Minimum Sideway Jumps

Minimum Sideway Jumps

What is this problem about?

The Minimum Sideway Jumps problem places a frog on a 3-lane road with obstacles. The frog starts in lane 2 at position 0 and must reach the end. At each step, the frog can move forward for free or jump sideways (changing lanes), with each sideways jump costing 1. Obstacles block certain lane-position combinations. Find the minimum number of sideways jumps needed. This Minimum Sideway Jumps interview question is a DP or greedy problem on a lane-based movement model.

Why is this asked in interviews?

Companies like Microsoft, Oracle, and Google ask this because it tests the ability to maintain local state (current lane, current position) and apply greedy or DP reasoning to minimize cost over a sequence of decisions. The array, greedy, and dynamic programming interview pattern is well-suited here. The frog's constraint of needing to switch lanes when blocked adds the required complexity.

Algorithmic pattern used

The approach is DP with 3 states (one per lane). Define dp[lane] as the minimum jumps to be in that lane at the current position. Process positions left to right. At each position, first invalidate lanes blocked by obstacles. Then, for unblocked lanes, compute the minimum jumps to switch from any other non-blocked lane (cost: dp[otherLane] + 1). Update all lane costs simultaneously. Final answer: min(dp[1], dp[2], dp[3]) at the last position.

Example explanation

Obstacles: [0,1,2,3,0] (obstacle array where 0 means no obstacle, 1/2/3 means lane 1/2/3 is blocked). Start: lane 2, jumps = 0. Position 1: lane 1 blocked. Frog is in lane 2 (no jump needed), dp = [∞, 0, ∞]. Position 2: lane 2 blocked. Must jump from lane 2 to lane 1 or 3. dp = [1, ∞, 1]. Continue until end. Minimum jumps = 2 (example-specific).

Common mistakes candidates make

• Updating lane costs in-place during the same position sweep (causes dependency errors).
• Not initializing the starting lane correctly.
• Forgetting that jumping sideways is only needed when the current lane is blocked.
• Missing the greedy insight: always jumping to the nearest unblocked lane minimizes cost.

Interview preparation tip

Lane-based DP problems benefit from a clear state table visualization. Before coding, draw a grid of positions × lanes and fill in DP values for a small example by hand. This reveals the transition logic before you commit to code. For this problem, a key insight is that you only need O(1) space (just 3 values for 3 lanes), regardless of road length — recognize and exploit such space compression in interviews.