XPost: comp.lang.lisp   
   From: 433-929-6894@kylheku.com   
      
   On 2024-01-03, Julieta Shem  wrote:   
   > Why do they say ``overlapping subproblems'' when it seems that what is   
   > meant is a duplicate problem?  For instance, the interval [0, 10]   
   > overlaps with the interval [5, 15], but they're not the same.  AFAICT,   
   > memoization is only useful when at least two of the subproblems are   
   > exactly the same.   
      
   The famous example is Fibonacci. If you calculate fib(7) recursively,   
   fib(3), and others, will show up more than once in the recursion:   
      
                 fib(7)   
                /      \   
          fib(6)        fib(5)   
          /    \       /      \   
      fib(4)   fib(5) fib(4)   fib(3)   
              /     \         /     \   
            fib(4) fib(3)   
           /     \ /    \   
      
   Why is that called overlapping? Because the left subtree fib(6)   
   and fib(5) are not the same, but they contain some common content   
   (nodes that are exactly the same like another copy of fib(5), and   
   multiple fib(4) and so on).   
      
   It's just in contrast to divide-and-conquer, where the problem   
   space is being strictly partitioned; no part or sub-part of the   
   left tree occcurs in the right or vice versa.   
      
   [0, 10] and [5, 15] overlap, and they have [5, 10] in common.   
   If that can be solved as a sub-problem, such that we can solve   
   [0, 4], [5, 10] and [11, 15], and put them together,   
   that would be better than solving [5, 10] twice and doing   
   the same thing.   
      
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