Interval DP — Filling dp[i][j] by Increasing Length

dp[i][j] = best over split point k of ( dp[i][k] + dp[k+1][j] + cost(i,k,j) ). Watch the table fill diagonal by diagonal and the optimal split k highlight.

step 0

dp[i][j] table (upper triangle)

sequence
current cell (i,j) left dp[i][k] right dp[k(+1)][j] finished

Trying split points k

Fill order is by increasing interval length: length 1 (the diagonal), then 2, then 3, … up to the whole range.
Press Step to begin. We fill dp[i][j] for every contiguous range, shortest first, trying every split point k.
Self-contained visualization. Split-point problems partition [i..j] into [i..k] and [k+1..j]; burst balloons uses the last-element framing with halves (i,k) and (k,j) on the padded array. The answer is the top-right corner dp[0][n-1]. See junior.md and professional.md for the proof that increasing-length order is a valid topological fill.