CCC '19 S5 - Triangle: The Data Structure

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Points: 20 (partial)
Time limit: 1.0s
Memory limit: 512M

Problem type
Canadian Computing Competition: 2019 Stage 1, Senior #5

In a parallel universe, the most important data structure in computer science is the triangle. A triangle of size M consists of M rows, with the i^{th} row containing i elements. Furthermore, these rows must be arranged to form the shape of an equilateral triangle. That is, each row is centred around a vertical line of symmetry through the middle of the triangle. For example, the diagram below shows a triangle of size 4:

A triangle contains sub-triangles. For example, the triangle above contains ten sub-triangles of size 1, six sub-triangles of size 2 (two of which are the triangle containing (3,1,2) and the triangle containing (4,6,1)), three sub-triangles of size 3 (one of which contains (2,2,1,1,4,2)). Note that every triangle is a sub-triangle of itself.

You are given a triangle of size N and must find the sum of the maximum elements of every sub-triangle of size K.

Input Specification

The first line contains two space-separated integers N and K (1 \le K \le N \le 3\,000).

Following this are N lines describing the triangle. The i^{th} of these lines contains i space-separated integers a_{i,j} (0 \le a_{i,j} \le 10^9), representing the i^{th} row of the triangle.

For 4 of the 15 available marks, N \le 1000.

Output Specification

Output the integer sum of the maximum elements of every sub-triangle of size K.

Sample Input

4 2
1 2
4 2 1
6 1 4 2

Output for Sample Input



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