Problem2546--Bear and Cavalry

2546: Bear and Cavalry

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Time Limit : 2.000 sec  Memory Limit : 256 MiB

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Description

time limit per test
3 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

Would you want to fight against bears riding horses? Me neither.

Limak is a grizzly bear. He is general of the dreadful army of Bearland. The most important part of an army is cavalry of course.

Cavalry of Bearland consists of n warriors and n horses. i-th warrior has strength wi and i-th horse has strength hi. Warrior together with his horse is called a unit. Strength of a unit is equal to multiplied strengths of warrior and horse. Total strength of cavalry is equal to sum of strengths of all n units. Good assignment of warriors and horses makes cavalry truly powerful.

Initially, i-th warrior has i-th horse. You are given q queries. In each query two warriors swap their horses with each other.

General Limak must be ready for every possible situation. What if warriors weren't allowed to ride their own horses? After each query find the maximum possible strength of cavalry if we consider assignments of all warriors to all horses that no warrior is assigned to his own horse (it can be proven that for n≥2 there is always at least one correct assignment).

Note that we can't leave a warrior without a horse.

Input

The first line contains two space-separated integers, n and q (2≤n≤30 000, 1≤q≤10 000).

The second line contains n space-separated integers, w1,w2,...,wn (1≤wi≤106) − strengths of warriors.

The third line contains n space-separated integers, h1,h2,...,hn (1≤hi≤106) − strengths of horses.

Next q lines describe queries. i-th of them contains two space-separated integers ai and bi (1≤ai,bi≤n, ai≠bi), indices of warriors who swap their horses with each other.

Output

Print q lines with answers to queries. In i-th line print the maximum possible strength of cavalry after first i queries.

Examples
Input
4 2
1 10 100 1000
3 7 2 5
2 4
2 4
Output
5732
7532
Input
3 3
7 11 5
3 2 1
1 2
1 3
2 3
Output
44
48
52
Input
7 4
1 2 4 8 16 32 64
87 40 77 29 50 11 18
1 5
2 7
6 2
5 6
Output
9315
9308
9315
9315
Note

Clarification for the first sample:

Warriors:1101001000

Horses:3725

After first query situation looks like the following:

Warriors:1101001000

Horses:3527

We can get 1·2+10·3+100·7+1000·5=5732 (note that no hussar takes his own horse in this assignment).

After second query we get back to initial situation and optimal assignment is 1·2+10·3+100·5+1000·7=7532.

Clarification for the second sample. After first query:

Warriors:7115

Horses:231

Optimal assignment is 7·1+11·2+5·3=44.

Then after second query 7·3+11·2+5·1=48.

Finally 7·2+11·3+5·1=52.

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