Class 51: Introduction Graphs

Held Wednesday, May 3, 2000

Overview

Today we move on to a new data structure, the graph. Graphs are particularly useful in modeling a number of ``real'' problems.

Notes

• In case you've forgotten, here's what's due for the project:
  • Working, integrated, code
  • A 3-5 page essay that discusses the design of your part (one per group)
  • A 1 page reflective essay about what it was like to work on a big, distributed project
  • Optional: distribute 100 units of extra credit to fellow students.

Contents

• Modeling
• Introduction to Graphs
  • Some Terminology
• Traveling Salescritter Problem

Summary

• Common graph problems
• The traveling salescritter problem
• Reachabliity: Can you get there from here?

Modeling

• As you have (hopefully) noted, computer science is not (always) done for its own sake. Often, we write computer programs to help us with ``real world'' problems.
• Sometimes, we answer real-world questions by modeling them with one or more data structures.
  • For example, we might model the consumer's model of a store with a series of queues (plus some methods for choosing how long between service and entry in each queue).
• There are a number of problems that require nonlinear data structures, and which need fewer restrictions than trees require.
  • Suppose we are building a new campus network. Determine the least-cost way to connect building so that the network stays active, even if we cut any one connection. (Alternately, given the existing system, determine if any one connection will disconnect parts of the network.)
  • In the telephone system, find the least congested route between two phones, given connections between switching stations.
  • On the Web, determine if there is a way to get to one page from another, just by following normal links.
  • ``Six degrees of Kevin Bacon''.
  • While driving, find the shortest path from one city to another.
  • As a traveling salescritter who needs to visit a number of cities, find the shortest path that includes all the cities.
  • On the Internet, determine how many connections between computers you can eliminate and still make sure that every computer can reach every other computer. (The Internet was originally funded by the military; they wanted it robust even if we were attacked.)
  • Determine an ordering of courses so that you always take prerequisite courses first.
• We often draw pictures to help us think about these problems. I may ask you to draw pictures.
• These, and many other problems, can be modeled by a data structure known as the graph.
• Previously, we've used data structures to improve particular tasks. Here, we use the data structure to motivate the algorithms.

Introduction to Graphs

• Graphs are data structures that contain labeled nodes which are connected by edges.
• Sound familiar? We often think of lists and trees similarly (or at least implementations of lists and trees).
• How do graphs differ from lists and trees?
  • The nodes are labeled, so that we can refer to them by name.
  • Each node may have multiple edges connected to it.
  • In graphs, we don't always distinguish between successors and predecessors.
  • Graphs don't necessarily have a unique start node.
  • Graphs don't necessarily have designated end nodes.
  • In a graph, a node may be its own ``descendant''.
  • The graph may have independent subparts.
• Just as each list is a tree (although not a very balanced tree), each tree is a graph.

Some Terminology

• As with lists and trees, we can make the edges unidirectional or bidirectional.
  • If the edges are unidirectional, the graph is called a directed graph.
  • If the edges are bidirectional, the graph is called an undirected graph.
• In some uses of graphs, we may associate a numeric weight to each edge. Graphs with weights on edges are called weighted graphs. Typically, a weight represents the cost to get from one node to another.
• Note that in some graphs it is possible to follow a sequence of edges and return to the place you started. That path (sequence of edges) is called a cycle. Graphs with cycles are called cyclic graphs.
  • If there are no cycles in a graph, it is an acyclic graph.
  • If you don't say whether or not a graph is cyclic, you are implying that you will deal with either type.
  • Note that directed acyclic graphs (DAGs) have one or more roots and one or more leaves.
• Because graphs can be disconnected, we often refer to connected components of graphs. In a connected component, it is possible to reach every node from every other node.

Traveling Salescritter Problem

• Many of you seemed to have heard vague "rumors" about the traveling salescritter problem (TSP), so we'll begin with that problem.
• The problem is to find the shortest path through a graph that visits every node in the graph.
  • Just as a salescritter must visit every location in its teritory, so must this algorithm visit every node.
• As you might guess, this problem is typically considered for weighted graphs (sometimes directed, sometimes undirected).
• Is there a solution? Certainly.

  List every path that visits all the cities
  Find the shortest such path
  

• Is this a good algorithm? No. It is O(n!). How bad is that?
  • 10! = 3,628,800
  • 20! = 2,432,902,008,176,640,000
• If we could check and compare the length of one trillion paths each second, checking 20! paths would take us 2,432,902 seconds
  • That's 40548 minutes
  • That's 675 hours
  • That's about one month
• Surpisingly, no significantly better algorithm is known. That is, all known algorithms are O(n!). Some just have better constants.
• However, it turns out that if you're willing to accept approximate answers (e.g., a path no worse than two times as long as the best path), then there are much faster solutions.