Sightseeing trip
Time Limit:1000MS  Memory Limit:65536K
Total Submit:317 Accepted:133 Special Judged

Description
There is a travel agency in Adelton town on Zanzibar island. It has decided to offer its clients, besides many other attractions, sightseeing the town. To earn as much as possible from this attraction, the agency has accepted a shrewd decision: it is necessary to find the shortest route which begins and ends at the same place. Your task is to write a program which finds such a route.

In the town there are N crossing points numbered from 1 to N and M two-way roads numbered from 1 to M. Two crossing points can be connected by multiple roads, but no road connects a crossing point with itself. Each sightseeing route is a sequence of road numbers y_1, ..., y_k, k>2. The road y_i (1<=i<=k-1) connects crossing points x_i and x_{i+1}, the road y_k connects crossing points x_k and x_1. All the numbers x_1,...,x_k should be different.The length of the sightseeing route is the sum of the lengths of all roads on the sightseeing route, i.e. L(y_1)+L(y_2)+...+L(y_k) where L(y_i) is the length of the road y_i (1<=i<=k). Your program has to find such a sightseeing route, the length of which is minimal, or to specify that it is not possible,because there is no sightseeing route in the town.

Input
The first line of input contains two positive integers: the number of crossing points N<=100 and the number of roads M<=10000. Each of the next M lines describes one road. It contains 3 positive integers: the number of its first crossing point, the number of the second one, and the length of the road (a positive integer less than 500).

Output
There is only one line in output. It contains either a string 'No solution.' in case there isn't any sightseeing route, or it contains the numbers of all crossing points on the shortest sightseeing route in the order how to pass them (i.e. the numbers x_1 to x_k from our definition of a sightseeing route), separated by single spaces. If there are multiple sightseeing routes of the minimal length, you can output any one of them.

Sample Input

5 7
1 4 1
1 3 300
3 1 10
1 2 16
2 3 100
2 5 15
5 3 20

Sample Output

1 3 5 2

 

點數是100個 題目意思是找一個最小權圈 以任意序輸出
從圖論的角度上考慮 應該是任意選一條邊（枚舉） 然后刪除邊 再以一個點為原點求Dijkstra 找出最小權圈 o(M*N^2)的復雜度 一個更加好的算法是限定枚舉的點為圈內序號最大的點 這樣就避免了對一個圈的多次枚舉(參考程序3)
如果直接搜就是任選一個點開始走回到原點則記錄長度 搜的時候必須要先對每個點的邊按照邊權進行排序 以備后面大量剪枝

程序1

 1//Solution
 2//by oyjpArt
 3//Algorithm:Search
 4#include <vector>
 5#include <iostream>
 6#include <algorithm>
 7using namespace std;
 8
 9const int N = 101;
10struct Node {int x, w; void set(int xx, int ww) {x =xx; w = ww; }};
11vector<Node> adj[N];
12int nv, ne, ans[N], na, S, rec[N];
13bool chk[N];
14int best;
15
16bool operator<(const Node& a, const Node& b) {
17 return a.w < b.w;
18}
19
20void search(int x, int sum, int depth, int father) {
21 int i;
22 if(x == S && chk[x]) {
23  if(sum < best) {
24   best = sum;  na = depth;
25   for(i = 0; i < depth; i++) ans[i] = rec[i];
26  }
27  return;
28 }
29 rec[depth] = x;
30 for(i = 0; i < adj[x].size(); ++i) if(adj[x][i].x != father) if(!chk[adj[x][i].x] || adj[x][i].x == S) {
31  chk[adj[x][i].x] = 1;
32  if(sum + adj[x][i].w < best) search(adj[x][i].x, sum + adj[x][i].w, depth+1, x);
33  chk[adj[x][i].x] = 0;
34 }
35} 
36
37int main() {
38 scanf("%d %d", &nv, &ne);
39 int i, u, v, w;
40 Node now;
41 for(i = 0; i < ne; i++) {
42  scanf("%d %d %d", &u, &v, &w);
43  --u; --v;
44  now.set(v, w);
45  adj[u].push_back(now);
46  now.x = u;
47  adj[v].push_back(now);
48 }
49 for(i = 0; i < nv; ++i) 
50  sort(adj[i].begin(), adj[i].end());
51 
52 best = 123456789;
53 for(i = 0; i < nv; ++i) {
54  memset(chk, 0, nv * sizeof(bool));
55  S = i;
56  search(i, 0, 0, -1);
57 }
58
59 if(best == 123456789) { printf("No solution.\n"); return 0; }
60 printf("%d", ans[0]+1);
61 for(i = 1; i < na; ++i) printf(" %d", ans[i]+1); putchar('\n');
62
63 return 0;
64}
65
66

