A[i,j]+A[k,l]≤A[i,l]+A[k,j]cap A open bracket i comma j close bracket plus cap A open bracket k comma l close bracket is less than or equal to cap A open bracket i comma l close bracket plus cap A open bracket k comma j close bracket
Look into Monge Arrays to see how these "Gnome" properties allow for faster shortest-path algorithms in geometric graphs. stefani_problem_stefani_problem
fkfk+1+fk+12=fk+1(fk+fk+1)f sub k f sub k plus 1 end-sub plus f sub k plus 1 end-sub squared equals f sub k plus 1 end-sub of open paren f sub k plus f sub k plus 1 end-sub close paren by definition: fk+1fk+2f sub k plus 1 end-sub f sub k plus 2 end-sub The identity is proven for all Resources for Further Study stefani_problem_stefani_problem
∑i=1k+1fi2=(∑i=1kfi2)+fk+12sum from i equals 1 to k plus 1 of f sub i squared equals open paren sum from i equals 1 to k of f sub i squared close paren plus f sub k plus 1 end-sub squared Substitute the inductive hypothesis: stefani_problem_stefani_problem
Algorithm Design & Discrete Mathematics Context: CSCI1570 (Brown University) - Lorenzo De Stefani 1. Problem Definition
