Maze Router (Lee's Algorithm)

In this post, let's talk about Maze Routing Algorithm which is a manifestation of Breadth First Search (BFS) Algorithm to find the shortest path between two nodes in a grid.

A crude version of this algorithm is also known as Lee's Algorithm. I will discuss Lee's Algorithm, and few improvements to that to improve the run time and memory.

Here's the problem statement. You need to connect the node S (source) and the node T (target or the destination) with the shortest possible path. These nodes are shown in red. The grids in blue represent a routing blockage, meaning you cannot route over these grids. You'll need to find a way around these to reach the destination node.

Problem Statement: Maze Router (Lee's Algorithm)

VLSI routes are laid orthogonal in X and Y direction. Diagonal (also known as X-routing) is usually forbidden. Let's say you need to start out from node S, you have 4 possible directions in which you can proceed:

4 possible directions from a given node

The number 1 represents the distance traveled from the source node. Once you have traveled a distance 1, here is how your grid looks like:

Grid after traveling a distance of 1 unit

Similarly, after traveling a distance of 2 units, the grid is shown below. 

Grid after 2 iterations of Lee's algorithm

Now you've hit a wall, and it will be apparent that you cannot hop over the wall (or the blockage) from the next figure. After multiple iterations of the Lee's algorithm, the grid would look something this:

Grid after 8 iterations of Lee's algorithm

Continue doing this till you hit the target or the destination node.

Grid after you hit the target node

Now you need to backtrace from the target to the source following successive lower integers to find the shortest path. Note that you may have many possible shortest choices, but all of them are guaranteed to be the shortest. Usually, there's a cost associated with turns (vias in a physical context), so practically, you may assign a weight or parameter to minimize the number of turns to choose among more than 1 possible shortest paths.

Back-tracing to find the shortest path

This embodiment of the Lee's algorithm has high complexity, especially if the grid size is higher than the one shown in the example above. Notice how much wasteful computation we had to perform over to the right. This can be minimized if we initiate the same computation from both the target and the source, and back-trace to the target and the source respectively once the two wavefronts (the one in green and one in yellow) intersect. This results in far less time complexity and much less wasteful computations.

Modification to the Lee's algorithm to start computation from both target and the source

One another possible improvement to the above algorithm is the memory required to save the distance numbers for each node. Imagine a 10x10 grid. The worst distance could be 100, and you'd require 7 bits to store numbers up to 100. That means a worst space complexity of 700 bits for 10x10 grid. For 20x20 grid, worst distance could be 400, requiring 9 bits per box and a total space complexity of 3600 bits. In order to reduce the complexity, it's possible to go only up to 3 while counting, and then counting down to 1, and so on.. Back-tracing is slightly more complicated, but it saves you a ton of space!