Home > Articles > Programming > Algorithms

This chapter is from the book

21.5 Euclidean Networks

In applications where networks model maps, our primary interest is often in finding the best route from one place to another. In this section, we examine a strategy for this problem: a fast algorithm for the source–sink shortest-path problem in Euclidean networks, which are networks whose vertices are points in the plane and whose edge weights are defined by the geometric distances between the points.

These networks satisfy two important properties that do not necessarily hold for general edge weights. First, the distances satisfy the triangle inequality: The distance from s to d is never greater than the distance from s to x plus the distance from x to d. Second, vertex positions give a lower bound on path length: No path from s to d will be shorter than the distance from s to d. The algorithm for the source–sink shortest-paths problem that we examine in this section takes advantage of these two properties to improve performance.

Often, Euclidean networks are also symmetric: Edges run in both directions. As mentioned at the beginning of the chapter, such networks arise immediately if, for example, we interpret the adjacency-matrix or adjacency-lists representation of an undirected weighted Euclidean graph (see Section 20.7) as a weighted digraph (network). When we draw an undirected Euclidean network, we assume this interpretation to avoid proliferation of arrowheads in the drawings.

The basic idea is straightforward: Priority-first search provides us with a general mechanism to search for paths in graphs. With Dijkstra's algorithm, we examine paths in order of their distance from the start vertex. This ordering ensures that, when we reach the sink, we have examined all paths in the graph that are shorter, none of which took us to the sink. But in a Euclidean graph, we have additional information: If we are looking for a path from a source s to a sink d and we encounter a third vertex v, then we know that not only do we have to take the path that we have found from s to v, but also the best that we could possibly do in traveling from v to d is first to take an edge v-w and then to find a path whose length is the straight-line distance from w to d (see Figure 21.18). With priority-first search, we can easily take into account this extra information to improve performance. We use the standard algorithm, but we use the sum of the following three quantities as the priority of each edge v-w: the length of the known path from s to v, the weight of the edge v-w, and the distance from w to t. If we always pick the edge for which this number is smallest, then, when we reach t, we are still assured that there is no shorter path in the graph from s to t. Furthermore, in typical networks we reach this conclusion after doing far less work than we would were we using Dijkstra's algorithm.

21fig18.gifFigure 21.18. Edge relaxation (Euclidean)

To implement this approach, we use a standard PFS implementation of Dijkstra's algorithm (Program 21.1, since Euclidean graphs are normally sparse, but also see Exercise 21.73) with two changes: First, instead of initializing wt[s] at the beginning of the search to 0.0, we set it to the quantity dist(s, d), where distance is a method that returns the distance between two vertices. Second, we define the priority P to be the function

(wt[v] + e.wt() + distance(w, d) - distance(v, d))

instead of the function (wt[v] + e.wt()) that we used in Program 21.1 (recall that v and w are local variables that are set to the values e.v() and e.w(), respectively). These changes, to which we refer as the Euclidean heuristic, maintain the invariant that the quantity wt[v] - distance(v, d) is the length of the shortest path through the network from s to v, for every tree vertex v (and therefore wt[v] is a lower bound on the length of the shortest possible path through v from s to d). We compute wt[w] by adding to this quantity the edge weight (the distance to w) plus the distance from w to the sink d.

Priority-first search with the Euclidean heuristic solves the source–sink shortest-paths problem in Euclidean graphs.

Proof: The proof of Property 21.2 applies: At the time that we add a vertex x to the tree, the addition of the distance from x to d to the priority does not affect the reasoning that the tree path from s to x is a shortest path in the graph from s to x, since the same quantity is added to the length of all paths to x. When d is added to the tree, we know that no other path from s to d is shorter than the tree path, because any such path must consist of a tree path followed by an edge to some vertex w that is not on the tree, followed by a path from w to d (whose length cannot be shorter than the distance from w to d); and, by construction, we know that the length of the path from s to w plus the distance from w to d is no smaller than the length of the tree path from s to d. ▪

In Section 21.6, we discuss another simple way to implement the Euclidean heuristic. First, we make a pass through the graph to change the weight of each edge: For each edge v-w, we add the quantity distance(w, d) - distance(v, d). Then, we run a standard shortest-path algorithm, starting at s (with wt[s] initialized to distance(s, d)) and stopping when we reach d. This method is computationally equivalent to the method that we have described (which essentially computes the same weights on the fly) and is a specific example of a basic operation known as reweighting a network. Reweighting plays an essential role in solving the shortest-paths problems with negative weights; we discuss it in detail in Section 21.6.

