What Two Standards Below Have Been Developed to Replace the Spanning Tree Protocol?

Spanning Tree

SDN in the Data Centre

Paul Göransson , ... Timothy Culver , in Software Defined Networks (Second Edition), 2017

8.4.2 Multiple Spanning Tree Protocol

The Multiple Spanning Tree Protocol(MSTP) was introduced to achieve better network link utilization with spanning tree applied science when there are multiple VLANs present. Each VLAN would operate nether its own spanning tree. The improved use of the links was to have i VLAN's spanning tree use unused links from another VLANs when reasonable to do then. MSTP was originally introduced as IEEE 802.1s. It is at present part of IEEE 802.1Q-2005 [vii]. It was necessary to accept a big number of VLANs in order to reach a well distributed utilization level across the network links as the only distribution was at VLAN-granularity. MSTP predates the Shortest Path Bridging protocol discussed below, which does not suffer this limitation.

Read full chapter

URL:

https://www.sciencedirect.com/scientific discipline/article/pii/B9780128045558000089

Network Architectures and Overlay Networks

Casimer DeCusatis , in Handbook of Cobweb Optic Data Advice (Quaternary Edition), 2013

13.1 STP and MC-LAG

STP is a Layer 2 switching protocol used by classic Ethernet [ane] , which ensures loop-gratuitous network topologies past always creating a single path tree structure through the network. In the effect of a link failure or reconfiguration, the network halts all traffic while the spanning tree algorithm recalculates the allowed loop-free paths through the network. (STP creates a loop-gratis topology using Mutlichassis EtherChannel (MCEC), also referred to as virtual Port Channels (vPCs) for Cisco switches [2].) The changing requirements of cloud data center networks are forcing designers to reexamine the role of STP. One of the drawbacks of an STP is that in blocking redundant ports and paths, a spanning tree reduces the aggregate available network bandwidth significantly. Additionally, STP can event in circuitous and suboptimal advice paths through the network, adding latency and degrading application performance. A spanning tree cannot be easily segregated into smaller domains to provide amend scalability, error isolation, or multitenancy. Finally, the fourth dimension taken to recompute the spanning tree and propagate the changes in the event of a failure tin can vary widely, and sometimes become quite large (seconds to minutes). This is highly disruptive for elastic applications and virtual machine migrations, and tin pb to cascaded system level failures.

To help overcome the limitations of STP, several enhancements accept been standardized. These include the Multiple Spanning Tree Protocol (MSTP), which configures a carve up spanning tree for each virtual local area network (VLAN) group and blocks all but 1 of the possible alternate paths within each spanning tree. Too, the link aggregation grouping (LAG) standard (IEEE 802.3ad) [iii] allows two or more physical links to be bonded into a single logical link, either between two switches or between a server and a switch. Since LAG introduces a loop in the network, STP has to exist disabled on network ports using LAGs. It is possible for one finish of the link-aggregated port group to be dual-homed into 2 different devices to provide device level redundancy. The other terminate of the group is still single-homed and continues to run normal LAG. This extension to the LAG specification is called MC-LAG, and is standardized every bit IEEE 802.1ax (2008). As shown in Figure 13.1, MC-LAG can be used to create a loop-complimentary topology without relying on STP; because STP views the LAG every bit a single link, it will not exclude redundant links within the LAG [four]. For instance, it is possible for a pair of network interface cards (NICs) to exist dual-homed into a pair of access switches (using NIC teaming), and so one can apply MC-LAG to interconnect the admission switches with a pair of cadre switches.

Effigy 13.i. MC-LAG configuration without STP (left) and with STP (right).

Most MC-LAG systems allow dual homing across only two paths; in practice, MC-LAG systems are limited to dual cadre switches because it is extremely difficult to maintain a coherent state between more than than two devices with submicrosecond refresh times. Unfortunately, the hashing algorithms that are associated with MC-LAG are not standardized; care needs to be taken to ensure that the 2 switches on the same tier of the network are from the same vendor (switches from different vendors can be used on dissimilar tiers of the network). Equally a relatively mature standard, MC-LAG has been deployed extensively, does not require new forms of data encapsulation, and works with existing network management systems and multicast protocols.

Read full chapter

URL:

https://www.sciencedirect.com/scientific discipline/article/pii/B9780124016736000131

Spanning-Tree Protocol

Dale Liu , ... Luigi DiGrande , in Cisco CCNA/CCENT Test 640-802, 640-822, 640-816 Preparation Kit, 2009

Publisher Summary

Spanning-Tree Protocol (STP) creates the "fully-connected and loop-free" topology by finding loops and blocking interfaces. The original STP is very matured, widely implemented, robust, and vendor neutral. It is necessary for a span to understand enough about the rest of the network topology then that information technology can recognize when one of its ports is part of a loop. In guild to accomplish this, the STP chooses a single reference point in the network and and then finds all redundant paths to that indicate. In one case found, certain ports on redundant paths are blocked by the bridge so that a spanning tree is created that is a fully connected and loop-free topology. The reference point in the network is the root bridge. Span Protocol Data Units (Bpdus) are the frames sent by bridges to a well-known multicast accost out each port every two seconds. All bridges that are running STP listen for this accost and know to process the Bpdu. Bpdus are addressed to a link-local accost, which ensures that the bridge does not forward the Bpdu to other ports. Thus, a bridge only communicates through Bpdus with its direct connected neighbor bridges.

