Problem:
Score: 400 points

Problem Statement

There is a cave.

The cave has N rooms and M passages. The rooms are numbered 1 to N, and the passages are numbered 1 to M. Passage i connects Room A_i and Room B_i bidirectionally. One can travel between any two rooms by traversing passages. Room 1 is a special room with an entrance from the outside.

It is dark in the cave, so we have decided to place a signpost in each room except Room 1. The signpost in each room will point to one of the rooms directly connected to that room with a passage.

Since it is dangerous in the cave, our objective is to satisfy the condition below for each room except Room 1.

If you start in that room and repeatedly move to the room indicated by the signpost in the room you are in, you will reach Room 1 after traversing the minimum number of passages possible.

Determine whether there is a way to place signposts satisfying our objective, and print one such way if it exists.

Constraints

All values in input are integers.

2 \leq N \leq 10^5

1 \leq M \leq 2 \times 10^5

1 \leq A_i, B_i \leq N\ (1 \leq i \leq M)

A_i \neq B_i\ (1 \leq i \leq M)

One can travel between any two rooms by traversing passages.

Input

Input is given from Standard Input in the following format:

N M
A_1 B_1
:
A_M B_M

Output

If there is no way to place signposts satisfying the objective, print No.

Otherwise, print N lines. The first line should contain Yes, and the i-th line (2 \leq i \leq N) should contain the integer representing the room indicated by the signpost in Room i.

Sample Input 1

4 4
1 2
2 3
3 4
4 2

Sample Output 1

Yes
1
2
2

If we place the signposts as described in the sample output, the following happens:

Starting in Room 2, you will reach Room 1 after traversing one passage: (2) \to 1. This is the minimum number of passages possible.

Starting in Room 3, you will reach Room 1 after traversing two passages: (3) \to 2 \to 1. This is the minimum number of passages possible.

Starting in Room 4, you will reach Room 1 after traversing two passages: (4) \to 2 \to 1. This is the minimum number of passages possible.

Thus, the objective is satisfied.

Sample Input 2

6 9
3 4
6 1
2 4
5 3
4 6
1 5
6 2
4 5
5 6

Sample Output 2

Yes
6
5
5
1
1

If there are multiple solutions, any of them will be accepted.

Solution in Haskell (GHC 8.8.3) (judge verdict: Time Limit Exceeded)

import Control.Monad
import qualified Data.ByteString.Char8 as BS
import Data.List
import Data.Maybe
import qualified Data.Vector as V
import qualified Data.Vector.Unboxed as VU
import qualified Data.Vector.Unboxed.Mutable as VUM

readInt = fst . fromJust . BS.readInt

readIntList = map readInt . BS.words

getIntList = readIntList <$> BS.getLine

getIntNList n = map readIntList <$> replicateM (fromIntegral n) BS.getLine

paint :: VUM.IOVector Int -> V.Vector (Int, Int) -> [Int] -> [Int] -> IO (V.Vector (Int, Int), [Int])
paint table v (i : is) js = do
  let (xs, ys) = V.partition (\(a, b) -> a == i) v
  let ys' = V.filter (\(a, b) -> b /= i) ys
  let js' = V.map (\(a, b) -> b) xs
  V.mapM (\n -> VUM.write table (n - 2) i) js'
  let js'' = js ++ (V.toList js')
  if is /= []
    then paint table ys' is js''
    else return (ys', js'')

createLine :: [[Int]] -> [(Int, Int)]
createLine [] = []
createLine (ab : abs) = (a, b) : (b, a) : createLine abs
  where
    a = head ab
    b = last ab

padding :: VUM.IOVector Int -> V.Vector (Int, Int) -> [Int] -> IO ()
padding table v is = do
  (v', js) <- paint table v is []
  let v'' = V.filter (\(a, b) -> (a `notElem` js) || (b `notElem` js)) v'
  t <- VU.freeze table
  when (v'' /= V.empty && VU.any (== 0) t) $ padding table v'' js

main = do
  [n, m] <- getIntList
  abs <- getIntNList m
  let lines = createLine abs
  let v = V.fromList lines
  table <- VUM.replicate (n - 1) 0 :: IO (VUM.IOVector Int)
  padding table v [1]
  t <- VU.freeze table
  if VU.length (VU.filter (== 0) t) /= 0
    then putStrLn "No"
    else do
      putStrLn "Yes"
      VU.mapM_ print t