c++ - 加油站任务变化的DP算法

Question

我遇到一个问题，我从 A 点到 B 点，距离定义为 l。在我的车里，我有一辆坦克，只能让我行驶 b 公里。在 n 个地方有加油站和加满我的油箱的费用(我只能加满)。我从装满水箱开始。

从高速公路起点到终点花费最少的最有效算法是什么？

我最近的想法:

使用窗口滑动最小算法找到(长度为b)成本最小的站点。代码:

int n, range, targetDist;
scanf("%d %d %d", &n, &targetDist, &range);
int di, ci;
for (int k = 1; k <= n; k++)
{
    scanf("%d %d", &di, &ci);
    D[k] = di;
    C[k] = ci;
}
D[0] = C[0] = bestcost[0] = 0;
for (int i = 1; i < n+1; ++i)
{
    bestcost[i] = 1 << 30;
    for (int j = 0; j < i; ++j)
    {
        int xx = bestcost[j] + C[i];
        if (D[i] - D[j] <= range && xx < bestcost[i])
        {
            bestcost[i] = xx;
            printf("(%d, %d)\n", i, bestcost[i]);
        }
    }
}

输入是:

3 8 4
2 1
4 2
6 3

输出是:

(1, 1)
(2, 2)
(3, 4) 

所以它对应于到 (i, cost(i)) 的成本——为了到达第 i 个站，我必须支付 cost(i)。

如何使用该信息找到整个距离的最低成本？

Answer 1

我找到了一种使用滑动窗口算法在 O(n) 时间内完成此操作的方法:

#include <vector>
#include <algorithm>
#include <deque>
#include <stdio.h>


typedef unsigned long long int ulli;

using namespace std;

void sliding_window_minimum(vector<pair<ulli, ulli>> st, ulli n, ulli targetDist, ulli range)
{
    deque<pair<ulli, ulli>> minimize;
    ulli j = 0;
    for (ulli i = 0; i < n; i++)
    {
        if (st[i].first <= range)
        {
            while (!(minimize.empty()) && (minimize.back().second > st[i].second))
            {
                minimize.pop_back();
            }
            minimize.push_back(st[i]);
            j++;
        }
        else
        {
            break;
        }
    }
    for (ulli k = j; k < n; k++)
    {
        while (!(minimize.empty()) && ((st[k].first - minimize.front().first) > range))
        {
            minimize.pop_front();
        }
        if (minimize.empty()) {
            break;
        }
        ulli tempCost = st[k].second + minimize.front().second;
        while (!(minimize.empty()) && (minimize.back().second > tempCost))
        {
            minimize.pop_back();
        }
        minimize.push_back(make_pair(st[k].first, tempCost));
    }
    while (!(minimize.empty()) && ((targetDist - minimize.front().first) > range))
    {
        minimize.pop_front();
    }
    if (minimize.empty())
    {
        printf("NIE\n");
    }
    else
    {
        printf("%llu", minimize.front().second);
    }
}

int main()
{
    ulli n, d, b;
    scanf("%llu %llu %llu", &n, &d, &b);
    if (b >= d)
    {
        printf("%d", 0);
        return 0;
    }
    int temp = n;
    ulli d1, c1;
    vector<pair<ulli, ulli>> st;
    st.reserve(n+1);
    while (temp--)
    {
        scanf("%llu %llu", &d1, &c1);
        st.push_back(make_pair(d1, c1));
    }
    sliding_window_minimum(st, n, d, b);
}