Burst Balloons (H)
题目
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:
- You may imagine
nums[-1] = nums[n] = 1
. They are not real therefore you can not burst them. - 0 ≤
n
≤ 500, 0 ≤nums[i]
≤ 100
Example:
Input: [3,1,5,8]
Output: 167
Explanation: 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
题意
打气球游戏。每个气球有对应的分值,打爆一个气球i,能得到的积分为气球i、气球i-1、气球i+1积分的乘积。要求计算能够获得的最大积分。
思路
动态规划问题。dp[left, right]表示打爆从left到right所有气球能得到的最大积分。每一次我们在[left, right]这个区间中选择一个i,代表该区间中最后一个被打爆的气球,那么可以得到递推关系式:
\[dp[left,\ right]=\max_{left \le i \le right}(dp[left,\ i-1]+dp[i+1,\ right]+nums[left-1]*nums[i]*nums[right+1]) \]
代码实现
Java
记忆化搜索
class Solution {
public int maxCoins(int[] nums) {
int[][] memo = new int[nums.length][nums.length];
for (int i = 0; i < nums.length; i++) {
Arrays.fill(memo[i], -1);
}
return dfs(nums, 0, nums.length - 1, memo);
}
private int dfs(int[] nums, int left, int right, int[][] memo) {
if (right < left) {
return 0;
}
if (memo[left][right] != -1) {
return memo[left][right];
}
int max = 0;
for (int i = left; i <= right; i++) {
int x = dfs(nums, left, i - 1, memo)
+ dfs(nums, i + 1, right, memo)
+ nums[i] * (left == 0 ? 1 : nums[left - 1]) * (right == nums.length - 1 ? 1 : nums[right + 1]);
max = Math.max(max, x);
}
memo[left][right] = max;
return max;
}
}
动态规划
class Solution {
public int maxCoins(int[] nums) {
if (nums.length == 0) {
return 0;
}
int[][] dp = new int[nums.length][nums.length];
for (int right = 0; right < nums.length; right++) {
for (int left = right; left >= 0; left--) {
for (int i = left; i <= right; i++) {
int leftMax = i == left ? 0 : dp[left][i - 1];
int rightMax = i == right ? 0 : dp[i + 1][right];
int last = nums[i] * (left == 0 ? 1 : nums[left - 1]) * (right == nums.length - 1 ? 1 : nums[right + 1]);
dp[left][right] = Math.max(dp[left][right], leftMax + rightMax + last);
}
}
}
return dp[0][nums.length - 1];
}
}
来源:oschina
链接:https://my.oschina.net/u/4323284/blog/4422842