NUMPATH - Editorial

PROBLEM LINK:

Practice
Contest

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.

4 Likes

http://www.codechef.com/viewsolution/7049475

Can anyone suggest how to reduce time? I am exceeding the time limit. I am working with C.

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