{"problem":{"name":"Shortest Good Path","description":{"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","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":4000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"abc244_f"},"statements":[{"statement_type":"Markdown","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.)  \nFor $i = 1, 2, \\ldots, M$, the $i$\\-th edge connects Vertex $u_i$ and Vertex $v_i$.\nA 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:\n\n*   For all $i = 1, 2, \\dots, k$, it holds that $1 \\leq A_i \\leq N$.\n*   For all $i = 1, 2, \\ldots, k-1$, Vertex $A_i$ and Vertex $A_{i+1}$ are directly connected by an edge.\n\nAn empty sequence is regarded as a path of length $0$.\nLet $S = s_1s_2\\ldots s_N$ be a string 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:\n\n*   For all $i = 1, 2, \\ldots, N$, it holds that:\n    *   if $s_i = 0$, then $A$ has even number of $i$'s.\n    *   if $s_i = 1$, then $A$ has odd number of $i$'s.\n\nThere are $2^N$ possible $S$ (in other words, there are $2^N$ strings of length $N$ consisting of $0$ and $1$). Find the sum of \"the length of the shortest good path with respect to $S$\" over all those $S$.\nUnder the Constraints of this problem, it can be proved that, for any string $S$ of length $N$ consisting of $0$ and $1$, there is at least one good path with respect to $S$.\n\n## Constraints\n\n*   $2 \\leq N \\leq 17$\n*   $N-1 \\leq M \\leq \\frac{N(N-1)}{2}$\n*   $1 \\leq u_i, v_i \\leq N$\n*   The given graph is simple and connected.\n*   All values in input are integers.\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$ $M$\n$u_1$ $v_1$\n$u_2$ $v_2$\n$\\vdots$\n$u_M$ $v_M$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"abc244_f","tags":[],"sample_group":[["3 2\n1 2\n2 3","14\n\n*   For $S = 000$, the empty sequence $()$ is the shortest good path with respect to $S$, whose length is $0$.\n*   For $S = 100$, $(1)$ is the shortest good path with respect to $S$, whose length is $1$.\n*   For $S = 010$, $(2)$ is the shortest good path with respect to $S$, whose length is $1$.\n*   For $S = 110$, $(1, 2)$ is the shortest good path with respect to $S$, whose length is $2$.\n*   For $S = 001$, $(3)$ is the shortest good path with respect to $S$, whose length is $1$.\n*   For $S = 101$, $(1, 2, 3, 2)$ is the shortest good path with respect to $S$, whose length is $4$.\n*   For $S = 011$, $(2, 3)$ is the shortest good path with respect to $S$, whose length is $2$.\n*   For $S = 111$, $(1, 2, 3)$ is the shortest good path with respect to $S$, whose length is $3$.\n\nTherefore, the sought answer is $0 + 1 + 1 + 2 + 1 + 4 + 2 + 3 = 14$."],["5 5\n4 2\n2 3\n1 3\n2 1\n1 5","108"]],"created_at":"2026-03-03 11:01:14"}}