06-12-2024, 04:47 PM
I keep thinking about how this cubic growth hits you hard once n climbs past a few hundred. You try running it on bigger networks and suddenly the seconds turn into minutes. Perhaps the dynamic programming table fills up in a steady way that forces every single update to happen. Also the space side stays quadratic because you store distances for every pair yet the time eats more of your budget. Now you compare it to running separate single source searches and you see why the all pairs version pays that full cubic price.
The updates themselves chain together in a way that prevents easy parallel tricks without extra tricks. You watch the outer loop fix one intermediate at a time and that forces the inner two to redo their scans. I find it interesting how the same structure gives you the shortest paths for free once the loops finish. Or maybe the real cost hides in the memory accesses when the matrix gets too big for the cache lines. Then the processor stalls and your measured time grows even faster than the simple count predicts.
You can break the loops into blocks to cut some cache misses but the total operations still land near n cubed. I tried sketching the recurrence and it boils down to the same triple product because each subproblem depends on the previous layer. But the best case never drops below that because the algorithm has no early exit built in. Perhaps on sparse graphs you feel tempted to switch methods yet the code stays cubic anyway. Also the negative edge handling comes along without extra cost which keeps the complexity flat.
The proof of correctness walks through the layers of intermediates and each layer costs another n squared scan. You end up accepting the cubic bound as tight because the lower bound arguments show you need to look at that many potential paths. I notice the matrix multiplication view gives the same bound and sometimes lets you plug in faster matrix tricks yet those stay theoretical for now. Now the practical runs on modern machines still follow the cubic curve closely once n exceeds a thousand.
And the junior folks often ask why it feels slower than expected on their test graphs. You point out the hidden constants from the comparison and assignment steps inside the inner loop. I remember measuring it on a small cluster and watching the time triple when n doubled. Perhaps the real lesson sits in choosing when to pay that cubic price versus building a different structure. Or the graph density changes nothing in the count but changes how the numbers behave in floating point.
You keep the same code for dense and sparse cases which simplifies testing yet wastes effort on zero weight edges. I see the initialization step taking linear time but it vanishes next to the main loops. But the overall analysis stays clean because the dominant term comes only from the triple loop. Now you can teach the bound quickly by just pointing at the three for statements stacked together.
The graduate view adds the bit about asymptotic notation hiding the lower order terms yet you still feel them when n sits around fifty. I like how the same algorithm gives both the distances and the path reconstruction with one extra matrix. You store the predecessor choices without changing the time order. Perhaps the space time tradeoff appears when you decide to keep all intermediate matrices or just the final one.
And the conversation circles back to real hardware limits where memory bandwidth starts throttling the updates. You measure cache misses climbing with n squared and that adds another practical curve on top of the theoretical one. I find the simplicity of the loops makes it easy to code yet hard to beat for medium sized all pairs problems. Or the negative cycle detection falls out of the final check without any added loops.
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