The Algorithm Design Manual
Format: PDF / Kindle (mobi) / ePub
Most professional programmers that I’ve encountered are not well prepared to tackle algorithm design problems. This is a pity, because the techniques of algorithm design form one of the core practical technologies of computer science. Designing correct, efficient, and implementable algorithms for real-world problems requires access to two distinct bodies of knowledge: • Techniques – Good algorithm designers understand several fundamental algorithm design techniques, including data structures, dynamic programming, depth first search, backtracking, and heuristics. Perhaps the single most important design technique is modeling, the art of abstracting a messy real-world application into a clean problem suitable for algorithmic attack. • Resources – Good algorithm designers stand on the shoulders of giants. Rather than laboring from scratch to produce a new algorithm for every task, they can figure out what is known about a particular problem. Rather than re-implementing popular algorithms from scratch, they seek existing implementations to serve as a starting point. They are familiar with many classic algorithmic problems, which provide sufficient source material to model most any application. This book is intended as a manual on algorithm design, providing access to combinatorial algorithm technology for both students and computer professionals.
Now give a Θ(n)-time dynamic programming algorithm for this problem. To et partial credit, you may instead give a correct O(n log n) divide-and-conquer algorithm. 8-22.  Consider the problem of examining a string x = x1x2…xn from an alphabet of k symbols, and a multiplication table over this alphabet. Decide whether or not it is possible to parenthesize x in such a way that the value of the resulting expression is a, where a belongs to the alphabet. The multiplication table is neither
technique that might even work well beyond two dimensions. Start from an arbitrary point p in an arbitrary cell, hopefully close to the query point q. Construct the line (or ray) from p to q and identify which face of the cell this hits (a so-called ray shooting query). Such queries take constant time in triangulated arrangements. Proceeding to the neighboring cell through this face gets us one step closer to the target. The expected path length will be O(n1/d) for sufficiently regular
useful for computation. Since they are typically small and clean, they can prove the right foundation for simple applications. The most useful codes of this genre are described below. Most are available from the algorithm repository, http://www.cs.sunysb.edu/~algorith. Programming Challenges If you like the C code that appeared in the first half of the text, you should check out the programs I wrote for my book Programming Challenges [SR03]. Perhaps most useful are additional examples of
Θ(n2). Thus, the worst case for quicksort is worse than heapsort or mergesort. To justify its name, quicksort had better be good in the average case. Understanding why requires some intuition about random sampling. 4.6.1 Intuition: The Expected Case for Quicksort The expected performance of quicksort depends upon the height of the partition tree constructed by random pivot elements at each step. Mergesort ran in O(n log n) time because we split the keys into two equal halves, sorted them
“Skiena”) occurs in a given sorted array. Because sorting groups all the copies of k into a contiguous block, the problem reduces to finding the right block and then measures its size. The binary search routine presented above enables us to find the index of an element of the correct block (x) in O(lg n) time. The natural way to identify the boundaries of the block is to sequentially test elements to the left of x until we find the first one that differs from the search key, and then repeat this