An iconic algorithmic puzzle is the Rat in a Maze problem. The Rat Maze with Multiple Jumps variant, however, introduces an additional degree of difficulty. The rat can jump across several cells rather than take a single step at a time, which makes pathfinding and decision-making more difficult.
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:
- Try shorter jumps first.
- Move Right before Down for jumps of the same length.
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:
The memoization table (dp)
The recursion call stack
Note on Auxiliary Space
Some problem statements mention an expected auxiliary space of O(1).
However, this implementation explicitly allocates:
- An O(n²) memoization matrix
- Recursive stack frames
Achieving true O(1) auxiliary space would require:
- Pure in-place backtracking
- Destructive updates to the input matrix
- Bit-level state encoding
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:
- The memoization matrix (dp)
- The result matrix (res)
- The recursive call stack
Key Takeaway
The fundamental notion is the same whether you are an expert engineer assessing complexity and design trade-offs or a novice picturing the rat's path through the maze: remove unreachable paths as soon as possible and make sure traversal logic strictly adheres to the problem constraints.
The approach effectively finds a valid path while adhering to the necessary jump priority and movement restrictions by combining backtracking, memorization, and a two-phase search strategy.
Summary
By enabling variable-length hops based on cell values, Rat Maze with Multiple hops expands upon the conventional maze problem. The solution employs a two-phase method: a second traversal to reconstruct the valid path and a memoized depth-first search to ascertain reachability. By avoiding repeated computations and guaranteeing that the path complies with the jump-priority requirements of the issue, this approach increases efficiency.