K and Medians

题意翻译

给定奇数 $k$ 和长度分别为 $n,m$ 的序列 $a,b$，序列 $a$ 包含所有整数 $i$ 满足 $1\leq i\leq n$，$b$ 是给定的单调不减的序列。 现在你可以进行零次或若干次操作，每次操作选择序列 $a$ 的 $k$ 个整数，然后删除这 $k$ 个数里除了他们的中位数的其他 $(k-1)$ 个数。 求能否通过零次或若干次操作从序列 $a$ 的到序列 $b$。能就输出 `YES`，不能输出 `NO`。 $3\leq n\leq200000;3\leq k\leq n;1\leq m<n.$ $1\leq b_1<b_2<...<b_m\leq n.$

题目描述

Let's denote the median of a sequence $ s $ with odd length as the value in the middle of $ s $ if we sort $ s $ in non-decreasing order. For example, let $ s = [1, 2, 5, 7, 2, 3, 12] $ . After sorting, we get sequence $ [1, 2, 2, \underline{3}, 5, 7, 12] $ , and the median is equal to $ 3 $ . You have a sequence of $ n $ integers $ [1, 2, \dots, n] $ and an odd integer $ k $ . In one step, you choose any $ k $ elements from the sequence and erase all chosen elements except their median. These elements do not have to go continuously (gaps are allowed between them). For example, if you have a sequence $ [1, 2, 3, 4, 5, 6, 7] $ (i.e. $ n=7 $ ) and $ k = 3 $ , then the following options for the first step are possible: - choose $ [1, \underline{2}, 3] $ ; $ 2 $ is their median, so it is not erased, and the resulting sequence is $ [2, 4, 5, 6, 7] $ ; - choose $ [2, \underline{4}, 6] $ ; $ 4 $ is their median, so it is not erased, and the resulting sequence is $ [1, 3, 4, 5, 7] $ ; - choose $ [1, \underline{6}, 7] $ ; $ 6 $ is their median, so it is not erased, and the resulting sequence is $ [2, 3, 4, 5, 6] $ ; - and several others. You can do zero or more steps. Can you get a sequence $ b_1 $ , $ b_2 $ , ..., $ b_m $ after several steps? You'll be given $ t $ test cases. Solve each test case independently.

输入输出格式

输入格式

The first line contains a single integer $ t $ ( $ 1 \le t \le 1000 $ ) — the number of test cases. The first line of each test case contains three integers $ n $ , $ k $ , and $ m $ ( $ 3 \le n \le 2 \cdot 10^5 $ ; $ 3 \le k \le n $ ; $ k $ is odd; $ 1 \le m < n $ ) — the length of the sequence you have, the number of elements you choose in each step and the length of the sequence you'd like to get. The second line of each test case contains $ m $ integers $ b_1, b_2, \dots, b_m $ ( $ 1 \le b_1 < b_2 < \dots < b_m \le n $ ) — the sequence you'd like to get, given in the ascending order. It's guaranteed that the total sum of $ n $ over all test cases doesn't exceed $ 2 \cdot 10^5 $ .

输出格式

For each test case, print YES if you can obtain the sequence $ b $ or NO otherwise. You may print each letter in any case (for example, YES, Yes, yes, yEs will all be recognized as positive answer).

输入输出样例

输入样例 #1

4
3 3 1
1
7 3 3
1 5 7
10 5 3
4 5 6
13 7 7
1 3 5 7 9 11 12

输出样例 #1

NO
YES
NO
YES

说明

In the first test case, you have sequence $ [1, 2, 3] $ . Since $ k = 3 $ you have only one way to choose $ k $ elements — it's to choose all elements $ [1, \underline{2}, 3] $ with median $ 2 $ . That's why after erasing all chosen elements except its median you'll get sequence $ [2] $ . In other words, there is no way to get sequence $ b = [1] $ as the result. In the second test case, you have sequence $ [1, 2, 3, 4, 5, 6, 7] $ and one of the optimal strategies is following: 1. choose $ k = 3 $ elements $ [2, \underline{3}, 4] $ and erase them except its median; you'll get sequence $ [1, 3, 5, 6, 7] $ ; 2. choose $ 3 $ elements $ [3, \underline{5}, 6] $ and erase them except its median; you'll get desired sequence $ [1, 5, 7] $ ; In the fourth test case, you have sequence $ [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] $ . You can choose $ k=7 $ elements $ [2, 4, 6, \underline{7}, 8, 10, 13] $ and erase them except its median to get sequence $ b $ .

题意翻译

给定奇数 $k$ 和长度分别为 $n,m$ 的序列 $a,b$，序列 $a$ 包含所有整数 $i$ 满足 $1\leq i\leq n$，$b$ 是给定的单调不减的序列。 现在你可以进行零次或若干次操作，每次操作选择序列 $a$ 的 $k$ 个整数，然后删除这 $k$ 个数里除了他们的中位数的其他 $(k-1)$ 个数。 求能否通过零次或若干次操作从序列 $a$ 的到序列 $b$。能就输出 `YES`，不能输出 `NO`。 $3\leq n\leq200000;3\leq k\leq n;1\leq m<n.$ $1\leq b_1<b_2<...<b_m\leq n.$