The Euclidean heuristic affects the performance but not the correctness of Dijkstra's algorithm for the source–sink shortest-paths computation. As discussed in the proof of Property 21.2, using the standard algorithm to solve the source–sink problem amounts to building an SPT that has all vertices closer to the start than the sink d. With the Euclidean heuristic, the SPT contains just the vertices whose path from s plus distance to d is smaller than the length of the shortest path from s to d. We expect this tree to be substantially smaller for many applications because the heuristic prunes a substantial number of long paths. The precise savings is dependent on the structure of the graph and the geometry of the vertices. Figure 21.19 shows the operation of the Euclidean heuristic on our sample graph, where the savings are substantial. We refer to the method as a heuristic because there is no guarantee that there will be any savings at all: It could always be the case that the only path from source to sink is a long one that wanders arbitrarily far from the source before heading back to the sink (see Exercise 21.80).

21fig19.gifFigure 21.19. Shortest path in a Euclidean graph

Figure 21.20 illustrates the basic underlying geometry that describes the intuition behind the Euclidean heuristic: If the shortest-path length from s to d is z, then vertices examined by the algorithm fall roughly within the ellipse defined as the locus of points x for which the distance from s to x plus the distance from x to d is equal to z. For typical Euclidean graphs, we expect the number of vertices in this ellipse to be far smaller than the number of vertices in the circle of radius z that is centered at the source (those that would be examined by Dijkstra's algorithm).

21fig20.gifFigure 21.20. Euclidean heuristic cost bounds

Precise analysis of the savings is a difficult analytic problem and depends on models of both random point sets and random graphs (see reference section). For typical situations, we expect that, if the standard algorithm examines X vertices in computing a source–sink shortest path, the Euclidean heuristic will cut the cost to be proportional to 325fig01.gif, which leads to an expected running time proportional to V for dense graphs and proportional to 240fig01.gif for sparse graphs. This example illustrates that the difficulty of developing an appropriate model or analyzing associated algorithms should not dissuade us from taking advantage of the substantial savings that are available in many applications, particularly when the implementation (add a term to the priority) is trivial.

The proof of Property 21.11 applies for any function that gives a lower bound on the distance from each vertex to d. Might there be other functions that will cause the algorithm to examine even fewer vertices than the Euclidean heuristic? This question has been studied in a general setting that applies to a broad class of combinatorial search algorithms. Indeed, the Euclidean heuristic is a specific instance of an algorithm called A * (pronounced "ay-star"). This theory tells us that using the best available lower-bound function is optimal; stated another way, the better the bound function, the more efficient the search. In this case, the optimality of A* tells us that the Euclidean heuristic will certainly examine no more vertices than Dijkstra's algorithm (which is A* with a lower bound of 0). The analytic results just described give more precise information for specific random network models.

We can also use properties of Euclidean networks to help build efficient implementations of the abstract–shortest-paths ADT, trading time for space more effectively than we can for general networks (see Exercises 21.48 through 21.50). Such algorithms are important in applications such as map processing, where networks are huge and sparse. For example, suppose that we want to develop a navigation system based on shortest paths for a map with millions of roads. We perhaps can store the map itself in a small onboard computer, but the distances and paths matrices are much too large to be stored (see Exercises 21.39 and 21.40); therefore, the all-paths algorithms of Section 21.3 are not effective. Dijkstra's algorithm also may not give sufficiently short response times for huge maps. Exercises 21.77 through 21.78 explore strategies whereby we can invest a reasonable amount of preprocessing and space to provide fast responses to source–sink shortest-paths queries.


  • 21.68 Find a large Euclidean graph online—perhaps a map with an underlying table of locations and distances between them, telephone connections with costs, or airline routes and rates.

