### PROBLEM LINK:

**Author:** Vineet Paliwal

**Tester:** Roman Rubanenko

**Editorialist:** Jingbo Shang

### DIFFICULTY:

Medium

### PREREQUISITES:

Dynamic Programming, Suffix Sum, Fenwick Tree

### PROBLEM:

Given a Directed-Acyclic-Graph (DAG) G = (V, E) in which node i has edges to nodes in [i + 1, i + N*], find how many paths are there between S* and T.

### EXPLANATION:

This DAG is really special and the order of 1 … V is exactly same as its topo order in which edges are only existed from previous nodes to their later ones.

Use F* to state the number of different paths starting from node i to node T.

```
Initially F[T] = 1, F[others] = 0.
```

The transmission can be described as following:

```
For i = T - 1 downto 1
F* = \sum_{v = i + 1} ^ {i + N*}
```

To speed up this transmission procedure, we can use a Fenwick Tree to get the sum. But we can achieve it in a simpler way as following, using suffix sum.

```
suffixSum[] = 0;
suffixSum[T] = 1;
For i = T - 1 downto 1
F* = G[i + N*];
G* = G[i + 1] + F*
```

To answer each query, just directly output the F[S*]. Therefore, the time complexity is O(N + Q) in total.

### AUTHOR’S AND TESTER’S SOLUTIONS:

Author’s solution can be found here.

Tester’s solution can be found here.