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: Accepted)

{-# LANGUAGE BangPatterns #-}
import           Control.Exception           (assert)
import           Control.Monad
import           Control.Monad.Primitive
import qualified Control.Monad.ST            as ST
import qualified Data.Array.IO               as IO
import qualified Data.Array.ST               as ST
import qualified Data.Array.Unboxed          as A
import           Data.Bits
import qualified Data.ByteString.Char8       as BS
import           Data.Char
import           Data.Foldable
import           Data.List
import qualified Data.Map.Strict             as Map
import           Data.Maybe
import qualified Data.Sequence               as Seq
import qualified Data.Set                    as Set
import qualified Data.Vector                 as VB
import qualified Data.Vector.Mutable         as VBM
import qualified Data.Vector.Unboxed         as V
import qualified Data.Vector.Unboxed.Mutable as VM
import           Debug.Trace

readInt = fst . fromJust . BS.readInt
readIntList = map readInt . BS.words
getInt = readInt <$> BS.getLine
getIntList = readIntList <$> BS.getLine
getIntVec n = V.unfoldrN n (BS.readInt . BS.dropWhile isSpace) <$> BS.getLine

readInteger = fst . fromJust . BS.readInteger
readIntegerList = map readInteger . BS.words
getInteger = readInteger <$> BS.getLine
getIntegerList = readIntegerList <$> BS.getLine

inf :: Int
inf = 10^18

-- (node, dist)
type Graph_dijk = VB.Vector (V.Vector (Int, Int))

dijkstra :: Int -> Graph_dijk -> Int -> IO (V.Vector Int)
dijkstra s g numV = do

  let inf = 10^18 :: Int
  d <- VM.new numV
  VM.set d inf
  VM.write d s 0

  -- tuple represent (min distance, node id)
  let set = Set.singleton (0, s)

      loop :: Set.Set (Int, Int) -> IO ()
      loop set | Set.null set = return ()
               | otherwise = do
                   let ((mdist, v), set') = Set.deleteFindMin set
                       edges = g VB.! v
                   d_v <- VM.read d v
                   if d_v < mdist
                     then loop set'
                     else do set'' <- loop2 v edges 0 set'
                             loop set''

      loop2 :: Int -> V.Vector (Int, Int) -> Int -> Set.Set (Int, Int)
            -> IO (Set.Set (Int, Int))
      loop2 v edges j set
        | j >= V.length edges = return set
        | otherwise = do
            let e = edges V.! j
                v_to = fst e
                cost = snd e
            d_v <- VM.read d v
            d_to <- VM.read d v_to
            let cond = d_to > d_v + cost
                set' = if cond
                       then (d_v + cost, v_to) `Set.insert` set
                       else set
            when cond $ VM.write d v_to (d_v + cost)
            loop2 v edges (j+1) set'

  loop set
  V.freeze d

main = do
  [n, m] <- getIntList

  g <- VBM.new n
  VBM.set g ([] :: [(Int, Int)])

  replicateM_ m $ do
    [a, b] <- getIntList
    let (a', b') = (a-1, b-1)
    xs <- VBM.read g a'
    ys <- VBM.read g b'
    VBM.write g a' ((b', 1):xs)
    VBM.write g b' ((a', 1):ys)

  g' <- VB.freeze g
  let g'' = VB.map V.fromList g'

  costs <- dijkstra 0 g'' n

  let getMark i = let ci = costs V.! i
                      edges = V.toList (g'' VB.! i)
                  in findNext edges (ci - 1)

      findNext [] _ = -1
      findNext ((x, _):es) c | costs V.! x == c = (x + 1)
                             | otherwise = findNext es c

  putStrLn "Yes"

  forM_ [1..(n-1)] $ \i -> do
    let j = getMark i
    print j