  • 21.69 Using the strategies described in Exercises 17.71 through 17.73, write programs that generate random Euclidean graphs by connecting vertices arranged in a 055fig01.gif grid.

  • 21.70 Show that the partial SPT computed by the Euclidean heuristic is independent of the value that we use to initialize wt[s]. Explain how to compute the shortest-path lengths from the initial value.

  • 21.71 Show, in the style of Figure 21.10, what is the result when you use the Euclidean heuristic to compute a shortest path from 0 to 6 in the network defined in Exercise 21.1.

  • 21.72 Describe what happens if the method distance(s, t), used for the Euclidean heuristic, returns the actual shortest-path length from s to t for all pairs of vertices.

  • 21.73 Develop a class implementation for shortest paths in dense Euclidean graphs that is based upon a graph representation that supports the edge existence test and an implementation of Dijkstra's algorithm (Program 20.6, with an appropriate priority function).

  • 21.74 Run empirical studies to test the effectiveness of the Euclidean shortest-path heuristic for various Euclidean networks (see Exercises 21.9, 21.68, 21.69, and 21.80). For each graph, generate V/10 random pairs of vertices, and print a table that shows the average distance between the vertices, the average length of the shortest path between the vertices, the average ratio of the number of vertices examined with the Euclidean heuristic to the number of vertices examined with Dijkstra's algorithm, and the average ratio of the area of the ellipse associated with the Euclidean heuristic with the area of the circle associated with Dijkstra's algorithm.

  • 21.75 Develop a class implementation for the source–sink shortest-paths problem in Euclidean graphs that is based on the bidirectional search described in Exercise 21.35.

  • 21.76 Use a geometric interpretation to provide an estimate of the ratio of the number of vertices in the SPT produced by Dijkstra's algorithm for the source–sink problem to the number of vertices in the SPTs produced in the two-way version described in Exercise 21.75.

  • 21.77 Develop a class implementation for shortest paths in Euclidean graphs that performs the following preprocessing step in the constructor: Divide the map region into a W-by-W grid, and then use Floyd's all-pairs shortest-paths algorithm to compute a W2-by-W2 matrix, where row i and column j contain the length of a shortest path connecting any vertex in grid square i to any vertex in grid square j. Then, use these shortest-path lengths as lower bounds to improve the Euclidean heuristic. Experiment with a few different values of W such that you expect a small constant number of vertices per grid square.

  • 21.78 Develop an implementation of the all-pairs shortest-paths ADT for Euclidean graphs that combines the ideas in Exercises 21.75 and 21.77.

  • 21.79 Run empirical studies to compare the effectiveness of the heuristics described in Exercises 21.75 through 21.78, for various Euclidean networks (see Exercises 21.9, 21.68, 21.69, and 21.80).

  • 21.80 Expand your empirical studies to include Euclidean graphs that are derived by removal of all vertices and edges from a circle of radius r in the center, for r = 0.1, 0.2, 0.3, and 0.4. (These graphs provide a severe test of the Euclidean heuristic.)

  • 21.81 Give a direct implementation of Floyd's algorithm for an implementation of the network ADT for implicit Euclidean graphs defined by N points in the plane with edges that connect points within d of each other. Do not explicitly represent the graph; rather, given two vertices, compute their distance to determine whether an edge exists and, if one does, what its length is.

  • 21.82 Develop an implementation for the scenario described in Exercise 21.81 that builds a neighbor graph and then uses Dijkstra's algorithm from each vertex (see Program 21.1).

  • 21.83 Run empirical studies to compare the time and space needed by the algorithms in Exercises 21.81 and 21.82, for d = 0.1, 0.2, 0.3, and 0.4.

  • 21.84 Write a client program that does dynamic graphical animations of the Euclidean heuristic. Your program should produce images like Figure 21.19 (see Exercise 21.38). Test your program on various Euclidean networks (see Exercises 21.9, 21.68, 21.69, and 21.80).

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.


Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.


If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information

Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.


This site is not directed to children under the age of 13.


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure

Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact

Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice

We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020