Source
My school hosted a local ICPC format contest yesterday, this problem appeared in it. Since it was for local only, I cannot provide link for submission (sorry!)
Statement
Given a weighted undirected graph $$$G$$$ with $$$n$$$ vertices and $$$m$$$ edges. Edge $$$e_i$$$ is a tuple $$$(u_i, v_i, A_i, B_i)$$$ denotes there is an edge connects $$$u_i$$$ with $$$v_i$$$, with cost $$$A_i$$$ and $$$B_i$$$. Consider path $$$u \rightarrow v$$$ is a sequence of edges $$$E: \{e_1, e_2,... e_k\}$$$, cost of $$$E$$$ is defined as product of (sum of all $$$A_x$$$) and (sum of all $$$B_x$$$) with $$$x$$$ is an edge in $$$E$$$. Calculate shortest path from node $$$1$$$ to all other nodes.
Contraints
- $$$1 \leq n, m \leq 2000$$$
- $$$1 \leq A_i, B_i \leq 2000$$$
There is no additional guarantees
Sample Tests
Input 1
4 4 1 2 2 4 3 4 4 1 4 2 1 1 1 3 3 1
Output 1
8 3 14
Output shortest path from $$$1$$$ to $$$i (2 \leq i \leq n)$$$:
- Shortest path from node $$$1 \rightarrow$$$ node $$$2$$$ is the edge $$$(1,2)$$$, with cost $$$2 \times 4 = 8$$$
- Shortest path from node $$$1 \rightarrow$$$ node $$$3$$$ is the edge $$$(1,3)$$$, with cost $$$3 \times 1 = 3$$$
- Shortest path from node $$$1 \rightarrow$$$ node $$$4$$$ are $$$(1,3),(3,4)$$$ with cost $$$(3+4) \times (1+1) = 14$$$. The other path is $$$(1,2),(2,4)$$$ has cost $$$(2+1) \times (4+1) = 15$$$ is not the shortest path.
Input 2
3 2 1 2 2 5 2 1 3 3
Output 2
9 -1
$$$-1$$$ means there is no path from $$$1$$$ to that node.