Minimum Number of Jumps

View each jump count as a BFS layer over indices, but compute layers without an explicit queue. end marks the furthest index reachable using the current number of jumps, and farthest records the furthest index reachable after one more jump from any index scanned in that layer. Scan through n-2, update farthest with i + arr[i], and when i equals end, consume one jump and move end to farthest. Once end reaches or passes n-1, the answer is known. If a layer ends with farthest equal to end, no later index can be reached, so return -1 or the platform's failure sentinel. This is the same correctness idea as breadth-first search: the first layer that can reach the last index uses the fewest jumps. Handle a single element as zero jumps, leading zero in longer input as impossible, and large jump values safely. Time is O(n), space O(1).