{"problem":{"name":"K Derangement","description":{"content":"You are given positive integers $N$ and $K$. Find the lexicographically smallest permutation $A = (A_1, A_2, \\ldots, A_N)$ of the integers from $1$ through $N$ that satisfies the following condition: ","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"arc144_c"},"statements":[{"statement_type":"Markdown","content":"You are given positive integers $N$ and $K$. Find the lexicographically smallest permutation $A = (A_1, A_2, \\ldots, A_N)$ of the integers from $1$ through $N$ that satisfies the following condition:\n\n*   $\\lvert A_i - i\\rvert \\geq K$ for all $i$ ($1\\leq i\\leq N$).\n\nIf there is no such permutation, print `-1`.\nWhat is lexicographical order on sequences?The following is an algorithm to determine the lexicographical order between different sequences $S$ and $T$.\nBelow, let $S_i$ denote the $i$\\-th element of $S$. Also, if $S$ is lexicographically smaller than $T$, we will denote that fact as $S \\lt T$; if $S$ is lexicographically larger than $T$, we will denote that fact as $S \\gt T$.\n\n1.  Let $L$ be the smaller of the lengths of $S$ and $T$. For each $i=1,2,\\dots,L$, we check whether $S_i$ and $T_i$ are the same.\n2.  If there is an $i$ such that $S_i \\neq T_i$, let $j$ be the smallest such $i$. Then, we compare $S_j$ and $T_j$. If $S_j$ is less than $T_j$ (as a number), we determine that $S \\lt T$ and quit; if $S_j$ is greater than $T_j$, we determine that $S \\gt T$ and quit.\n3.  If there is no $i$ such that $S_i \\neq T_i$, we compare the lengths of $S$ and $T$. If $S$ is shorter than $T$, we determine that $S \\lt T$ and quit; if $S$ is longer than $T$, we determine that $S \\gt T$ and quit.\n\n## Constraints\n\n*   $2\\leq N\\leq 3\\times 10^5$\n*   $1\\leq K\\leq N - 1$\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$ $K$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"arc144_c","tags":[],"sample_group":[["3 1","2 3 1\n\nTwo permutations satisfy the condition: $(2, 3, 1)$ and $(3, 1, 2)$. For instance, the following holds for $(2, 3, 1)$:\n\n*   $\\lvert A_1 - 1\\rvert = 1 \\geq K$\n*   $\\lvert A_2 - 2\\rvert = 1 \\geq K$\n*   $\\lvert A_3 - 3\\rvert = 2 \\geq K$"],["8 3","4 5 6 7 8 1 2 3"],["8 6","\\-1"]],"created_at":"2026-03-03 11:01:14"}}