A Practical Introduction to Data Structures and Algorithm Analysis

512 Chap. 15 Lower BoundsABCGDEFFigure 15.1 Illustration of using a poset to model our current knowledge of therelationships among a collection of objects. A directed acyclic graph (DAG) isused to draw the poset (assume all edges are directed downward). In this example,our knowledge is such that we don’t know how A or B relate to any of the otherobjects. However, we know that both C and G are greater than E and F. Further,we know that C is greater than D, and that E is greater than F.current knowledge about the objects, such that zero or more of the order relationsbetween pairs of elements are known. We can represent this knowledge by drawingdirected acyclic graphs (DAGs) showing the known relationships, as illustrated byFigure 15.1.Initially, we know nothing about the relative order of the elements in L, ortheir relationship to K. So initially, we can view the n elements in L as being inn separate partial orders. Any comparison between two elements in L can affectthe structure of the partial orders. This is somewhat similar to the UNION/FINDalgorithm implemented using parent pointer trees, described in Section 6.2.Now, every comparison between elements in L can at best combine two of thepartial orders together. Any comparison between K and an element, say A, in L canat best eliminate the partial order that contains A. Thus, if we spend m comparisonscomparing elements in L we have at least n − m partial orders. Every such partialorder needs at least one comparison against K to make sure that K is not somewherein that partial order. Thus, any algorithm must make at least n comparisons in theworst case.15.2.2 Searching in Sorted ListsWe will now assume that list L is sorted. In this case, is linear search still optimal?Clearly no, but why not? Because we have additional information to work with thatwe do not have when the list is unsorted. We know that the standard binary searchalgorithm has a worst case cost of O(log n). Can we do better than this? We canprove that this is the best possible in the worst case with a proof similar to that usedto show the lower bound on sorting.Again we use the decision tree to model our algorithm. Unlike when searchingan unsorted list, comparisons between elements of L tell us nothing new about their