【正文】 Data Structures and Algorithm 習題答案Preface ii 1 Data Structures and Algorithms 1 2 Mathematical Preliminaries 5 3 Algorithm Analysis 17 4 Lists, Stacks, and Queues 23 5 Binary Trees 32 6 General Trees 40 7 Internal Sorting 46 8 File Processing and External Sorting 54 9Searching 58 10 Indexing 64 11 Graphs 69 12 Lists and Arrays Revisited 76 13 Advanced Tree Structures 82 i ii Contents 14 Analysis Techniques 88 15 Limits to Computation 94 Preface Contained herein are the solutions to all exercises from the textbook A Practical Introduction to Data Structures and Algorithm Analysis, 2nd edition. For most of the problems requiring an algorithm I have given actual code. In a few cases I have presented pseudocode. Please be aware that the code presented in this manual has not actually been piled and tested. While I believe the algorithms to be essentially correct, there may be errors in syntax as well as semantics. Most importantly, these solutions provide a guide to the instructor as to the intended answer, rather than usable programs. 1 Data Structures and Algorithms Instructor’s note: Unlike the other chapters, many of the questions in this chapter are not really suitable for graded work. The questions are mainly intended to get students thinking about data structures issues. This question does not have a specific right answer, provided the student keeps to the spirit of the question. Students may have trouble with the concept of “operations.” This exercise asks the student to expand on their concept of an integer representation. A good answer is described by Project , where a singlylinked list is suggested. The most straightforward implementation stores each digit in its own list node, with digits stored in reverse order. Addition and multiplication are implemented by what amounts to gradeschool arithmetic. For addition, simply march down in parallel through the two lists representing the operands, at each digit appending to a new list the appropriate partial sum and bringing forward a carry bit as necessary. For multiplication, bine the addition function with a new function that multiplies a single digit by an integer. Exponentiation can be done either by repeated multiplication (not really practical) or by the traditional Θ(log n)time algorithm based on the binary representation of the exponent. Discovering this faster algorithm will be beyond the reach of most students, so should not be required. A sample ADT for character strings might look as follows (with the normal interpretation of the function names assumed). Chap. 1 Data Structures and Algorithms // Concatenate two strings String strcat(String s1, String s2)。 // Return the length of a string int length(String s1)。 // Extract a substring, starting at ‘start’, // and of length ‘length’ String extract(String s1, int start, int length)。 // Get the first character char first(String s1)。 // Compare two strings: the normal C++ strcmp function. Some // convention should be indicated for how to interpret the // return value. In C++, this is 1 for s1s2。 0 for s1=s2。 // and 1 for s1s2. int strcmp(String s1, String s2) // Copy a string int strcpy(String source, String destination) The answer to this question is provided by the ADT for lists given in Chapter 4. One’s pliment stores the binary representation of positive numbers, and stores the binary representation of a negative number with the bits inverted. Two’s pliment is the same, except that a negative number has its bits inverted and then one is added (for reasons of efficiency in hardware implementation). This representation is the physical implementation of an ADT defined by the normal arithmetic operations, declarations, and other support given by the programming language for integers. An ADT for twodimensional arrays might look as follows. Matrix add(Matrix M1, Matrix M2)。 Matrix multiply(Matrix M1, Matrix M2)。 Matrix transpose(Matrix M1)。 void setvalue(Matrix M1, int row, int col, int val)。 int getvalue(Matrix M1, int row, int col)。 List getrow(Matrix M1, int row)。 One implementation for the sparse matrix is described in Section Another implementation is a hash table whose search key is a concatenation of the matrix coordinates. Every problem certainly does not have an algorithm. As discussed in Chapter 15, there are a number of reasons why this might be the case. Some problems don’t have a sufficiently clear definition. Some problems, such as the halting problem, are nonputable. For some problems, such as one typically studied by artificial intelligence researchers, we simply don’t know a solution. We must assume that by “algorithm” we mean something posed of steps are of a nature that they can be performed by a puter. If so, than any algorithm can be expressed in C++. In particular, if an algorithm can be expressed in any other puter programming language, then it can be expressed in C++, since all (sufficiently general) puter programming languages pute the same set of functions. The primitive operations are (1) adding new words to the dictionary and (2) searching the dictionary for a given word. Typically, dictionary access involves some sort of preprocessing of the word to arrive at the “root” of the word. A twenty page document (single spaced) is likely to contain about 20,000 words. A user may be willing to wait a few seconds between individual “hits” of misspelled words, or perhaps up to a minute for the whole document to be processed. This means that a check for an individual word can take about 1020 ms. Users will typically insert individual words into the dictionary interactively, so this process can take a couple of seconds. Thus, search must be mu