Construct Good Path

AtCoder

IDabc244_g

Time2000ms

Memory256MB

Difficulty—

You are given a simple connected undirected graph with $N$ vertices and $M$ edges. (A graph is said to be simple if it has no multi-edges and no self-loops.) For $i = 1, 2, \ldots, M$, the $i$\-th edge connects Vertex $u_i$ and Vertex $v_i$. A sequence $(A_1, A_2, \ldots, A_k)$ is said to be a **path** of length $k$ if both of the following two conditions are satisfied: * For all $i = 1, 2, \dots, k$, it holds that $1 \leq A_i \leq N$. * For all $i = 1, 2, \ldots, k-1$, Vertex $A_i$ and Vertex $A_{i+1}$ are directly connected with an edge. An empty sequence is regarded as a path of length $0$. You are given a sting $S = s_1s_2\ldots s_N$ of length $N$ consisting of $0$ and $1$. A path $A = (A_1, A_2, \ldots, A_k)$ is said to be a **good path** with respect to $S$ if the following conditions are satisfied: * For all $i = 1, 2, \ldots, N$, it holds that: * if $s_i = 0$, then $A$ has even number of $i$'s. * if $s_i = 1$, then $A$ has odd number of $i$'s. Under the Constraints of this problem, it can be proved that there is at least one good path with respect to $S$ of length at most $4N$. Print a good path with respect to $S$ of length at most $4N$. ## Constraints * $2 \leq N \leq 10^5$ * $N-1 \leq M \leq \min\lbrace 2 \times 10^5, \frac{N(N-1)}{2}\rbrace$ * $1 \leq u_i, v_i \leq N$ * The given graph is simple and connected. * $N, M, u_i$, and $v_i$ are integers. * $S$ is a string of length $N$ consisting of $0$ and $1$. ## Input Input is given from Standard Input in the following format: $N$ $M$ $u_1$ $v_1$ $u_2$ $v_2$ $\vdots$ $u_M$ $v_M$ $S$ [samples]

Samples

Input #1

Output #1

9
2 5 6 5 6 3 1 3 6

The path $(2, 5, 6, 5, 6, 3, 1, 3, 6)$ has a length no greater than $4N$, and

*   it has odd number ($1$) of $1$
*   it has odd number ($1$) of $2$
*   it has even number ($2$) of $3$
*   it has even number ($0$) of $4$
*   it has even number ($2$) of $5$
*   it has odd number ($3$) of $6$

so it is a good path with respect to $S = 110001$.

Input #2

Output #2

0

An empty path $()$ is a good path with respect to $S = 000000$. Alternatively, paths like $(1, 2, 3, 1, 2, 3)$ are also accepted.

API Response (JSON)

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    "name": "Construct Good Path",
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      "content": "You are given a simple connected undirected graph with $N$ vertices and $M$ edges. (A graph is said to be simple if it has no multi-edges and no self-loops.)   For $i = 1, 2, \\ldots, M$, the $i$\\-th e",
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      "time_limit": 2000,
      "memory_limit": 262144
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