Farmer John's cows have grown tired of his daily request that they sort themselves before leaving the barn each morning. They have just completed their PhDs in quantum physics, and are ready to speed things up a bit.

This morning, as usual, Farmer John's N cows (1 \leq N \leq 10^5), conveniently numbered 1 \dots N, are scattered throughout the barn at N distinct locations, also numbered 1 \dots N, such that cow i is at location p_i. But this morning there are also M wormholes (1 \leq M \leq 10^5), numbered 1 \dots M, where wormhole i bidirectionally connects location a_i with location b_i, and has a width w_i (1\le a_i,b_i\le N, a_i\neq b_i, 1\le w_i\le 10^9).

At any point in time, two cows located at opposite ends of a wormhole may choose to simultaneously swap places through the wormhole. The cows must perform such swaps until cow i is at location i for 1 \leq i \leq N.

The cows are not eager to get squished by the wormholes. Help them maximize the width of the least wide wormhole which they must use to sort themselves. It is guaranteed that it is possible for the cows to sort themselves.

SCORING:

Test cases 3-5 satisfy N,M\le 1000.
Test cases 6-10 satisfy no additional constraints.

Problem credits: Dhruv Rohatgi

#4149 upload 2 done 1 how_to_reg 1

Wormhole Sort timer 2s memory 512MB

#4149

Wormhole Sort 2s 512MB