The Rat in a Maze problem is a classic algorithmic puzzle. However, the variation Rat Maze with Multiple Jumps adds another level of complexity. Instead of moving just one step at a time, the rat can jump across multiple cells, making pathfinding and decision-making more challenging.
This article explores the problem from two perspectives:
The Fresher's Guide – Focused on foundational intuition and core logic.
The Experienced Engineer's Breakdown – Focused on edge cases, architectural patterns, and optimization considerations.
The Fresher's Guide: Understanding the Core Logic
As a fresher, your primary goal is to understand how the algorithm explores options and tracks its path. This problem is solved using a combination of Backtracking and Dynamic Programming (Memoization).
The Core Rules of the Game
The Starting Grid
You start at the top-left cell (0,0) and want to reach the bottom-right cell (n-1, n-1).
The Jumping Power
The number inside mat[i][j] tells you the maximum number of cells you can jump. If a cell contains 3, you can jump 1, 2, or 3 steps.
Dead Ends
A cell with 0 represents a wall. You cannot land on it or jump from it.
The Priority Rule
If multiple jumps are possible:
Always try the shortest jumps first.
For jumps of the same length, move Right before Down.
The Algorithmic Flow: Two Phases
The JavaScript solution approaches the problem by dividing it into two distinct phases.
Phase 1: Can We Reach the End? (canReach)
Think of this as a scouting mission.
The rat explores all possible jumps. To avoid recalculating the same cell repeatedly, it stores results in a memoization table (dp).
If a cell is marked true, the rat already knows it can reach the destination from there.
If a cell is marked false, further exploration is unnecessary.
Phase 2: Building the Path (build)
Once the scouting phase confirms that a path exists, the rat traverses the maze again.
Following the priority rules:
The path is marked with 1s in the result matrix.
The Experienced Engineer's Breakdown
For an experienced developer, correctness is only one aspect of the solution. It is equally important to evaluate execution flow, architectural decisions, complexity trade-offs, and optimization opportunities.
Tie-Breaking and Preference Order
The problem statement specifies:
Shortest possible jumps first.
For the same jump length, Right before Down.
The nested loop structure directly enforces these requirements.
for (let step = 1; step <= maxJump; step++) {
// 1. Shortest step length evaluated first
// 2. Right evaluated before Down
if (j + step < n && canReach(i, j + step)) { ... }
if (i + step < n && canReach(i + step, j)) { ... }
}
This ensures that the generated path always respects the required traversal order.
Architectural Analysis: The Two-Phase Pattern
The solution uses a Two-Phase Pass approach:
Memoized Validation DFS (canReach)
Greedy Reconstruction DFS (build)
Advantages
Clear separation of concerns.
Reachability validation is isolated from path reconstruction.
Reconstruction becomes simpler because viable paths are already known.
Avoids repeatedly evaluating impossible branches during path construction.
Considerations
The expected time complexity is:
O(n² × max_element)
Phase 1 performs the majority of the computation while caching results in the dp matrix.
The space complexity is:
O(n²)
due to:
Note on Auxiliary Space
Some problem statements mention an expected auxiliary space of O(1).
However, this implementation explicitly allocates:
Achieving true O(1) auxiliary space would require:
While possible, such approaches are often avoided in production systems because they reduce readability and may introduce side effects.
Implementation
The following solution implements the two-phase approach discussed above.
class Solution {
shortestDist(mat) {
const n = mat.length;
// dp array stores true/false reachability or undefined if unvisited
const dp = Array.from({ length: n }, () =>
Array(n).fill(undefined));
const res = Array.from({ length: n }, () =>
Array(n).fill(0));
// ---- Phase 1: Check Reachability via Memoized DFS ----
function canReach(i, j) {
if (i >= n || j >= n || mat[i][j] === 0)
return false;
if (i === n - 1 && j === n - 1)
return true;
if (dp[i][j] !== undefined)
return dp[i][j];
const maxJump = mat[i][j];
// Evaluate based on step size priority
for (let step = 1; step <= maxJump; step++) {
if (canReach(i, j + step))
return dp[i][j] = true;
if (canReach(i + step, j))
return dp[i][j] = true;
}
return dp[i][j] = false;
}
// Base Edge Case Check
if (mat[0][0] === 0 || !canReach(0, 0)) {
return [[-1]];
}
// ---- Phase 2: Construct Path Greedily ----
function build(i, j) {
res[i][j] = 1;
if (i === n - 1 && j === n - 1)
return true;
const maxJump = mat[i][j];
for (let step = 1; step <= maxJump; step++) {
// Right first
if (j + step < n &&
canReach(i, j + step)) {
if (build(i, j + step))
return true;
}
// Down second
if (i + step < n &&
canReach(i + step, j)) {
if (build(i + step, j))
return true;
}
}
res[i][j] = 0;
return false;
}
build(0, 0);
return res;
}
}
Complexity Analysis
Time Complexity
O(n² × max_element)
Each cell is evaluated at most once due to memoization, while exploring up to max_element possible jumps.
Space Complexity
O(n²)
Additional space is used by:
Key Takeaway
Whether you are a fresher visualizing the rat's movement through the maze or an experienced engineer evaluating complexity and design trade-offs, the core idea remains the same: eliminate unreachable paths as early as possible and ensure that traversal logic strictly follows the problem constraints.
By combining Backtracking, Memoization, and a two-phase search strategy, the solution efficiently identifies a valid path while respecting the required jump priorities and movement rules.
Summary
Rat Maze with Multiple Jumps extends the traditional maze problem by allowing variable-length jumps based on cell values. The solution uses a two-phase approach: a memoized depth-first search to determine reachability and a second traversal to reconstruct the valid path. This design improves efficiency by avoiding repeated calculations while ensuring that the path adheres to the problem's jump-priority rules.