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[i]], find how many paths are there between S[i] 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[i] 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[i] = \sum_{v = i + 1} ^ {i + N[i]}
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[i] = G[i + N[i]];
G[i] = G[i + 1] + F[i]
To answer each query, just directly output the F[S[i]]. 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.