Jerry @DOA&INPAC, SJTU

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Dijkstra算法与堆(C++)

Dijkstra算法用于解决单源最短路径问题,通过逐个收录顶点来确保已收录顶点的路径长度为最短。

    

图片来自陈越姥姥的数据结构课程:https://mooc.study.163.com/learn/1000033001?tid=1000044001#/learn/content?type=detail&id=1000112011&cid=1000126096

Dijkstra算法的时间复杂度,取决于“V=未收录顶点中dist最小者”的算法。这一步可以用线性查找实现,也可以用最小堆实现。

线性查找的算法就不用多说了。最小堆的算法有一个问题:最小堆是以未收录顶点的dist作为key来建立的,但是每一轮循环都会把部分顶点的dist值改变,也就会破坏最小堆的有序性,怎么解决?

显然应该在每一轮循环中把最小堆重新调整成有序。现在问题又来了:

1. 复杂度还合算吗?

建堆的时间复杂度是O(N),pop一个元素的时间复杂度是O(logN);线性查找的时间复杂度也是O(N)。建堆还额外使用了O(N)的空间。

看似一点都不合算。但我又想到每一轮循环中的建堆操作,很可能只需要调整少量元素,而对于其他元素,只需要进行访问。然而线性查找连调整都不需要,只有交换。再然而,循环过程中堆会变小,使建堆的时间复杂度中的常数变小。至于到底哪个更快,还得实践出真知。

所以只能从消除建堆操作入手。这样又是另一种算法了,参考资料[1]给出了详细说明,这种算法中每一轮的时间复杂度为O(logN),总时间复杂度为O(|E|log|V|)(V为顶点,E为边)。

2. 如何利用STL进行堆操作?

STL <algorithm> 头文件提供了 std::is_heap 、 std::is_heap_until (这两个需要C++11)、 std::make_heap 、 std::push_heap 、 std::pop_heap 和 std::sort_heap 等函数模板用于堆操作。

现有一道单源最短路径的题:https://pintia.cn/problem-sets/994805342720868352/problems/994805523835109376,Dijkstra算法的变形而已。

