Minimum forward-jump dynamic programming ($D[j]=1+\min \{D[i]:0\le i<j,i+A[i]\ge j\}$)

For nonnegative maximum jump lengths in an array, let $D[0]=0$ and let unreachable positions have value $\infty$. Classifying a route to $j$ by its penultimate position proves this dynamic programming recurrence. Increasing $j$ is a valid dependency order because every jump advances. Scanning all previous positions uses $\Theta (n^2)$ time complexity and $\Theta (n)$ space complexity. Store a minimizing predecessor to reconstruct a shortest route.