By Khuller S.

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The idea is to assume the opposite of what we want to prove, then show that this leads to a contradictory statement; thus what we want to prove cannot be false, and so must be true. (This assumes that every statement is either true or false, but, as we see in chapter seven, when looked at closely there are usually shades of gray. ) We can even tell when nz first becomes bigger than rn + s. It's bigger for all n larger than the positive root of the quadratic equation n 2 - rn - s. That is, for all n larger than (r + -/-r2 + 4s)/2.

Further, we want to know if we're wasting time trying to improve the algorithm; perhaps it cannot be improved? In sum, we want to predict how bad an algorithm can get, and we want to determine how hard the problem is. Upper Bounds Analysis first involves figuring out bounds on the number of operations an algorithm performs given an input of size n. Usually we will first find its worst cost, that is, the maximum number of times it performs the chosen operation. Since the algorithm is proof that we can solve the problem using at most that number of operations, even in the worst case, this gives an upper bound on the problem's worst cost as a function of n.

If one of the books is as big as the Encyclopxdia Britannica then we should take weight into account as well. Okay, pretend we have a reasonable measure of input size. Now we have to estimate the difficulty of each instance of the problem in terms of 8 1 OVERVIEW its size. We want a function of the problem's size that reflects the effort necessary to solve the problem. So far we have assumed that the first thing to look at is the program's speed. But, our machine may have little memory, so we could look for programs that use the least memory.

### Advanced algorithms by Khuller S.

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