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What does spiral order mean?
LeetCode problem 54 asks you to “return all elements of the matrix in spiral order.” The traversal begins at the upper-left and follows the outside perimeter clockwise before moving inward. For a nonempty rectangular matrix with m rows and n columns, the output contains exactly m × n elements. The problem’s stated constraints are 1 ≤ m, n ≤ 10 and -100 ≤ matrix[i][j] ≤ 100 (LeetCode, Spiral Matrix).
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Example: a 3 × 3 matrix
For [[1,2,3],[4,5,6],[7,8,9]], read the perimeter first, then the center:
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For [[1,2,3,4],[5,6,7,8],[9,10,11,12]], the spiral order is:
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[1, 2, 3, 4, 8, 12, 11, 10, 9, 5, 6, 7]How to trace the traversal one step at a time
A scrub-through visualizer can make the boundary method easy to follow. The visualizer described here is a trace specification, not a claim about a particular tested tool: each step should identify the current cell and direction, show the unvisited rectangle’s boundaries, and add the cell’s value to the output.
- Initialize: set
top = 0,bottom = m - 1,left = 0, andright = n - 1. Start with an empty output. - Move right: visit cells in row
top, from columnleftthroughright. Then movetopdown by one. - Check for remaining rows: if
top > bottom, stop. Otherwise, move down columnright, from rowtopthroughbottom, then moverightleft by one. - Check for remaining columns: if
left > right, stop. Otherwise, move left across rowbottom, from columnrightdown throughleft, then movebottomup by one. - Check for remaining rows again: if
top > bottom, stop. Otherwise, move up columnleft, from rowbottomup throughtop, then moveleftright by one. - Repeat: continue while
top ≤ bottomandleft ≤ right.
For a useful scrubber, make the active edge visually distinct, mark each visited cell, and update the four boundary values after the edge finishes. Display the output list alongside the matrix so a reader can match every move to its appended value. At the end, the list should contain m × n entries.
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Boundary-shrinking implementation
This implementation follows the trace directly. Its crossed-bound checks matter: after the top row or right column has been consumed, the remaining region might be only one row or one column. Traversing a bottom or left edge without checking can visit cells a second time.
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function spiralOrder(matrix) {
const m = matrix.length;
const n = matrix[0].length;
const result = [];
let top = 0;
let bottom = m - 1;
let left = 0;
let right = n - 1;
while (top <= bottom && left <= right) {
for (let col = left; col <= right; col++) {
result.push(matrix[top][col]);
}
top++;
if (top > bottom) break;
for (let row = top; row <= bottom; row++) {
result.push(matrix[row][right]);
}
right--;
if (left > right) break;
for (let col = right; col >= left; col--) {
result.push(matrix[bottom][col]);
}
bottom--;
if (top > bottom) break;
for (let row = bottom; row >= top; row--) {
result.push(matrix[row][left]);
}
left++;
}
return result;
}The snippet assumes a nonempty rectangular matrix, matching LeetCode 54’s stated constraints. The four bounds describe the as-yet-unvisited rectangle; they are not indices that should be reset when a lap ends. A boundary-method hint and caution about one-row and one-column cases appear in the LeetCode community discussion.
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Why the checks prevent repeats
- If the top row was the last row, incrementing
topcrossesbottom; there is no right edge left to traverse. - If the right column was the last column, decrementing
rightcrossesleft; there is no bottom edge left to traverse. - After the bottom edge is consumed, its boundary moves inward. The final check ensures there is still a row for the upward pass.
Alternative: simulate direction with a visited grid
Instead of shrinking a rectangle, track the current (row, column) and one of four directions. Visit exactly m × n cells. After each visit, inspect the next cell: if it is outside the matrix or already visited, rotate clockwise and move in the new direction. A separate visited grid makes the stopping rule explicit and leaves the input values unchanged. The cited LeetCode Wiki (Doocs) implementation gives this approach O(mn) time and O(mn) auxiliary space (LeetCode Wiki (Doocs), 54. Spiral Matrix).
Which traversal model should you use?
| Question | Shrinking boundaries | Direction simulation |
|---|---|---|
| How is the path represented? | Four edges of the unvisited rectangle | Current cell and one of four directions |
| How are repeats avoided? | Move bounds inward and check for crossed bounds before later edges | Turn when the next cell is out of bounds or already visited |
| Extra memory established by the cited material | No like-for-like complexity figure is established in the cited sources | O(mn) auxiliary space for the visited grid |
| Does traversal require changing input values? | The boundary method reads values and appends them; it does not need to change the matrix | A separate visited grid avoids changing matrix values |
| Visual explanation | Shows the unvisited rectangle contracting after each edge | Shows the next move, turns, and visited cells |
Choose shrinking boundaries when you want to explain the perimeter-and-inward pattern. Choose a visited-grid simulation when making each turn and the exact number of visits explicit is more useful. These are two models of the same clockwise order.
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How to handle a single row or column
These cases expose the boundary method’s main duplicate-visit risk. A single-row matrix is exhausted by the first left-to-right pass; a single-column matrix is exhausted by the top-to-bottom pass. The crossed-bound checks stop later passes from adding those cells again.
[[1,2,3]]produces[1,2,3].[[1],[2],[3]]produces[1,2,3].[[1]]produces[1].
How to check an implementation
Before accepting a traversal, verify it against the matrix’s shape as well as its values:
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- The output length is
m × n. - Every cell is appended once, including when the last unvisited region is a single row, column, or cell.
- The first pass starts at the upper-left and moves right; each completed turn is clockwise.
- The input matrix remains unchanged if your implementation uses a separate output and visited grid or only reads values.
A tempting shortcut is to mark visited cells by changing their values. One cited variant adds 300 to mark a cell, relying on LeetCode 54’s value bounds of -100 through 100. That assumption is specific to those constraints, alters the matrix during traversal, and is not a general-purpose marking strategy.
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