c++ - 曼哈顿度量的最小距离

Question

我正在尝试找到 Manhattan metric 中的最小距离(x,y)。我正在寻找有关此的信息。但我还没有发现任何东西。

#include<bits/stdc++.h>
using namespace std;
#define st first
#define nd second

pair<int, int> pointsA[1000001];
pair<int, int> pointsB[1000001];

int main() {
    int n, t;
    unsigned long long dist;

    scanf("%d", &t);

    while(t-->0) {
        dist = 4000000000LL;
        scanf("%d", &n);

        for(int i = 0; i < n; i++) {
            scanf("%d%d", &pointsA[i].st, &pointsA[i].nd);
        }

        for(int i = 0; i < n; i++) {
            scanf("%d%d", &pointsB[i].st, &pointsB[i].nd);
        }

        for(int i = 0; i < n ;i++) {
            for(int j = 0; j < n ; j++) {
                if(abs(pointsA[i].st - pointsB[j].st) + abs(pointsA[i].nd - pointsB[j].nd) < dist) {
                    dist = abs(pointsA[i].st - pointsB[j].st) + abs(pointsA[i].nd - pointsB[j].nd);
                }
            }
            printf("%lld\n", dist);
        }
    }
}

我的代码可以在 O(n^2) 内运行，但速度太慢。我不知道它是否有用，但点 A 中的 y 始终 > 0，点 B 中的 y 始终 < 0。我的代码比较了与下一个的实际距离并选择了最小的。

例如:

输入:

2
3
-2 2
1 3
3 1
0 -1
-1 -2
1 -2
1
1 1
-1 -1

输出:

5
4

Answer 1

我的解决方案(为简单起见，我不关心 manhattan_dist 中的溢出，因此它不适用于 unsigned long long):

#include <cstdlib>
#include <cstdio>
#include <cassert>
#include <vector>
#include <limits>
#include <algorithm>

typedef std::pair<int, int> Point;
typedef std::vector<std::pair<int, int> > PointsList;

static inline bool cmp_by_x(const Point &a, const Point &b)
{
    if (a.first < b.first) {
        return true;
    } else if (a.first > b.first) {
        return false;
    } else {
        return a.second < b.second;
    }
}

static inline bool cmp_by_y(const Point &a, const Point &b)
{
    if (a.second < b.second) {
        return true;
    } else if (a.second > b.second) {
        return false;
    } else {
        return a.first < b.first;
    }
}

static inline unsigned manhattan_dist(const Point &a, const Point &b)
{
    return std::abs(a.first - b.first) +
        std::abs(a.second - b.second);
}

int main()
{
    unsigned int n_iter = 0;
    if (scanf("%u", &n_iter) != 1) {
        std::abort();
    }
    for (unsigned i = 0; i < n_iter; ++i) {
        unsigned int N = 0;
        if (scanf("%u", &N) != 1) {
            std::abort();
        }
        if (N == 0) {
            continue;
        }
        PointsList pointsA(N);
        for (PointsList::iterator it = pointsA.begin(), endi = pointsA.end(); it != endi; ++it) {
            if (scanf("%d%d", &it->first, &it->second) != 2) {
                std::abort();
            }
            assert(it->second > 0);
        }
        PointsList pointsB(N);
        for (PointsList::iterator it = pointsB.begin(), endi = pointsB.end(); it != endi; ++it) {
            if (scanf("%d%d", &it->first, &it->second) != 2) {
                std::abort();
            }
            assert(it->second < 0);
        }

        std::sort(pointsA.begin(), pointsA.end(), cmp_by_y);
        std::sort(pointsB.begin(), pointsB.end(), cmp_by_y);
        const PointsList::const_iterator min_a_by_y = pointsA.begin();
        const PointsList::const_iterator max_b_by_y = (pointsB.rbegin() + 1).base();
        assert(*max_b_by_y == pointsB.back());

        unsigned dist = manhattan_dist(*min_a_by_y, *max_b_by_y);
        const unsigned diff_x = std::abs(min_a_by_y->first - max_b_by_y->first);
        const unsigned best_diff_y = dist - diff_x;

        const int max_y_for_a = max_b_by_y->second + dist;
        const int min_y_for_b = min_a_by_y->second - dist;
        PointsList::iterator it;
        for (it = pointsA.begin() + 1; it != pointsA.end() && it->second <= max_y_for_a; ++it) {
        }
        if (it != pointsA.end()) {
            pointsA.erase(it, pointsA.end());
        }

        PointsList::reverse_iterator rit;
        for (rit = pointsB.rbegin() + 1; rit != pointsB.rend() && rit->second >= min_y_for_b; ++rit) {
        }
        if (rit != pointsB.rend()) {
            pointsB.erase(pointsB.begin(), (rit + 1).base());
        }
        std::sort(pointsA.begin(), pointsA.end(), cmp_by_x);
        std::sort(pointsB.begin(), pointsB.end(), cmp_by_x);

        for (size_t j = 0; diff_x > 0 && j < pointsA.size(); ++j) {
            const Point &cur_a_point = pointsA[j];
            assert(max_y_for_a >= cur_a_point.second);
            const int diff_x = dist - best_diff_y;
            const int min_x = cur_a_point.first - diff_x + 1;
            const int max_x = cur_a_point.first + diff_x - 1;

            const Point search_term = std::make_pair(max_x, std::numeric_limits<int>::min());
            PointsList::const_iterator may_be_near_it = std::lower_bound(pointsB.begin(), pointsB.end(), search_term, cmp_by_x);

            for (PointsList::const_reverse_iterator rit(may_be_near_it); rit != pointsB.rend() && rit->first >= min_x; ++rit) {
                const unsigned cur_dist = manhattan_dist(cur_a_point, *rit);
                if (cur_dist < dist) {
                    dist = cur_dist;
                }
            }
        }
        printf("%u\n", dist);
    }
}

我的机器上的基准测试(Linux + i7 2.70 GHz + gcc -Ofast -march=native):

$ make bench
time ./test1 < data.txt  > test1_res

real    0m7.846s
user    0m7.820s
sys     0m0.000s
time ./test2 < data.txt  > test2_res

real    0m0.605s
user    0m0.590s
sys     0m0.010s

test1 是你的变体，test2 是我的变体。