2010 Athena lecture

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Lecture : Integer Programming , Spring 2010

2 Euclidean Algorithm and Hermite Normal Form 9 2.1 Sizes of Rational Numbers and Polynomial Complexity . . . . . . . . . . . . 9 2.2 Computing the GCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Computing the Hermite Normal Form . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Lattices and the Hermite Normal Form . . . . . . . . . . . . . . . . . . . . . 15 2.5 Dual ...

Lecture 9 — March 4 , 2010

Last lecture we defined the Least Common Ancestor (LCA) and Range Min Query (RMQ) problems. Recall that an LCA query asks for the deepest node in a given rooted tree which is an ancestor of each of two query nodes, while an RMQ query asks for the minimal value stored in some contiguous subarray of a given array. We showed last lecture that an LCA query is nothing more than an RMQ query on an Eu...

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Lecture 7 – February 26 , 2010

For integers i and j, define T [i : j] as the substring of T starting from the ith character and ending with the jth character inclusive. We will often omit j and write T [i :] to denote the suffix of T starting at the ith character. Finally, we let the symbol ◦ denote concatenation. As a simple illustration of our notation, (abcde[0 : 2]) ◦ (cde[1 :]) = abcde. Now we can formally state the str...