codeforces545E Paths and Trees

本博客探讨了一个在给定连通无向图中,寻找包含特定顶点u的最优树结构的问题,该树满足任意点到u的距离等于原图中到该点的最短距离,并且在所有满足条件的树中选择总权重最小的树。通过实施Dijkstra算法进行最短路径搜索,并在遍历过程中优化连接路径,最终实现问题求解。

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题目:给定一个连通无向图,求包含u的一棵树,满足树上任意一点到u的距离等于原图中到那个点的最短距离。如果有多种这样的树,找到总权值最小的树。

题解:如果u连了i再连到j,相比u直接连到j,如果都是最短路,肯定优先选择u-i和i-j两条边。所以先跑一遍最短路,然后枚举每个点,找到那个点从哪条边连接会更优。

这题很明显爆longlong。。原来的dijkstra的模板是默认int的。。改了好几次都WA。。后来把距离的数据类型给define掉了。。以后不能再犯了。。

代码:

#include<bits/stdc++.h>
using namespace std;
#define ll long long
#define pp pair<ll,int>
#define inf LONG_LONG_MAX/3
#define ll long long
using namespace std;
int n,m,cnt,last[1000011];
int x,y,z;
ll dis[1000005];
struct data{int to,next,id;ll v;}e[1000011];
inline void read(int &m){
    int x=0,f=1;char ch=getchar();
    while(ch<'0'||ch>'9'){if(ch=='-')f=-1;ch=getchar();}
    while(ch>='0'&&ch<='9'){x=x*10+ch-'0';ch=getchar();}
    m=x*f;
}
void add(int u,int v,int w,int i){
    e[++cnt].to=v;e[cnt].next=last[u];last[u]=cnt;e[cnt].v=w;e[cnt].id=i;
}
priority_queue<pp,vector<pp>,greater<pp> > q;
void dijkstra(int s){
    for(int i=1;i<=n;i++) dis[i]=inf;
    dis[s]=0;q.push(make_pair(0,s));
    while(!q.empty())
    {
        int now=q.top().second;q.pop();
        for(int i=last[now];i;i=e[i].next){
            ll d=e[i].v+dis[now];
            if(dis[e[i].to]>d){
                dis[e[i].to]=d;
                q.push(make_pair(dis[e[i].to],e[i].to));
            }
        }
    }
}
vector<int> ans;
ll sum;
int main()
{
    int i,u,v,w;
    read(n);read(m);
    for(i=1;i<=m;i++){
        read(u);read(v);read(w);
        add(u,v,w,i);add(v,u,w,i);
    }
    read(u);
    dijkstra(u);
    for(v=1;v<=n;v++){
        if(v==u) continue;
        ll Mi=LONG_LONG_MAX,id;
        for(int i=last[v];i;i=e[i].next){
            int now=e[i].to;
            if(dis[v]==dis[now]+e[i].v&&Mi>e[i].v){
                Mi=e[i].v;
                id=e[i].id;
            }
        }
        sum+=Mi;
        ans.push_back(id);
    }
    printf("%I64d\n",sum);
    sort(ans.begin(),ans.end());
    for(int i=0;i<ans.size();i++)
        printf("%d%c",ans[i]," \n"[i==n]);
}


### Codeforces 887E Problem Solution and Discussion The problem **887E - The Great Game** on Codeforces involves a strategic game between two players who take turns to perform operations under specific rules. To tackle this challenge effectively, understanding both dynamic programming (DP) techniques and bitwise manipulation is crucial. #### Dynamic Programming Approach One effective method to approach this problem utilizes DP with memoization. By defining `dp[i][j]` as the optimal result when starting from state `(i,j)` where `i` represents current position and `j` indicates some status flag related to previous moves: ```cpp #include <bits/stdc++.h> using namespace std; const int MAXN = ...; // Define based on constraints int dp[MAXN][2]; // Function to calculate minimum steps using top-down DP int minSteps(int pos, bool prevMoveType) { if (pos >= N) return 0; if (dp[pos][prevMoveType] != -1) return dp[pos][prevMoveType]; int res = INT_MAX; // Try all possible next positions and update 'res' for (...) { /* Logic here */ } dp[pos][prevMoveType] = res; return res; } ``` This code snippet outlines how one might structure a solution involving recursive calls combined with caching results through an array named `dp`. #### Bitwise Operations Insight Another critical aspect lies within efficiently handling large integers via bitwise operators instead of arithmetic ones whenever applicable. This optimization can significantly reduce computation time especially given tight limits often found in competitive coding challenges like those hosted by platforms such as Codeforces[^1]. For detailed discussions about similar problems or more insights into solving strategies specifically tailored towards contest preparation, visiting forums dedicated to algorithmic contests would be beneficial. Websites associated directly with Codeforces offer rich resources including editorials written after each round which provide comprehensive explanations alongside alternative approaches taken by successful contestants during live events. --related questions-- 1. What are common pitfalls encountered while implementing dynamic programming solutions? 2. How does bit manipulation improve performance in algorithms dealing with integer values? 3. Can you recommend any online communities focused on discussing competitive programming tactics? 4. Are there particular patterns that frequently appear across different levels of difficulty within Codeforces contests?
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