We can make the following expression: (-5 + -3) * -8



Solution :



title-img




                        Solution in C++ :

int solve(vector<int>& A) {
    int N = A.size();

    int t[N][N][2];

    for (int i = 0; i < N; i++)
        for (int j = 0; j < N; j++) t[i][j][0] = INT_MAX, t[i][j][1] = INT_MIN;

    for (int i = 0; i < N; i++) t[i][i][0] = t[i][i][1] = A[i];

    for (int L = 2; L <= N; L++)
        for (int i = 0; i + L - 1 < N; i++) {
            int j = i + L - 1;

            for (int k = i; k < j; k++) {
                int a = t[i][k][0];
                int b = t[i][k][1];

                int c = t[k + 1][j][0];
                int d = t[k + 1][j][1];

                vector<int> P = {a, b};
                vector<int> Q = {c, d};

                for (int m = 0; m < 2; m++)
                    for (int n = 0; n < 2; n++) {
                        t[i][j][0] = min(P[m] * Q[n], t[i][j][0]);
                        t[i][j][1] = max(P[m] * Q[n], t[i][j][1]);

                        t[i][j][0] = min(P[m] + Q[n], t[i][j][0]);
                        t[i][j][1] = max(P[m] + Q[n], t[i][j][1]);

                        t[i][j][0] = min(P[m] - Q[n], t[i][j][0]);
                        t[i][j][1] = max(P[m] - Q[n], t[i][j][1]);
                    }
            }
        }

    return t[0][N - 1][1];
}
                    


                        Solution in Java :

import java.util.*;

class Solution {
    class Node {
        int max;
        int min;
        public Node(int max, int min) {
            this.max = max;
            this.min = min;
        }
    }
    public int solve(int[] nums) {
        int len = nums.length;
        Node[][] dp = new Node[len][len];

        for (int i = len - 1; i >= 0; i--) {
            for (int j = i; j < len; j++) {
                dp[i][j] = new Node(Integer.MIN_VALUE, Integer.MAX_VALUE);
                if (i == j) {
                    dp[i][j] = new Node(nums[i], nums[i]);
                } else {
                    int max = Integer.MIN_VALUE;
                    int min = Integer.MAX_VALUE;

                    for (int index = i; index < j; index++) {
                        max = Math.max(max, dp[i][index].max + dp[index + 1][j].max);
                        max = Math.max(max, dp[i][index].max - dp[index + 1][j].min);
                        max = Math.max(max,
                            Math.max(dp[i][index].max * dp[index + 1][j].max,
                                dp[i][index].min * dp[index + 1][j].min));
                        min = Math.min(min, dp[i][index].min + dp[index + 1][j].min);
                        min = Math.min(min, dp[i][index].min - dp[index + 1][j].max);
                        min = Math.min(min,
                            Math.min(dp[i][index].max * dp[index + 1][j].min,
                                dp[i][index].min * dp[index + 1][j].max));
                    }
                    dp[i][j] = new Node(max, min);
                }
            }
        }

        return dp[0][len - 1].max;
    }
    /**
    HashMap<String, int[]> cache;
    public int solve(int[] nums) {
        int len = nums.length;
        cache = new HashMap<>();
        return process(nums, 0, len - 1)[0];
    }

    //[MAX, MIN]
    public int[] process(int[] nums, int i, int j) {
        //System.out.println("I" + i + "j" + j);
        // if (i > j) {
        //     return new int[]{Integer.MIN_VALUE, Integer.MAX_VALUE};
        // }
        // if (i >= nums.length || j >= nums.length) {
        //     return new int[]{Integer.MIN_VALUE, Integer.MAX_VALUE};
        // }
        if (i == j) {
            return new int[]{nums[i], nums[i]};
        }

        String key = i + "#" + j;
        if (cache.containsKey(key)) {
            return cache.get(key);
        }

        int max = Integer.MIN_VALUE;
        int min = Integer.MAX_VALUE;
        for (int index = i; index < j; index++) {
            int[] res1 = process(nums, i, index);
            int[] res2 = process(nums, index + 1, j);
            max = Math.max(max, res1[0] + res2[0]);
            max = Math.max(max, res1[0] - res2[1]);
            max = Math.max(max, Math.max(res1[0] * res2[0], res1[1] * res2[1]));
            min = Math.min(min, res1[1] + res2[1]);
            min = Math.min(min, res1[1] - res2[0]);
            min = Math.min(min, Math.min(res1[0] * res2[1], res1[1] * res2[0]));
        }
        int[] res = new int[]{max, min};
        cache.put(key, res);
        return res;
    }
    */
}
                    


                        Solution in Python : 
                            
class Solution:
    def solve(self, nums):
        @cache
        def dp(i, j):
            if i == j:
                return nums[i], nums[i]
            elif i + 1 == j:
                return max(nums[i] * nums[j], nums[i] + nums[j], nums[i] - nums[j]), min(
                    nums[i] * nums[j], nums[i] + nums[j], nums[i] - nums[j]
                )
            else:
                maxx = -1e9
                minn = -maxx
                for k in range(i, j):
                    a, b = dp(i, k)
                    c, d = dp(k + 1, j)
                    maxx = max(maxx, a * c, b * d, a + c, a - d)
                    minn = min(minn, a * d, b * c, b + d, b - c)
                return maxx, minn

        a, b = dp(0, len(nums) - 1)
        return a
                    


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