We didn’t get time to see Fredman’s “trick” after all. What Fredman shows in this paper is the following result:

The min-sum product (MSP) of two matrices can be computed using only comparisons.

(He uses instead of our .) Using the same approach of breaking up the MSP of two matrices into “slabs” of width , we can compute the MSP of using comparisons. Setting gives us a total of comparisons.

Note this is just the number of *comparisons*, not the actual time to write down the answer. But it shows that the comparisons are not the bottleneck.

In the paper, he then optimizes things to remove the term, and uses the trick to get an algorithm whose runtime actually beats by a little bit. But I wanted to highlight the proof of the above fact, which is one paragraph long, with a very clever idea:

A natural question: can you beat this bound and get comparisons?

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