Platforms Jumping
Contest: Codeforces Round 598 (Div. 3)
Difficulty rating: 1700
Tags: greedy

There is a river of width $$$n$$$. The left bank of the river is cell $$$0$$$ and the right bank is cell $$$n + 1$$$ (more formally, the river can be represented as a sequence of $$$n + 2$$$ cells numbered from $$$0$$$ to $$$n + 1$$$). There are also $$$m$$$ wooden platforms on a river, the $$$i$$$-th platform has length $$$c_i$$$ (so the $$$i$$$-th platform takes $$$c_i$$$ consecutive cells of the river). It is guaranteed that the sum of lengths of platforms does not exceed $$$n$$$.

You are standing at $$$0$$$ and want to reach $$$n+1$$$ somehow. If you are standing at the position $$$x$$$, you can jump to any position in the range $$$[x + 1; x + d]$$$. However you don't really like the water so you can jump only to such cells that belong to some wooden platform. For example, if $$$d=1$$$, you can jump only to the next position (if it belongs to the wooden platform). You can assume that cells $$$0$$$ and $$$n+1$$$ belong to wooden platforms.

You want to know if it is possible to reach $$$n+1$$$ from $$$0$$$ if you can move any platform to the left or to the right arbitrary number of times (possibly, zero) as long as they do not intersect each other (but two platforms can touch each other). It also means that you cannot change the relative order of platforms.

Note that you should move platforms until you start jumping (in other words, you first move the platforms and then start jumping).

For example, if $$$n=7$$$, $$$m=3$$$, $$$d=2$$$ and $$$c = [1, 2, 1]$$$, then one of the ways to reach $$$8$$$ from $$$0$$$ is follow:

The first example: $$$n=7$$$.

Input format:
The first line of the input contains three integers $$$n$$$, $$$m$$$ and $$$d$$$ ($$$1 \le n, m, d \le 1000, m \le n$$$) — the width of the river, the number of platforms and the maximum distance of your jump, correspondingly.

The second line of the input contains $$$m$$$ integers $$$c_1, c_2, \dots, c_m$$$ ($$$1 \le c_i \le n, \sum\limits_{i=1}^{m} c_i \le n$$$), where $$$c_i$$$ is the length of the $$$i$$$-th platform.

Output format:
If it is impossible to reach $$$n+1$$$ from $$$0$$$, print NO in the first line. Otherwise, print YES in the first line and the array $$$a$$$ of length $$$n$$$ in the second line — the sequence of river cells (excluding cell $$$0$$$ and cell $$$n + 1$$$).

If the cell $$$i$$$ does not belong to any platform, $$$a_i$$$ should be $$$0$$$. Otherwise, it should be equal to the index of the platform ($$$1$$$-indexed, platforms are numbered from $$$1$$$ to $$$m$$$ in order of input) to which the cell $$$i$$$ belongs.

Note that all $$$a_i$$$ equal to $$$1$$$ should form a contiguous subsegment of the array $$$a$$$ of length $$$c_1$$$, all $$$a_i$$$ equal to $$$2$$$ should form a contiguous subsegment of the array $$$a$$$ of length $$$c_2$$$, ..., all $$$a_i$$$ equal to $$$m$$$ should form a contiguous subsegment of the array $$$a$$$ of length $$$c_m$$$. The leftmost position of $$$2$$$ in $$$a$$$ should be greater than the rightmost position of $$$1$$$, the leftmost position of $$$3$$$ in $$$a$$$ should be greater than the rightmost position of $$$2$$$, ..., the leftmost position of $$$m$$$ in $$$a$$$ should be greater than the rightmost position of $$$m-1$$$.

See example outputs for better understanding.

Constraints:
Time limit: 1.0 s
Memory limit: 256.0 MB

Note:
Consider the first example: the answer is $$$[0, 1, 0, 2, 2, 0, 3]$$$. The sequence of jumps you perform is $$$0 \rightarrow 2 \rightarrow 4 \rightarrow 5 \rightarrow 7 \rightarrow 8$$$.

Consider the second example: it does not matter how to place the platform because you always can jump from $$$0$$$ to $$$11$$$.

Consider the third example: the answer is $$$[0, 0, 0, 0, 1, 1, 0, 0, 0, 0]$$$. The sequence of jumps you perform is $$$0 \rightarrow 5 \rightarrow 6 \rightarrow 11$$$.

Example input:
7 3 2
1 2 1

Example output:
YES
0 1 0 2 2 0 3

Example input:
10 1 11
1

Example output:
YES
0 0 0 0 0 0 0 0 0 1

Editorial:
This problem has a very easy idea but requires terrible implementation. Firstly, let's place all platforms as rightmost as we can. Thus, we will have the array, in which the first $$$n - \sum\limits_{i=1}^{m} c_i$$$ elements are zeros and other elements are $$$1$$$, $$$2$$$, ..., $$$m$$$.
Now, let's start the algorithm. Firstly, we need to jump to the position $$$d$$$ or less. If we could jump to the position $$$d$$$ then we don't need to jump to some position to the left from $$$d$$$. But if we cannot do it, let's take the leftmost platform to the right from the position $$$d$$$ and move it in such a way that its left border will be at the position $$$d$$$. Now we can jump to the position $$$d$$$ and then jump by $$$1$$$ right to reach the position $$$d + c_1 - 1$$$. Let's repeat the same algorithm and continue jumping.
If after some move we can jump to the position at least $$$n+1$$$ then we are done.
Time complexity: $$$O(n^2)$$$ but I'm sure it can be implemented in $$$O(n \log n)$$$ or $$$O(n)$$$.