Greedy algorithms make locally optimal choices at each step, hoping to find a global optimum. They're particularly effective for problems where making the best choice at each moment leads to the overall best solution.
The key insight for jump problems is that we don't need to explore every possible path. Instead, we can use a range-based approach similar to breadth-first search (BFS). Think of it as expanding our reach in "waves" - each wave represents positions we can reach with the same number of jumps.
For example, if we start at position 0 and can jump 2 steps, our first "wave" covers positions 1 and 2. From those positions, we can reach even further positions in our second wave. The brilliant part is that we only need to track:
This approach works because if we can reach a position with fewer jumps, we never need to consider reaching it with more jumps. The greedy choice - always extending our reach as far as possible - naturally leads to the minimum number of jumps.
The algorithm processes positions left to right, constantly updating how far we can reach. When we exhaust our current range (reach the boundary), we "use up" one jump and extend to our new maximum range. This continues until we can reach the target position.
1def max_reach_in_steps(arr, max_steps):
2 """Find maximum index reachable within given steps."""
3 current_reach = 0
4 next_reach = 0
5 steps_used = 0
6
7 for i in range(len(arr)):
8 # Update how far we can go with one more step
9 next_reach = max(next_reach, i + arr[i])
10
11 # If we've used up our current range
12 if i == current_reach and steps_used < max_steps:
13 steps_used += 1
14 current_reach = next_reach # Extend our reach
15
16 return current_reach