Class 50: Introduction to Graphs

Held

Summary

• Generalizing the Tree ADT: a graph ADT
• Applications of graphs
• Important graph algorithms
• Implementing graphs

Contents

Notes

• The current version of your project components are due today. How do you turn them in? Use CVS.
• When you use CVS for the rest of this week,
  • Commit only compilable, working code.
  • Do not commit code that you've modified from other people's part of the project.
  • Commit code only when it's been tested.
• There is a growing list of visitors for the project demonstration. We'll spend some time on Friday discussing the structure of the demonstration.
  • We also need to decide where to hold the demonstration.
• We'll be trying a different teaching/learning style in class tomorrow. You folks will be at the board, leading design discussions for algorithms. Last year's class liked the technique and suggested we try it again.
• The annual Math/CS Spring picnic is ths Sunday (mother's day; don't forget to call your mothers) at noon. There's a signup list.
  • If you're on board, entry your number
  • If you're not on board, the department will cover your food.
• There's a cool statistics talk tomorrow at 4:30 in Science 2413: Statistical Lessons from Basketball. Learn whether the ``hot hand'' is myth or statistical reality.
• Don't forget, attendance is required on the last day of class.
• It seems that I have to say it once again. Group work is a significant responsibility. You need to contribute to your groups! This means showing up to meetings, informing group members about meetings, and doing work when you come to meetings and between meetings.

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.
  • 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.
• 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 came up with similar definitions for 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 have a unique start node.
  • Graphs don't necessarily have a unique end node.
  • In a graph, a node may be its own ``descendant''.
  • The graph may have independent subparts.
• 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.
• Just as each list is a tree (although not a very balanced tree), each tree is a 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.
• When writing recursive graph algorithms that are not restricted to acyclic graphs, you may need to mark the nodes in the graph to ensure that you don't repeatedly use the same node.
• 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.

Modeling with Graphs

• We can phrase a number of modeling problems in terms of graphs.
• In the telephone system, find the least congested route between two phones, given connections between switching stations.
  • In a nonnegatively weighted (by congestion) undirected graph, find a path from node A to node B with lowest maximum edge.
• On the Web, determine if there is a way to get to one page from another, just by following normal links.
  • In a directed graph, determine whether node B is reachable from node A.
• While driving, find the shortest path from one city to another.
  • In a weighted directed graph (or simply a directed graph) (or even an undirected graph), find the shortest path from node A to node B.
• Determine someone's ``Kevin Bacon Degree''
  • In an undirected graph, find the shortest path from from node A to node B.
  • Movies are nodes.
  • Actors are edges. An actor connects all pairs of movies in which (s)he has appeared.
• As a traveling salesperson who needs to visit a number of cities, find the shortest path that includes all the cities.
  • In an undirected graph, find the shortest path that includes all nodes (often called a tour).
• On the Internet, determine how many connections between computers you can eliminate and still make sure that every computer can reach every other computer.
  • In a (possibly weighted) graph, find a minimum spanning tree. A spanning three is a set of edges such that every node is reachable from every other node, and the removal of any edge from the tree eliminates the reachability property. A minimum spanning tree is the smallest such tree.
• Determine an ordering of classes so that you always take prerequisite courses first.
• Number the nodes in a directed acyclic graph in such a way that if there is a path from A to B, then the number assigned to A (the term in which you take A) is smaller than the number assigned to B.

Graph Algorithms

• One advantage of modeling problems with graphs is that we can then solve more general problems on the graphs an then use those solutions to solve a wide variety of ``real-world'' questions.
• That is, we will develop a number of graph algorithms to solve typical problems.
• What are some of the core graph algorithms/problems? In no particular order,
  • Reachability. Can you get to B from A?
  • Shortest path (min-cost path). Find the path from B to A with the minimum cost (determined as some simple function of the edges traversed in the path).
  • Minimum spanning tree. Find the ``smallest'' subset of the edges in which all the nodes are connected.
  • Traveling salesman. Find the smallest cost path through all the nodes.
  • Visit all nodes. Traversal.
  • Transitive closure. Determine all pairs of nodes that can reach each other.
  • Topological sort. Number the nodes in such a way that any node has a smaller number than all of its successors.
• We'll consider most of these algorithms in the next few days.
  • There are also many variations of each of these. For example, some versions of traveling salesperson require a cycle (returning to the start), rather than a path.

Implementing Graphs

• How do we implement graphs? There are many techniques, which may depend on our intended use of the graph.
• The simplest one is to use some sort of Node structure, and to treat the graph as a collection of nodes.
  • Each node will need a list of nodes for its neighbors.
• If we can number the nodes, we can use an adjacency matrix: if there is an edge from A to B, then M[A,B] is true.
  • In a weighted graph, we give M[A,B] the weight of the edge. We use some special value (perhaps the largest integer or a negative integer) fo ``not there'').
• We can make a list of all the node pairs in the graph.
• Which implementation should we use? It depends a lot on the algorithm(s) we intend to use.
  • If we need all the neighbors of a node, we want the simple ``collection of nodes'' implementation.
  • If we need all the edges of the graph, we use the list of edges.
  • If we need to quickly determine whether there is an edge between two nodes, we use the adjacency matrix.
• As we study graph algorithms over the next few days, think about what core operations we're using.

History

• Created Monday, January 11, 1999
• Added short summary on Friday, January 22, 1999
• Filled in the details on Monday, May 3, 1999. Many were based on outline 48 of CS152 98S.
• Added a few more introductory notes on Tuesday, May 4, 1999.