A Practical Introduction to Data Structures and Algorithm Analysis

84 Chap. 3 Algorithm AnalysisThe analysis techniques used to measure space requirements are similar to thoseused to measure time requirements. However, while time requirements are normallymeasured for an algorithm that manipulates a particular data structure, spacerequirements are normally determined for the data structure itself. The concepts ofasymptotic analysis for growth rates on input size apply completely to measuringspace requirements.Example 3.16 What are the space requirements for an array of n integers?If each integer requires c bytes, then the array requires cn bytes,which is Θ(n).Example 3.17 Imagine that we want to keep track of friendships betweenn people. We can do this with an array of size n × n. Each row of the arrayrepresents the friends of an individual, with the columns indicating who hasthat individual as a friend. For example, if person j is a friend of personi, then we place a mark in column j of row i in the array. Likewise, weshould also place a mark in column i of row j if we assume that friendshipworks both ways. For n people, the total size of the array is Θ(n 2 ).A data structure’s primary purpose is to store data in a way that allows efficientaccess to those data. To provide efficient access, it may be necessary to store additionalinformation about where the data are within the data structure. For example,each node of a linked list must store a pointer to the next value on the list. All suchinformation stored in addition to the actual data values is referred to as overhead.Ideally, overhead should be kept to a minimum while allowing maximum access.The need to maintain a balance between these opposing goals is what makes thestudy of data structures so interesting.One important aspect of algorithm design is referred to as the space/time tradeoffprinciple. The space/time tradeoff principle says that one can often achieve areduction in time if one is willing to sacrifice space or vice versa. Many programscan be modified to reduce storage requirements by “packing” or encoding information.“Unpacking” or decoding the information requires additional time. Thus, theresulting program uses less space but runs slower. Conversely, many programs canbe modified to pre-store results or reorganize information to allow faster runningtime at the expense of greater storage requirements. Typically, such changes in timeand space are both by a constant factor.