# Best Time to Buy and Sell Stock IV

## 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.

Example
Given prices = [4,4,6,1,1,4,2,5], and k = 2, return 6.

Note
You may not engage in multiple transactions at the same time
(i.e., you must sell the stock before you buy again).

Challenge
O(nk) time.


## 题解1

f[i][j] 代表第 i 天为止交易 k 次获得的最大收益，那么将问题分解为前 x 天交易 k-1 次，第 x+1 天至第 i 天交易一次两个子问题，于是动态方程如下：

f[i][j] = max(f[x][j - 1] + profit(x + 1, i))


### Python

class Solution:
"""
@param k: an integer
@param prices: a list of integer
@return: an integer which is maximum profit
"""
def maxProfit(self, k, prices):
if prices is None or len(prices) <= 1 or k <= 0:
return 0

n = len(prices)
# k >= prices.length / 2 ==> multiple transactions Stock II
if k >= n / 2:
profit_max = 0
for i in xrange(1, n):
diff = prices[i] - prices[i - 1]
if diff > 0:
profit_max += diff
return profit_max

f = [[0 for i in xrange(k + 1)] for j in xrange(n + 1)]
for j in xrange(1, k + 1):
for i in xrange(1, n + 1):
for x in xrange(0, i + 1):
f[i][j] = max(f[i][j], f[x][j - 1] + self.profit(prices, x + 1, i))

return f[n][k]

# calculate the profit of prices(l, u)
def profit(self, prices, l, u):
if l >= u:
return 0
valley = 2**31 - 1
profit_max = 0
for price in prices[l - 1:u]:
profit_max = max(profit_max, price - valley)
valley = min(valley, price)
return profit_max


### C++

class Solution {
public:
/**
* @param k: An integer
* @param prices: Given an integer array
* @return: Maximum profit
*/
int maxProfit(int k, vector<int> &prices) {
if (prices.size() <= 1 || k <= 0) return 0;

int n = prices.size();
// k >= prices.length / 2 ==> multiple transactions Stock II
if (k >= n / 2) {
int profit_max = 0;
for (int i = 1; i < n; ++i) {
int diff = prices[i] - prices[i - 1];
if (diff > 0) {
profit_max += diff;
}
}
return profit_max;
}

vector<vector<int> > f = vector<vector<int> >(n + 1, vector<int>(k + 1, 0));
for (int j = 1; j <= k; ++j) {
for (int i = 1; i <= n; ++i) {
for (int x = 0; x <= i; ++x) {
f[i][j] = max(f[i][j], f[x][j - 1] + profit(prices, x + 1, i));
}
}
}

return f[n][k];
}

private:
int profit(vector<int> &prices, int l, int u) {
if (l >= u) return 0;

int valley = INT_MAX;
int profit_max = 0;
for (int i = l - 1; i < u; ++i) {
profit_max = max(profit_max, prices[i] - valley);
valley = min(valley, prices[i]);
}

return profit_max;
}
};


### Java

class Solution {
/**
* @param k: An integer
* @param prices: Given an integer array
* @return: Maximum profit
*/
public int maxProfit(int k, int[] prices) {
if (prices == null || prices.length <= 1 || k <= 0) return 0;

int n = prices.length;
if (k >= n / 2) {
int profit_max = 0;
for (int i = 1; i < n; i++) {
if (prices[i] - prices[i - 1] > 0) {
profit_max += prices[i] - prices[i - 1];
}
}
return profit_max;
}

int[][] f = new int[n + 1][k + 1];
for (int j = 1; j <= k; j++) {
for (int i = 1; i <= n; i++) {
for (int x = 0; x <= i; x++) {
f[i][j] = Math.max(f[i][j], f[x][j - 1] + profit(prices, x + 1, i));
}
}
}

return f[n][k];
}

private int profit(int[] prices, int l, int u) {
if (l >= u) return 0;

int valley = Integer.MAX_VALUE;
int profit_max = 0;
for (int i = l - 1; i < u; i++) {
profit_max = Math.max(profit_max, prices[i] - valley);
valley = Math.min(valley, prices[i]);
}
return profit_max;
}
};