以下为实现代码。三种算法用宏定义选择,已选择优先队列算法。

  1 #include <iostream>
  2 #include <limits>
  3 #include <vector>
  4 #include <queue>
  5 #include <algorithm>
  6 #include <utility>
  7 #include <functional>
  8 
  9 //#define LINEAR
 10 //#define HEAP
 11 #define QUEUE
 12 
 13 struct Path
 14 {
 15     Path() = default;
 16     Path(int _city, int _dist)
 17         : city(_city), dist(_dist)
 18     {
 19         ;
 20     }
 21     int city;
 22     int dist;
 23     bool operator<(const Path& _rhs) const
 24     {
 25         return dist < _rhs.dist;
 26     }
 27     bool operator>(const Path& _rhs) const
 28     {
 29         return dist > _rhs.dist;
 30     }
 31 };
 32 
 33 struct City
 34 {
 35     std::vector<Path> paths;
 36     int team;
 37     int dist = std::numeric_limits<int>::max();
 38     bool collected = false;
 39     int team_max = 0;
 40     int dist_count = 0;
 41 };
 42 
 43 #ifdef HEAP
 44 class Comparator
 45 {
 46 public:
 47     Comparator(std::vector<City>& _cities)
 48         : cities_(&_cities)
 49     {
 50         ;
 51     }
 52     bool operator()(int _lhs, int _rhs)
 53     {
 54         return (*cities_)[_lhs].dist > (*cities_)[_rhs].dist;
 55     }
 56 private:
 57     std::vector<City>* cities_;
 58 };
 59 #endif
 60 
 61 int main()
 62 {
 63     int n, m, src, dst;
 64     std::cin >> n >> m >> src >> dst;
 65     std::vector<City> cities(n);
 66     for (auto& city : cities)
 67         std::cin >> city.team;
 68     for (int cnt = 0; cnt != m; ++cnt)
 69     {
 70         int src, dst, dist;
 71         std::cin >> src >> dst >> dist;
 72         cities[src].paths.emplace_back(dst, dist);
 73         cities[dst].paths.emplace_back(src, dist);
 74     }
 75 
 76     {
 77         auto& city = cities[src];
 78         cities[src].collected = true;
 79         cities[src].dist = 0;
 80         cities[src].dist_count = 1;
 81         cities[src].team_max = cities[src].team;
 82     }
 83 #ifdef QUEUE
 84     std::priority_queue<Path, std::vector<Path>, std::greater<Path>> queue;
 85 #endif
 86     for (const auto& path : cities[src].paths)
 87     {
 88         cities[path.city].dist = path.dist;
 89         cities[path.city].dist_count = 1;
 90         cities[path.city].team_max = cities[src].team + cities[path.city].team;
 91 #ifdef QUEUE
 92         queue.emplace(path.city, path.dist);
 93 #endif
 94     }
 95 
 96 #ifdef HEAP
 97     std::vector<int> heap;
 98     heap.reserve(n - 1);
 99     for (int i = 0; i != n; ++i)
100         if (i != src)
101             heap.push_back(i);
102     Comparator comp(cities);
103     std::make_heap(heap.begin(), heap.end(), comp);
104 #endif
105 
106     while (1)
107     {
108 #ifdef LINEAR
109         int min_dist = std::numeric_limits<int>::max();
110         int index = -1;
111         for (int i = 0; i != n; ++i)
112             if (!cities[i].collected && cities[i].dist < min_dist)
113                 min_dist = cities[i].dist, index = i;
114         if (index == -1)
115             break;
116         auto& city = cities[index];
117 #endif
118 #ifdef HEAP
119         if (heap.empty())
120             break;
121         auto& city = cities[heap[0]];
122 #endif
123 #ifdef QUEUE
124         if (queue.empty())
125             break;
126         Path temp;
127         while (1)
128         {
129             temp = queue.top();
130             queue.pop();
131             if (!cities[temp.city].collected)
132                 break;
133         }
134         auto& city = cities[temp.city];
135 #endif
136         city.collected = true;
137         for (const auto& path : city.paths)
138         {
139             if (!cities[path.city].collected)
140             {
141                 auto& dest = cities[path.city];
142                 if (city.dist + path.dist < cities[path.city].dist)
143                 {
144                     dest.dist = city.dist + path.dist;
145                     dest.dist_count = city.dist_count;
146                     dest.team_max = city.team_max + dest.team;
147                 }
148                 else if (city.dist + path.dist == cities[path.city].dist)
149                 {
150                     dest.dist = city.dist + path.dist;
151                     dest.dist_count += city.dist_count;
152                     if (city.team_max + dest.team > dest.team_max)
153                         dest.team_max = city.team_max + dest.team;
154                 }
155 #ifdef QUEUE
156                 queue.emplace(path.city, dest.dist);
157 #endif
158             }
159         }
160 #ifdef LINEAR
161         if (index == dst)
162             break;
163 #endif
164 #ifdef HEAP
165         if (heap[0] == dst)
166             break;
167         std::pop_heap(heap.begin(), heap.end(), comp);
168         heap.pop_back();
169         std::make_heap(heap.begin(), heap.end(), comp);
170 #endif
171 #ifdef QUEUE
172         if (temp.city == dst)
173             break;
174 #endif
175     }
176 
177     {
178         auto& city = cities[dst];
179         std::cout << cities[dst].dist_count << ' ' << cities[dst].team_max;
180     }
181 
182     return 0;
183 }

测试结果:

线性查找版

最小堆版

优先队列版

平台显示线性查找版的时间6ms,内存512KB;最小堆版的时间5ms,内存512KB;优先队列版的时间3ms,内存424KB。我认为时间都太短了,数据量不够大,不足以说明问题。

如果仅从理论上分析的话,我认为优先队列的算法是最优的。

 

参考资料:

[1] dijkstra + heap 优化 https://blog.csdn.net/sentimental_dog/article/details/51955765

 

posted on 2019-07-13 12:52  Jerry_SJTU  阅读(1132)  评论(0编辑  收藏  举报