A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 15.4 Adversarial Lower Bounds Proofs 517Figure 15.2 An example of building a binomial tree. Pairs of elements arecombined by choosing one of the parents to be the root of the entire tree. Giventwo trees of size four, one of the roots is chosen to be the root for the combinedtree of eight nodes.to 2 k . The second tree is in positions 2 k + 1 to 2 k+1 . The root of each subtree is inthe final array position for that subtree.To join two trees, we simply compare the roots of the subtrees. If necessary,swap the subtrees so that tree with the the larger root element becomes the secondsubtree. This trades space (only need space for the data values, no pointers) fortime (in the worst case, all of the data swapping might cost O(n log n), though thisdoes not affect the number of comparisons required).Because the binomial tree’s root has log n children, and building the tree requiresn − 1 comparisons, the total number of comparisons required by this algorithmis n + ⌈log n⌉ − 2. This is clearly better than our previous algorithm. But isit optimal?We now go back to trying to improve the lower bounds proof. To do this,we introduce the concept of an adversary. The adversary’s job is to make analgorithm’s cost as high as possible. Imagine that the adversary keeps a list of allpossible inputs. We view the algorithm as asking the adversary for informationabout the algorithm’s input. The adversary may never lie, but it is permitted to“rearrange” the input as it sees fit in order to drive the total cost for the algorithmas high as possible. In particular, when the algorithm asks a question, the adversaryanswers in a way that is consistent with at least one remaining input. The adversarythen crosses out all remaining inputs inconsistent with that answer. Keep in mindthat there is not really an entity within the computer program that is the adversary,and we don’t actually modify the program. The adversary operates merely as ananalysis device, to help us reason about the program.As an example of an adversary, consider the standard game of Hangman. PlayerA picks a word and tells player B how many letters the word has. Player B guessesvarious letters. If B guesses a letter in the word, then A will indicate which position(s)in the word have the letter. Player B is permitted to make only so manyguesses of letters not in the word before losing.