P07 — Dynamic programming — every solution explained

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1. Ways to climb n stairs taking 1 or 2 steps
2. DECODE WAYS -- the dated question. 'A'=1 ... 'Z'=26. dp[i] = the number of ways to decode the FIRST i characters
3. The same thing as plain recursion -- exponential, but it SCORES on small inputs. Write this if the recurrence will not come to you
4. Fibonacci, bottom-up. The shape every 1-D DP reduces to
5. Cheapest way to the top when each step charges a toll
6. Largest total from non-adjacent elements
7. Largest sum of any contiguous subarray (Kadane)
8. Largest product of any contiguous subarray. Track BOTH ends -- a negative flips them
9. Fewest coins summing to amount; -1 if impossible
10. How many distinct COMBINATIONS of coins sum to amount. Coin loop outside, amount inside -- swap them and you count permutations instead
11. Length of the longest strictly increasing subsequence. O(n^2) is fine here
12. Can the array be split into two parts with equal sums? Subset sum over one dimension
13. Ways to reach step n taking 1, 2 or 3 steps