Koa and the Beach (Hard Version)

题意翻译

这是问题 B 的困难版本。它与 B1 的唯一区别在于数据规模的大小。 Koa 要从海岸游到岛上。海岸的位置是 $0$，岛的位置是 $n+1$。 除了海岸和岛上，在 $[1,n]$ 之间的位置 $i$ 都有一个深度 $d_i$。海里有潮汐，具体来讲，给定值 $k$，则有下标**从** $0$ **开始**的数组 $p=[0,1,2,...,k-1,k,k-1,...,1]$。在时间 $t$，海里的潮汐高度为 $p_{t \mod 2k}$。 在任意时间，如果 Koa 在位置 $x$，可以选择下一个时间停留位置 $x$，或游到位置 $x+1$。 Koa 能在时间 $t$ 到达位置 $i$ 当且仅当时间 $t$ 的潮汐高度与位置 $i$ 的深度之和不超过一个给定的值 $l$，即：$p_{t \mod 2k}+d_i \leq l$，否则她会溺水。海岸和岛上是安全的，不会受潮汐影响，所以在任何时间停留在那里都不会溺水。Koa 在游泳的时候不会受潮汐的影响（不会溺水）。 求出 Koa 是否能安全地到达岛上。 多组数据，数据组数 $t \leq 10^4$，$1 \leq n \leq 3\cdot 10^5,1 \leq k,l\leq 10^9,0 \leq d_i \leq 10^9$ 翻译 by Meatherm

题目描述

The only difference between easy and hard versions is on constraints. In this version constraints are higher. You can make hacks only if all versions of the problem are solved. Koa the Koala is at the beach! The beach consists (from left to right) of a shore, $ n+1 $ meters of sea and an island at $ n+1 $ meters from the shore. She measured the depth of the sea at $ 1, 2, \dots, n $ meters from the shore and saved them in array $ d $ . $ d_i $ denotes the depth of the sea at $ i $ meters from the shore for $ 1 \le i \le n $ . Like any beach this one has tide, the intensity of the tide is measured by parameter $ k $ and affects all depths from the beginning at time $ t=0 $ in the following way: - For a total of $ k $ seconds, each second, tide increases all depths by $ 1 $ . - Then, for a total of $ k $ seconds, each second, tide decreases all depths by $ 1 $ . - This process repeats again and again (ie. depths increase for $ k $ seconds then decrease for $ k $ seconds and so on ...). Formally, let's define $ 0 $ -indexed array $ p = [0, 1, 2, \ldots, k - 2, k - 1, k, k - 1, k - 2, \ldots, 2, 1] $ of length $ 2k $ . At time $ t $ ( $ 0 \le t $ ) depth at $ i $ meters from the shore equals $ d_i + p[t \bmod 2k] $ ( $ t \bmod 2k $ denotes the remainder of the division of $ t $ by $ 2k $ ). Note that the changes occur instantaneously after each second, see the notes for better understanding. At time $ t=0 $ Koa is standing at the shore and wants to get to the island. Suppose that at some time $ t $ ( $ 0 \le t $ ) she is at $ x $ ( $ 0 \le x \le n $ ) meters from the shore: - In one second Koa can swim $ 1 $ meter further from the shore ( $ x $ changes to $ x+1 $ ) or not swim at all ( $ x $ stays the same), in both cases $ t $ changes to $ t+1 $ . - As Koa is a bad swimmer, the depth of the sea at the point where she is can't exceed $ l $ at integer points of time (or she will drown). More formally, if Koa is at $ x $ ( $ 1 \le x \le n $ ) meters from the shore at the moment $ t $ (for some integer $ t\ge 0 $ ), the depth of the sea at this point — $ d_x + p[t \bmod 2k] $ — can't exceed $ l $ . In other words, $ d_x + p[t \bmod 2k] \le l $ must hold always. - Once Koa reaches the island at $ n+1 $ meters from the shore, she stops and can rest. Note that while Koa swims tide doesn't have effect on her (ie. she can't drown while swimming). Note that Koa can choose to stay on the shore for as long as she needs and neither the shore or the island are affected by the tide (they are solid ground and she won't drown there). Koa wants to know whether she can go from the shore to the island. Help her!

输入输出格式

输入格式

The first line of the input contains one integer $ t $ ( $ 1 \le t \le 10^4 $ ) — the number of test cases. Description of the test cases follows. The first line of each test case contains three integers $ n $ , $ k $ and $ l $ ( $ 1 \le n \le 3 \cdot 10^5; 1 \le k \le 10^9; 1 \le l \le 10^9 $ ) — the number of meters of sea Koa measured and parameters $ k $ and $ l $ . The second line of each test case contains $ n $ integers $ d_1, d_2, \ldots, d_n $ ( $ 0 \le d_i \le 10^9 $ ) — the depths of each meter of sea Koa measured. It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 3 \cdot 10^5 $ .

输出格式

For each test case: Print Yes if Koa can get from the shore to the island, and No otherwise. You may print each letter in any case (upper or lower).

输入输出样例

输入样例 #1

7
2 1 1
1 0
5 2 3
1 2 3 2 2
4 3 4
0 2 4 3
2 3 5
3 0
7 2 3
3 0 2 1 3 0 1
7 1 4
4 4 3 0 2 4 2
5 2 3
1 2 3 2 2

输出样例 #1

Yes
No
Yes
Yes
Yes
No
No

说明

In the following $ s $ denotes the shore, $ i $ denotes the island, $ x $ denotes distance from Koa to the shore, the underline denotes the position of Koa, and values in the array below denote current depths, affected by tide, at $ 1, 2, \dots, n $ meters from the shore. In test case $ 1 $ we have $ n = 2, k = 1, l = 1, p = [ 0, 1 ] $ . Koa wants to go from shore (at $ x = 0 $ ) to the island (at $ x = 3 $ ). Let's describe a possible solution: - Initially at $ t = 0 $ the beach looks like this: $ [\underline{s}, 1, 0, i] $ . - At $ t = 0 $ if Koa would decide to swim to $ x = 1 $ , beach would look like: $ [s, \underline{2}, 1, i] $ at $ t = 1 $ , since $ 2 > 1 $ she would drown. So Koa waits $ 1 $ second instead and beach looks like $ [\underline{s}, 2, 1, i] $ at $ t = 1 $ . - At $ t = 1 $ Koa swims to $ x = 1 $ , beach looks like $ [s, \underline{1}, 0, i] $ at $ t = 2 $ . Koa doesn't drown because $ 1 \le 1 $ . - At $ t = 2 $ Koa swims to $ x = 2 $ , beach looks like $ [s, 2, \underline{1}, i] $ at $ t = 3 $ . Koa doesn't drown because $ 1 \le 1 $ . - At $ t = 3 $ Koa swims to $ x = 3 $ , beach looks like $ [s, 1, 0, \underline{i}] $ at $ t = 4 $ . - At $ t = 4 $ Koa is at $ x = 3 $ and she made it! We can show that in test case $ 2 $ Koa can't get to the island.