Grid DP from above and left
Count Routes Across a Roof Garden
01. The question
A rectangular roof garden has no blocked cells. Count paths from its top-left cell to its bottom-right cell, moving only right or down. Original garden scenario; the standard dimensions-and-count contract is unchanged.
| r / c | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 2 | 3 | 4 |
| 2 | 1 | 3 | 6 | 10 |
This is the completed route-count DP table for a 3-by-4 garden, not an obstacle map.
Before revealing the idea, predict the result and describe what information you would need to track.
02. Worked example
Input[3, 4]
Output10
- Every cell along the top edge or left edge has one incoming route.
- The inner second row becomes [1,2,3,4].
- The final row becomes [1,3,6,10], giving ten routes.
Try your code
Define solve(data, parameter="") and return your result. Your code runs locally, not on a server.
Edit your test cases
Use 1-10 objects with data and an optional string parameter. Expected outputs are computed from the lesson. An optional expected field must be correct.
Input contract & reference code
Example data: [3,4]. Parameter: "". Numeric parameters arrive as strings; convert them with int(parameter) when needed.
The example and generated cases use the question’s exact variant. This checks behavior on a small test batch—not performance on every possible input.
Python downloads only when you run code, using compressed runtime assets. Code is not saved or uploaded. Enable JavaScript to run it.
Printed output
Five-second execution limit per batch. Standard-library Python only; additional packages and network access are unavailable. Stop ends execution without blocking the lesson.
03. The idea
Try sketching your own approach first. What would you track, and why would it work?
Reveal the idea
- Put one route at the starting cell.
- Visit cells in row-major order.
- Add routes from above and from the left, using zero for missing neighbors.
04. Watch it happen
Use Step to move slowly, or Play to follow the whole trace. Change a preset or load your own input.
Input & settings
Red = current item / blue = active window or cells / green = completed work. A value docks at its actual destination; the original input stays unchanged. Positions start at 0.
Working state / what changes
Ready to begin
Enable JavaScript to step through the animation. The worked example and solution below remain readable without it.
Python / why it changes
The line behind the step
def solve(data, parameter=''):
rows, columns = data
ways = [[0] * columns for _ in range(rows)]
ways[0][0] = 1
for row in range(rows):
for column in range(columns):
if row == column == 0:
continue
above = ways[row - 1][column] if row else 0
left = ways[row][column - 1] if column else 0
ways[row][column] = above + left
return ways[-1][-1]How to use this walkthrough
Hover or focus a control for its label. Play forms the input and follows values into the working state. Step shows one change; Back and the slider restore exact snapshots. Reduced motion uses instant updates. Saved snapshots and sorted heap displays are teaching overhead, not algorithm complexity.
05. Python solution
def solve(data, parameter=''):
rows, columns = data
ways = [[0] * columns for _ in range(rows)]
ways[0][0] = 1
for row in range(rows):
for column in range(columns):
if row == column == 0:
continue
above = ways[row - 1][column] if row else 0
left = ways[row][column - 1] if column else 0
ways[row][column] = above + left
return ways[-1][-1]06. Cost & edge cases
- Time
- O(rows × columns)
- Space
- O(rows × columns)
- A one-cell garden has one empty-move route.
- A single row or column has one route.
- Require exactly [rows,columns], with both dimensions integers from 1 to 8.