2023-2024年USACO首次月考題目出爐！白金級(jí)第二題真題及解析

Category: 國(guó)際競(jìng)賽, 翰林嚴(yán)選, 計(jì)算機(jī)國(guó)際競(jìng)賽 Date: 2024年1月2日上午11:25

2023-2024賽季的USACO考試已于12月18日正式結(jié)束，晉級(jí)賽的最新試題也已經(jīng)公布。近日，USACO官方公布了2023-2024賽季首場(chǎng)月賽的晉級(jí)分?jǐn)?shù)線。如果沒有晉級(jí)，12月可以當(dāng)做一個(gè)練手+熟悉賽制的環(huán)節(jié)，1、2月的月賽持續(xù)發(fā)力，USACO可以重復(fù)挑戰(zhàn)。

今天給大家分享白金級(jí)試題和解析，參賽的同學(xué)來(lái)看看解題思路，沒參賽的同學(xué)來(lái)看看難度如何？

USACO 2023年12月白金級(jí)第二題

To improve her mathematical knowledge, Bessie has been taking a graph theory course and finds herself stumped by the following problem. Please help her!

You are given a connected, undirected graph with vertices labeled 1…N and edges labeled 1…M(2≤N≤2?10⁵, N?1≤M≤4?10⁵). For each vertex v in the graph, the following process is conducted：?

Let S={v}and h=0.

While |S|<N,?

1.Out of all edges with exactly one endpoint in S, let e be the edge with the minimum label.
2.Add the endpoint of e not in S to S.
3.Set h=10h+e.
?
Return h ( mod 10⁹+7).
?
Determine all the return values of this process.?

INPUT FORMAT (pipe stdin)：

The first line contains N and M .?Then follow M lines, the eth containing the endpoints (a_e,b_e) of the eth edge (1≤a_e<b_e≤N). It is guaranteed that these edges form a connected graph, and at most one edge connects each pair of vertices.
?
OUTPUT FORMAT (pipe stdout)：

Output N lines, where the ith line should contain the return value of the process starting at vertex i.

SAMPLE INPUT：

3 2
1 2
2 3

SAMPLE OUTPUT：

12
12
21

SAMPLE INPUT：

5 6
1 2
3 4
2 4
2 3
2 5
1 5

SAMPLE OUTPUT：

1325
1325
2315
2315
5132

Consider starting at i=3. First, we choose edge 2, after which S={3,4} and h=2.Second, we choose edge 3, after which S={2,3,4}and h=23. Third, we choose edge 1, after which S={1,2,3,4} and h=231. Finally, we choose edge 5, after which S={1,2,3,4,5}and h=2315. The answer for i=3 is therefore 2315.

SAMPLE INPUT：

15 14
1 2
2 3
3 4
4 5
5 6
6 7
7 8
8 9
9 10
10 11
11 12
12 13
13 14
14 15

SAMPLE OUTPUT：

678925929
678925929
678862929
678787329
678709839
678632097
178554320
218476543
321398766
431520989
542453212
653475435
764507558
875540761
986574081

Make sure to output the answers modulo 109+7.

SCORING：

Input 4： N,M≤2000
Inputs 5-6： N≤2000
Inputs 7-10： N≤10000
Inputs 11-14： a_e+1=b_e
for all e
Inputs 15-23： No additional constraints.

Problem credits： Benjamin Q_i

部分代碼截圖：