Narrowest column widths for spanning header cells

A data grid sizes its columns to fit their content. Some header cells span several adjacent columns, like a "Q1" header over January, February and March. Every cell needs a minimum width to show its text without clipping.

Write narrowest_table(columns, cells) that returns the smallest total width the columns can have.

Input and output:

• columns: the number of columns, from 1 to 1,000.
• cells: up to 1,000 cells, each [first, span, width]. The cell covers the span columns that start at column first (columns count from 0), and it needs at least width pixels. span is at least 1, first + span is at most columns, and width is an integer from 0 to 1,000,000,000.
• Returns an integer.

Rules:

• Every column gets a whole number of pixels, 0 or more.
• For every cell, the widths of the columns it covers add up to at least the cell's width.
• Borders and padding are already included in the cell widths.
• Return the smallest possible sum of all the column widths.