CodeForces Add on a Tree

探讨在一棵树形结构中,通过特定操作能否使边权变为任意实数的问题。关键在于是否存在子节点出入度均为1的情况,这将决定是否能形成任意想要的图。

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题目链接:点击这里

就是说给你一颗树,然后你可以执行任意操作,操作的内容如下:在每次操作中你都可以选择任意两个叶子节点(叶子节点就是出入为1的结点),对于两个节点间的最短路径上的所有边你都能加上任意实数。问你能不能让给定的树的边呈现任意想要的实数
比如说1-2-3这个图,它的叶子节点是1和3,所以每次的操作都只能选1和3,所以只能给1和2与2和3同时加上一个数,这样就没法实现任意图,比如我想实现1和2边上数是0.1、2和3边上数是0.2就不行,因为这两条边永远是相等的。

 

这个图的1-2和2-5永远是相等的。

只要是有一个子节点的出度为1,入度也为1,那么这个图肯定不能组成任意图。以为,连接这个点的两条边肯定相等。

AC代码:

#include<bits/stdc++.h>
#define N 100005
using namespace std ;
typedef long long ll ;
vector <int> v[N] ;
int num = 0 ;
bool flag = 1 ;
int main()
{
	int i , j ;
	int a , b ;
	int len ;
	int n ;
	cin>>n ;
	for(i=1 ; i<n ; i++)
	{
	   	cin>>a>>b ;
	   	v[a].push_back(b) ;
	   	v[b].push_back(a) ;
	}
	for(i=1 ; i<=n ; i++)
	  if(v[i].size()==2)
	     flag=0 ;
	if(!flag)
	  printf("NO");
	else
	  printf("YES");  
}

 

### Codeforces Problem 1009 Details and Solution The referenced content discusses a set of problems related to number theory on the Codeforces platform with difficulties ranging from 2000 to 3000+. The focus is primarily on optimizing algorithms due to large input sizes where complexities such as \(O(n \sqrt{n})\) are not feasible without additional optimizations[^1]. For **Problem 1009**, while specific details about this exact problem may vary depending on updates or changes within the Codeforces database, it generally involves handling operations over tree structures efficiently given constraints like those described: - Input consists of two integers \(n\) (number of nodes) and \(m\) (number of queries), followed by detailed descriptions of each node's children. - Queries include three types: adding an edge between two vertices ("1 v u"), removing edges based on height conditions ("2 v h"), and finding paths satisfying certain properties ("3 k"). Given these parameters, efficient data structure usage becomes critical. Below outlines how one might approach solving similar structured questions effectively. #### Efficient Data Structures Utilization To handle up to \(10^5\) elements per dimension (\(n,m\)), advanced techniques must be employed: - Preprocessing using Depth First Search (DFS) traversal can help compute auxiliary information quickly during runtime. - Segment trees or Binary Indexed Trees could assist managing dynamic range updates/queries required under different operation modes mentioned above. Here’s some sample pseudo-code demonstrating initialization phase alongside query processing logic utilizing DFS precomputation technique combined with segment tree implementation for fast querying capabilities: ```python from collections import defaultdict class Node: def __init__(self): self.children = [] def dfs(node_id, parent=-1): global depth, subtree_size start_time[node_id] = time_counter[0] time_increases() subtree_size[node_id]=1 for child in adj_list[node_id]: if(child !=parent): depth[child]=depth[node_id]+1 dfs(child,node_id) subtree_size[node_id]+=subtree_size[child] # Initialize variables & read inputs here... adj_list=defaultdict(list) for _ in range(m): op,*params=input().split() params=list(map(int,params)) if(op=='1'): # Add Edge Logic Here Using Params[v,u] elif(op=='2'): pass else: # Operation Type '3' result=process_query_k(params[k]) print(result) ``` This code snippet provides foundational steps towards constructing solutions tailored specifically toward addressing challenges posed through complex graph manipulations involving numerous sequential actions against hierarchical datasets represented via adjacency lists. §§Related Questions§§ 1. What other common optimization strategies exist when dealing with computational limits exceeding standard algorithmic approaches? 2. How does implementing lazy propagation affect performance improvements inside segment-tree-based systems used across competitive programming platforms? 3. Can you provide examples illustrating alternative methods besides DFS preorder/postorder traversals that achieve equivalent results more suitably suited under particular circumstances? 4. In what ways do persistent data structures contribute differently compared traditional mutable ones concerning memory consumption versus speed tradeoffs seen regularly throughout high-level contests including CF rounds?
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