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Dynamic Programming: Why Knuth's improvement to Optimal Binary Search Tree O(n^2)?

You can find the exact running time analysis with a google search or just start to write your own analysis w.r.t for loops. But just note that in all of them sum in total is calculated by telescopic sum, I mean may be one of them is big but in each iteration for first loop takes O(n), and totally takes O(n2).

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