2017-11-22 13:03

Sky Soldiers

  • numbers
  • air
  • less
  • lines
  • each

Problem Description
An airplane carried k soldiers with parachute is plan to let these soldiers jump off the plane along a straight air route. The landing point of these soldiers is undetermined because of various weather conditions. However, the statisticians of the army can analysis the probability of landing in a certain place through landing history records. To make it simple, the statistician suggests that these sky soldiers will land on finite discrete points of a straight line.

This mission plans to place m provisions for the soldiers on the line for landing. These soldiers will be informed the direction of the nearest provision point by a special device after landing, and then walk to the point. The manager of this mission is asking you for help: to determine m points for provisions so that the expected sum of walking distance should be minimized. You can put provisions on any point of the landing line.

There are multiple test cases. For each case, the first line contains two integers k and m (1 ≤ k ≤ 1,000, 1 ≤ m ≤ 50), which represent the number of sky soldiers and the number of positions to place provisions separately.

The following k lines contain descriptions of landing parameters for the soldiers numbered from 1 to k. Each description consists of an integer L followed by L pairs of (x, p), which indicates that the probability of the soldier's landing on integer coordination x is p. It is guaranteed that all the p values are positive real numbers, and the sum of p in a single line is exactly 1. The same x may appear more than once on the same line which you should simply add up all the probability p of the pairs with equal x.
The number of places on which all the soldiers could land is no more than 1000 and it can not be less than m.
The input ends with k=m=0.

For each test case, output a line containing only one real number which indicates the minimum expected sum of distance these soldiers will move and should be rounded to two digits after the decimal point.

Sample Input
2 1
2 0 0.5 1 0.5
2 1 0.1 3 0.9
0 0

Sample Output

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