Modeling the Problem

To find out what is known about your current problem you need to have the vocabulary to formulate your problem. The following list of computing properties of abstract structures begins to enhance your algorithmic vocabularly. To become fluent in this vocabulary, browse throught the catalog (in Chapter 8) and study the input and output pictures for each problem. This should enable you to know where to look later when a problem arises in your application.

1. Permutations - arrangements or orderings of items.
   Example:
   • Traveling Sales Problem - least-cost order to visit n cities.
   • Bandwith Problems - order the vertices of a graph on a line so as to minimize the length of the longest edge.
   • Graph Isomorphism - order the vertices of one graph so that it is identical to another.

2. Subsets - selections from a set of items
   Example:
   • Vertex Cover of a Graph - minimum number of vertices needed to touch all edges of a graph.
   • Knapsack Problem - determine maximum number of items that can be taken on a trip given some constraints.
   • Set Packing - smallest number of subsets that together cover each item exactly once, useful for finding independant sets and scheduling airline flight crews to airplanes.
   • Set Partitions - divide the elements 1..n into nonempty subsets, useful for vertex coloring and connected components.
   • Boolean Logic Minimization - given a boolean function mapping the 2^k input vectors to the values 0 or 1, find a simplest circuit that exactly implements this function.

3. Trees - hierarchical relationships between items
   Example:
   • Searching - binary trees, red-black trees, B-trees, ...
   • Minimum Spanning tree problems - cheapest subset of edges that keeps the graph in one connected component. Useful for telephone site wiring, and network design.

4. Graphs - relationships between arbitrary pairs of objects
   Example:
   • Connected Components - finding pieces of graphs (or classes of items).
   • Shortest Path - finding the shortest paths in a graph. Useful for routing problems, image segmentation, speech recognition, finding the center of a graph (facility location).
   • Matching - find the largest-size of edges so that the edges don't intersect. Useful in task assignment. Stable matchings have applications in medical resident assigments to hospitals.
   • Eulerian Cycle - find the shortest tour of a graph visiting each edge at least once. Useful for routing garbage trucks, snow plows, postal workers.
   • Edge and Vertex Connectivity - smallest subset of vertices (or edges) whose deletion will disconnect a graph. Useful in determining network reliability (i.e.: telephone networks), and network flow.
   • Drawing graphs "Nicely" - useful criteria for nicely are minimum number of crossing edges (circuit board design), smallest area (electric wiring), edge length (network connections), aspect ratio (display).

5. Points - locations in some geometric space
   Example:
   • Convex Hull - determining the outer covering of a set of points. Useful for object recognition and image processing.
   • Voronoi Diagrams - separate space into regions around each point so that all the points in the region around p_i are closer to p_i than they are to any other point in S. Useful for facility locationing, finding largest empty circle, path planning and quality triangulations.
   • Nearest Neighbor Search - which point is closest to a give point. Useful in emergency vehicle dispatching, object classification, image compression.
   • Range Search - which points from a set lie within a given space. Useful in database and geographic information system applications.
   • Intersection Detection - determining if a pair of lines intersect. Useful in Virtual Reality and in VLSI layout designs.

6. Polygons - regions in some geometric space
   Example:
   • Polygon Partitioning - determining if a shape can be partitioned into a small number of simple (convex) pieces. Useful in image recognition and triangulation.
   • Simplifying Polygons - find a shape that is close to a given polygon. Useful in eliminating noise and fuzziness of images andd in data compression.
   • Motion Planning - determining the shortest path in the room without running into obstacles. Useful for robotics, rearragements, molecular docking, computer animations.

7. Strings - sequences of charaters or patterns
   Example:
   • String Matching - find all instances of a given pattern in the text. Useful for text processing, spell checkers and database queries.
   • Cryptography - encode or decode text. Useful for security on computer networks and spies.
   • Longest Common Substring - find the longest substring so that each string in a list of strings contains the substring. Useful for DNA sequencing, protein evaluation, edit distance.
   • Shortest Common Superstring - find the shortest string that contains each element of a list as a substring. Useful for sparse matrix compression, DNA sequencing, can be reduced to a traveling sales problem.