程序2

 1//Solution
 2//by oyjpArt
 3//Algorithm : Enumerate + Dijkstra
 4#include <stdio.h>
 5#include <string.h>
 6
 7const int N = 101, M = 20001, MAXINT = 2000000000;
 8int ne, nv;
 9struct E {
10 int x, w; E* next;
11 void set(int xx, int ww, E* nn) {x = xx; w = ww; next = nn;}
12}e[M], * head[N];
13int best, dist[N], q[N], ans[N], pre[N], na;
14bool chk[N];
15
16void Dijk(int st, int end, int ow) {
17 memset(chk, 0, sizeof(chk));
18 memset(dist, -1, sizeof(dist));
19 int qe = 1, qs = 0, i;
20 E * p;
21 for(i = 0; i < nv; ++i) if(i != st) {
22  for(p = head[st]; p != NULL; p = p->next) {
23   if(p->x == i && p->w > 0 && (dist[i] == -1 || dist[i] > p->w ) ) 
24    dist[i] = p->w;
25  }
26  if(dist[i] == -1) dist[i] = MAXINT;
27 }
28 q[0] = st;
29 dist[st] = 0;
30 chk[st] = 1;
31 for(i = 0; i < nv; ++i) pre[i] = st;
32 pre[st] = -1;
33 while(qs < qe) {
34  int cur = q[qs++];
35  chk[cur] = 1;
36  if(ow + dist[cur] >= best) return;
37  if(cur == end) {
38   if(dist[end] + ow < best) {
39    na = 0;
40    for(i = cur; i != -1; i = pre[i]) ans[na++] = i;
41    best = dist[end] + ow;
42   }
43   return;
44  }
45  int _min = MAXINT, mini = -1;
46  for(i = 0; i < nv; i++) if(!chk[i]) {
47   if(dist[i] < _min) {
48    _min = dist[i];
49    mini = i;
50   }
51  }
52  if(mini == -1) return;
53  q[qe++] = mini;
54  for(i = 0; i < nv; ++i) if(!chk[i]) {
55   for(p = head[mini]; p != NULL; p = p->next) if(p->x == i)  break;
56   if(p == NULL) continue;
57   if(p->w > 0 && p->w + dist[mini] < dist[i]) {
58    dist[i] = p->w + dist[mini];
59    pre[i] = mini;
60   }
61  }
62 }
63}
64
65int main() {
66 scanf("%d %d", &nv, &ne);
67 memset(head, NULL, nv * sizeof(E*));
68 int i, u, v, w;
69 for(i = 0; i < ne; ++i) {
70  scanf("%d %d %d", &u, &v, &w);
71  --u; --v;
72  e[2*i].set(u, w, head[v]);
73  head[v] = &e[2*i];
74  e[2*i+1].set(v, w, head[u]);
75  head[u] = &e[2*i+1];
76 }
77 E * p, * q;
78 best = MAXINT;
79 for(i = 0; i < nv; ++i) {
80  for(p = head[i]; p != NULL; p = p->next) {
81   int w = p->w;
82   int j = p->x;
83   for(q = head[i]; q != NULL; q = q->next) if(q->x == j) q->w = -q->w;
84   for(q = head[j]; q != NULL; q = q->next) if(q->x == i) q->w = -q->w;
85   Dijk(i, j, w);
86   for(q = head[i]; q != NULL; q = q->next) if(q->x == j) q->w = -q->w;
87   for(q = head[j]; q != NULL; q = q->next) if(q->x == i) q->w = -q->w;
88  }
89 }
90 if(best == MAXINT) printf("No solution.\n");
91 else {
92  printf("%d", ans[0] + 1);
93  for(i = 1; i < na; ++i) printf(" %d", ans[i] + 1); putchar('\n');
94 }
95 return 0;
96}
97//唉 不用vector代碼量增大好多。。暈倒
98

 1#include <stdio.h>
 2#include <string.h>
 3
 4const int N = 101;
 5const int MAXINT = 123456789;
 6int ne, nv;
 7int adj[N][N];
 8int pre[N][N];
 9int conn[N][N];
10int na, ans[N];
11int best;
12
13void floyd() {
14    int i, j, k, tmp, p;
15    for(k = 0; k < nv; ++k) {
16        for(i = 0; i < k; ++i) {
17            for(j = 0; j < k; ++j) if(conn[i][k] && conn[k][j] && j != i) {
18                if( (tmp = adj[i][j] + conn[k][i] + conn[j][k]) < best) {
19                    best = tmp;
20                    na = 1; ans[0] = k; p = i;
21                    while(p != -1) {
22                        ans[na++] = p;
23                        p = pre[p][j];
24                    }
25                }
26            } 
27        }
28        for(i = 0; i < nv; ++i) 
29            for(j = 0; j < nv; ++j) {
30                if(adj[i][j] > adj[i][k] + adj[k][j]) {
31                    adj[i][j] = adj[i][k] + adj[k][j];
32                    pre[i][j] = pre[i][k];
33                }
34            }
35    }
36}
37
38int main() {
39    int i, j, u, v, w;
40    memset(pre, -1, sizeof(pre));
41    scanf("%d %d", &nv, &ne);
42    for(i = 0; i < nv; ++i) {
43        for(j = i+1; j < nv; ++j) 
44            adj[i][j] = adj[j][i] = MAXINT;
45        adj[i][i] = 0;
46    }
47    for(i = 0; i < ne; ++i) {
48        scanf("%d %d %d", &u, &v, &w);
49        --u; --v;
50        if(w < adj[u][v])     
51            conn[u][v] = conn[v][u] = adj[u][v] = adj[v][u] = w;
52        pre[u][v] = v, pre[v][u] = u;
53    }
54    best = MAXINT;
55    floyd();
56    if(best == MAXINT) printf("No solution.\n");
57    else {
58        for(i = 0; i < na; ++i) {
59            printf("%d", ans[i] + 1);
60            if(i != na-1) putchar(' ');
61            else putchar('\n');
62        }
63    }
64
65    return 0;
66}
67