11-27-2019, 05:21 PM
Perhaps you start by clearing the whole grid to show no links at first. I always tell you to keep track of point labels separately so the grid stays clean. You add an edge by flipping one spot to show the tie exists. Or maybe flip the matching spot too if ties go both ways. Then you test by looking at that spot later to confirm.
Also the check happens in one step since you jump straight to the row and column pair. I notice this beats scanning other structures when graphs sit dense with ties. You remove an edge just by resetting the spot back. But watch out for memory use because the grid holds room for every possible pair even if unused. Perhaps you reuse the same grid across multiple runs by resetting only changed spots.
Now think about walking the graph by scanning each row for marks that indicate next points. I show you this scans every column per row so time grows with points squared. You gain speed on single queries though because no loops needed to find if two points touch. Or you combine it with other tools when graphs mix dense and sparse parts. Then you see tradeoffs appear in real projects where quick lookups matter more than saving bytes.
You handle weights by putting numbers instead of marks in the spots. I remember cases where negative weights need careful handling to avoid wrong paths. But the grid still works the same for storage. Perhaps you update a weight by overwriting the spot directly. Also this keeps updates simple without shifting other data around.
You compare this to edge lists in your head by noting how matrix wins on dense cases. I point out space waste hits hard when ties stay few. Then you choose based on expected density before coding anything. Or you test both on sample data to see which runs smoother. But the matrix stays easier for beginners to picture as a table of yes or no.
You extend it for directed ties by skipping the symmetric flip. I find this lets one way links appear without forcing return paths. Perhaps you mark self ties on the diagonal if loops allowed. Then you query those same spots for cycle checks later. Also you layer multiple matrices if you track different relation types between same points.
You debug by printing the grid rows to spot wrong marks. I always reset the structure after tests to prevent carryover errors. But errors show quick since one glance reveals mismatches. Or you count filled spots to verify edge totals match what you added. Then you scale up by allocating bigger grids when point count increases.
You integrate this into algorithms by feeding the grid into search routines that read rows for neighbors. I notice breadth first runs faster here on dense setups. Perhaps you cache frequent query results outside the grid to cut repeats. Also you handle dynamic point adds by resizing the grid and copying old marks. But resizing costs time so plan sizes ahead when possible.
You measure performance by timing single checks versus full scans. I tell you the constant time access feels worth the upfront space. Then you balance this against other options in mixed workloads. Or you profile memory to confirm grid fits in available room. But practice shows it shines for social network style ties where most points connect.
You refine by using bit packing if values stay binary. I find this shrinks space without losing speed much. Perhaps you store in files by writing rows sequentially for later loads. Then you reload by reading back into the grid shape. Also you version the structure if multiple graph states need keeping.
You explore variants like triangular storage for undirected cases to halve space. I show how that saves bytes but adds index math. But keep it simple first before optimizing. Or you test on university level problems like shortest paths where matrix helps. Then you see real gains in lookup heavy code.
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