Question
Say you have an array for which the ith element is the price of a given stock on day i.
Design an algorithm to find the maximum profit. You may complete at most k transactions.
For example, given prices = [4,4,6,1,1,4,2,5], and k = 2, return 6.
Answer
用DP解答。
local[i][j] 表示0～i的数，最多j次交易，并且最后一次交易必须包含prices[j]（即最后一天必须卖出），收益最大值。
global[i][j] 表示0~i的数，最多j次交易，收益最大值。
diff = prices[i] - prices[i-1]
local[i][j] = max(global[i-1][j-1] + max(diff,0), local[i-1][j]+diff)
global[i][j] = max(local[i][j], global[i-1][j])
local[i-1][j] + diff 表示第i-1天，买进prices[i-1]，第i天卖出。由于local[i-1][j]表示最后一次交易必须包含prices[i-1]，即prices[i-1]必须卖出。所以可以把第i-1天的交易和第i天的交易合并，即成了最多j次交易，最后一天交易必须卖出prices[i]。
global[i-1][j-1] 表示前i-1天，最多j-1次交易。最后一天比较diff，如果diff>0，则第i-1天买进，第i天卖出；如果diff<0，则第i天买进，第i天卖出。
class Solution {
/**
* @param k: An integer
* @param prices: Given an integer array
* @return: Maximum profit
*/
public int maxProfit(int k, int[] prices) {
// write your code here
if (prices == null || prices.length == 0 || k == 0) {
return 0;
}
int len = prices.length;
if (k >= len / 2) {
int profit = 0;
for (int i = 1; i < len; i++) {
if (prices[i] > prices[i - 1]) {
profit += (prices[i] - prices[i - 1]);
}
}
return profit;
}
int[][] local = new int[len][k + 1];
int[][] global = new int[len][k + 1];
for (int i = 0; i < len; i++) {
local[i][0] = 0;
global[i][0] = 0;
}
Arrays.fill(local[0], 0);
Arrays.fill(global[0], 0);
for (int i = 1; i < len; i++) {
for (int j = 1; j <= k; j++) {
int diff = prices[i] - prices[i - 1];
local[i][j] = Math.max(local[i - 1][j] + diff,
global[i - 1][j - 1] + Math.max(diff, 0));
global[i][j] = Math.max(local[i][j], global[i - 1][j]);
}
}
return global[len - 1][k];
}
};