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Shortest path problem.
By Loserinlife, history, 8 months ago, In English

Given a weighted undirected graph with N nodes and M edges (u, v, w) meaning node u and v is connected with weight w. Additionally, each node has a value a[i], and you can go from node i to node j with the cost of (a[i] + a[j]) % k, where k is a given constant. Find the shortest path from s to t

Constraints: N, M <= 2e5

a[i] < k <= 1e9

1 <= u, v, s, t <= n

w <= 1e9

Thanks

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8 months ago, # |
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I think that normal Dijkstra's algorithm can solve it.

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    8 months ago, # ^ |
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    Additionally, each node has a value a[i], and you can go from node i to node j with the cost of (a[i] + a[j]) % k, where k is a given constant. Find the shortest path from s to t

    This is for all $$$n(n-1)/2$$$ pairs of vertices, meaning that the graph actually has $$$m + n(n-1)/2 = O(n^2)$$$ edges. Dijkstra runs in $$$O(E \log V)$$$ which is not fast enough if $$$E = O(V^2)$$$ and $$$V = 200\ 000$$$.

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8 months ago, # |
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Check this problem Modulo Shortest Path.

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