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  • [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)
    }
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  • 原文地址:https://www.cnblogs.com/william-cheung/p/5371856.html
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