Sum of Paths
Contest: Codeforces Round 695 (Div. 2)
Difficulty rating: 2200
Tags: combinatorics, dp, math

There are $$$n$$$ cells, numbered $$$1,2,\dots, n$$$ from left to right. You have to place a robot at any cell initially. The robot must make exactly $$$k$$$ moves.

In one move, the robot must move one cell to the left or right, provided that it doesn't move out of bounds. In other words, if the robot was in the cell $$$i$$$, it must move to either the cell $$$i-1$$$ or the cell $$$i+1$$$, as long as it lies between $$$1$$$ and $$$n$$$ (endpoints inclusive). The cells, in the order they are visited (including the cell the robot is placed), together make a good path.

Each cell $$$i$$$ has a value $$$a_i$$$ associated with it. Let $$$c_0, c_1, \dots, c_k$$$ be the sequence of cells in a good path in the order they are visited ($$$c_0$$$ is the cell robot is initially placed, $$$c_1$$$ is the cell where the robot is after its first move, and so on; more formally, $$$c_i$$$ is the cell that the robot is at after $$$i$$$ moves). Then the value of the path is calculated as $$$a_{c_0} + a_{c_1} + \dots + a_{c_k}$$$.

Your task is to calculate the sum of values over all possible good paths. Since this number can be very large, output it modulo $$$10^9 + 7$$$. Two good paths are considered different if the starting cell differs or there exists an integer $$$i \in [1, k]$$$ such that the current cell of the robot after exactly $$$i$$$ moves is different in those paths.

You must process $$$q$$$ updates to $$$a$$$ and print the updated sum each time. Each update changes the value of exactly one cell. See the input format and the sample input-output for more details.

Input format:
The first line of the input contains three space-separated integers $$$n$$$, $$$k$$$ and $$$q$$$ ($$$2 \le n \le 5000$$$; $$$1 \le k \le 5000$$$; $$$1 \le q \le 2 \cdot 10^5$$$).

The second line of the input contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$).

$$$q$$$ lines follow. Each line contains two space-separated integers $$$i$$$ and $$$x$$$ ($$$1 \le i \le n$$$; $$$1 \le x \le 10^9$$$) indicating that you must change the value of $$$a_i$$$ to $$$x$$$.

Output format:
Print $$$q$$$ integers. The $$$i$$$-th integer should be the sum of values over all good paths after the first $$$i$$$ updates are performed. Since the answers may be large, print them modulo $$$10^9 + 7$$$.

Constraints:
Time limit: 3.0 s
Memory limit: 1024.0 MB

Note:
In the first example, the good paths are $$$(1, 2), (2, 1), (2, 3), (3, 2), (3, 4), (4, 3), (4, 5), (5, 4)$$$.

Initially the values of $$$a$$$ are $$$[3, 5, 1, 4, 2]$$$. After the first update, they become $$$[9, 5, 1, 4, 2]$$$. After the second update, they become $$$[9, 4, 1, 4, 2]$$$, and so on.

Example input:
5 1 5
3 5 1 4 2
1 9
2 4
3 6
4 6
5 2

Example output:
62
58
78
86
86

Example input:
5 2 5
3 5 1 4 2
1 9
2 4
3 6
4 6
5 2

Example output:
157
147
207
227
227

Editorial:
The main idea of the problem is to calculate the contribution of each cell to the answer.
Let $$$cnt_i$$$ denote the sum of the number of times cell $$$i$$$ appears in all good paths of length $$$k$$$. Then the answer is equal to $$$a_1*cnt_1 + a_2*cnt_2 + \dots + a_n*cnt_n$$$. We shall use dynamic programming to calculate these values.
Let $$$dp_{{i,j}}$$$ denote the number of good paths of length $$$j$$$ that end at cell $$$i$$$. Then, $$$dp_{{i,0}} = 1$$$ for all $$$i$$$ such that $$$1 \le i \le n$$$. Further, for all $$$i$$$ such that $$$1 \le i \le n$$$ and for all $$$j$$$ such that $$$1 \le j \le k$$$, $$$dp_{{i, j}}$$$ can be calculated as $$$dp_{{i,j}} = dp_{{{{i-1}},{{j-1}}}} + dp_{{{{i+1}},{{j-1}}}}$$$ because we can move to cell $$$i$$$ from either cell $$$i-1$$$ or cell $$$i+1$$$. Handle the cases where $$$i=1$$$ and $$$i=n$$$ separately to avoid out-of-bounds error.
Observe that $$$dp_{{i,j}}$$$ is also equal to the number of good paths of length $$$j$$$ that start at cell $$$i$$$.
Let $$$a_{{i,j}}$$$ denote the number of times cell $$$i$$$ appears after exactly $$$j$$$ moves in all valid paths of length $$$k$$$. Well $$$a_{{i,j}} = dp_{{i,j}} * dp_{{i,k-j}}$$$ because we can split a path of length $$$k$$$ into two paths of length $$$j$$$ and length $$$k-j$$$, with the first path ending at cell $$$i$$$ and the second path starting at cell $$$i$$$.
Since $$$cnt_i$$$ denotes the sum of the number of times cell $$$i$$$ appears in all good path of length $$$k$$$, cell $$$i$$$ can appear after exactly $$$0, 1, 2, \dots, k$$$ moves. This means that $$$cnt_i = \sum_{{j=0}}^{{k}} a_{{i,j}}$$$.
Extending the solution to account for updates is easy once we have calculated these values.