Knapsack 2 - Walkthrough

Let's trace value-based Knapsack with items $[(w=10, v=1), (w=20, v=2), (w=30, v=3)]$ and capacity $W=50$. The DP is $dp[v]$ = minimum weight to achieve value $v$. Start with $dp[0]=0$, all others $= \infty$. After item 1 ($v=1, w=10$): $dp[1]=10$. After item 2 ($v=2, w=20$): $dp[2]=20$, $dp[3]=30$. After item 3 ($v=3,$w$=30$): $dp[3]=\min(30, 30)=30$, $dp[4]=40$, $dp[5]=50$, $dp[6]=60$.

Answer: the maximum $v$ where $dp[v] \leq 50$ is $v=5$ (weight $50$). We took items 2 and 3.