[LeetCode] Burst Ballons

Given n balloons, indexed from 0 to n-1. Each balloon is painted with a number on it represented by array nums. You are asked to burst all the balloons. If the you burst balloon i you will get nums[left] * nums[i] * nums[right] coins. Here left and right are adjacent indices of i. After the burst, the left and right then becomes adjacent.

Find the maximum coins you can collect by bursting the balloons wisely.

Note: 
(1) You may imagine nums[-1] = nums[n] = 1. They are not real therefore you can not burst them.
(2) 0 ≤ n ≤ 500, 0 ≤ nums[i] ≤ 100

Example:

Given [3, 1, 5, 8]

Return 167

    nums = [3,1,5,8] --> [3,5,8] -->   [3,8]   -->  [8]  --> []
   coins =  3*1*5      +  3*5*8    +  1*3*8      + 1*8*1   = 167


Denote dp[i][j] the maximum coins we collect by bursting ballons i~j (0 <= i <= j < n) , then the state transition function is:
　　
　　dp[i][j] = max { dp[i][k-1] + score(k) + dp[k+1][j] } for k in range [i, j]
       score[k] = nums[i-1] * nums[k] * nums[j+1]
　　
Innitialization: for any i and j , dp[i][j] = 0 

Time Complexity: O(n^3),  Space Complexity: O(n^2)


Sample Code in Golang
  
  burst_ballons.go

package burst_ballons

func MaxCoins(nums []int) int {
    n := len(nums)
    
    ar := make([]int, n + 2)
    for i := 1; i <= n; i++ {
        ar[i] = nums[i - 1]
    }
    ar[0], ar[n + 1] = 1, 1

    dp := make([][]int, n + 2)
    for i := 0; i < n + 2; i++ {
        dp[i] = make([]int, n + 2)
    }

    max := func(x, y int) int {
        if x > y {
            return x
        }
        return y
    }

    for i := n; i > 0; i-- {
        for j := i; j < n + 1; j++ {
            for k := i; k <= j; k++ {
                scr := ar[i - 1] * ar[k] * ar[j + 1]
                tmp := dp[i][k - 1] + scr + dp[k + 1][j]
                dp[i][j] = max(dp[i][j], tmp)
            }
        }
    }

    return dp[1][n]
}

 

  burst_ballons_test.go

package burst_ballons

import "testing"

func check_answer(t *testing.T, v1, v2 int) {
    if v1 != v2 {
        t.Fatalf("wrong answer %d, expected %d", v1, v2)
    }
}

func TestMaxCoins(t *testing.T) {
    nums := []int{}
    n := MaxCoins(nums)
    check_answer(t, n, 0)
    
    nums = []int{5}
    n = MaxCoins(nums)
    check_answer(t, n, 5)

    nums = []int{3, 5, 8}
    n = MaxCoins(nums)
    check_answer(t, n, 152)

    nums = []int{3, 1, 5, 8}
    n = MaxCoins(nums)
    check_answer(t, n, 167)
}