Read total chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9781597493062000166

Languages and Issues

Raymond Greenlaw , H. James Hoover , in Fundamentals of the Theory of Computation: Principles and Practice, 1998

2.9.2 Spanning Copse

Permit One thousand = (V, E) be an undirected graph with edge weights from ℕ specified by the weighting office w : E ↦ ℕ. V represents the set of vertices, and E the set of edges. An instance of such a weighted graph is shown in Figure two.5. The graph has six vertices and 9 edges.

Figure 2.5. An case of a weighted, undirected graph used to illustrate concepts relating to spanning trees.

A spanning tree is a subgraph T of G that contains all the vertices of One thousand, and just enough edges from E and then that information technology connects all the vertices together but does not have any cycles. ¹¹ Figure two.6 illustrates a spanning tree of the graph shown in Figure ii.v. The cost of a spanning tree T is equal to the sum of the weights on the edges in the tree. The cost of the tree shown in Figure 2.half-dozen is nineteen. A minimum cost spanning tree is a spanning tree with least cost among all possible spanning copse. In that location are many well-known algorithms for efficiently calculating minimum toll spanning trees. The minimum cost spanning tree of the graph shown in Figure 2.five has a price of 17. Information technology should be articulate that this is indeed a minimum, since whatsoever spanning tree of this graph volition comprise five edges and the five least-cost edges sum to 17.

Figure two.six. A spanning tree of the graph shown in Figure 2.5. The cost of this spanning tree is 19.

To really talk near minimum cost spanning trees, we also need to augment the graph representation to include the border weighting function w. Nosotros could supply it as a table that maps each edge (set of two vertices) to its respective weight. Or nosotros could modify the representation to contain edge weights directly For instance, the adjacency matrix might be defined by

$(a_{ij})=w\text{if}\text{there}\text{is}\text{an}\text{edge}\text{of}\text{weight}\mathrm{due\; west}\text{betwixt}\text{vertex}{v}_i\text{and}{v}_j$

and thus directly supply the weights as entries in the adjacency matrix. Any similar approach will work, and the details are left every bit an exercise.

With graph representation schemes and spanning copse in our toolkit, we can at present proceed to ascertain decision, function, and search problems. We start with the simplest of these—the decision trouble.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9781558605473500066

Spanning Tree Attacks

Stacy Prowell , ... Mike Borkin , in Seven Deadliest Network Attacks, 2010

The Future of Spanning Tree Attacks

Root guard and BPDU guard are both very effective strategies to mitigate STP attacks but are both (at the time of writing) limited to Cisco hardware. These technologies may still allow an intruder to monitor BPDU frames and use these to discover network information. In one case you lot have established a perimeter, the intruder may seek to compromise a device within the perimeter. By capturing frames, an intruder can obtain the MAC addresses of devices inside the perimeter. By using other layer 2 attacks, the attacker may be able to compromise a device within the perimeter and then launch an STP-based assail such every bit a deprival of service.

Many unlike versions of STP exist with proprietary extensions, such every bit the portfast extension mentioned previously. It may be possible to exploit these extensions in ways not described hither. If an intruder tin fix all ports to forward and then they can strength cycles and trigger a denial of service. Many switch implementations provide diagnostic settings that copy traffic from all ports to a unmarried port, making information technology much easier to monitor and steal information.

STP attacks are themselves relatively new, having been initially proposed in Phrack ^GG magazine in 2002, and investigated at Black Hat Europe ^HH in 2005 with the introduction of Yersinia. Because of this it is likely that intruders have non notwithstanding begun to take full advantage of STP-based attacks. Nosotros can expect this sort of attack to become more than prevalent.

Read full chapter

URL:

https://world wide web.sciencedirect.com/scientific discipline/commodity/pii/B9781597495493000055

Fundamentals of algorithms

Chung-Yang (Ric) Huang , ... Kwang-Ting (Tim) Cheng , in Electronic Design Automation, 2009

4.three.7 Minimum spanning tree

Spanning trees are divers on connected, undirected graphs. Given a graph Chiliad = (V, E), a spanning tree connects all of the vertices in V by employ of some edges in E without producing cycles. A spanning tree has exactly (| 5 | − ane) edges. For example, the thickened edges shown in Effigy 4.18 class a spanning tree. The tree weight of a spanning tree is divers as the sum of the weights of the tree edges. At that place would exist many spanning copse in a connected, weighted graph with different tree weights. The minimum spanning tree (MST) problem searches for a spanning tree whose tree weight is minimized. The MST trouble can model the structure of a power network with a minimum wire length in an integrated circuit. Information technology can as well model the clock network, which connects the clock source to each terminal with the least number of clock delays. In this subsection, we present an algorithm for the MST trouble, Prim'south algorithm [Prim 1957].

Figure 4.18. An example of an MST returned by Prim's algorithm. The MST consists of the thickened edges. The society of choices is shown on the right.

Prim's algorithm builds an MST by maintaining a set of vertices and edges. This set initially includes a starting vertex. The algorithm then adds edges (along with vertices) one by i to the fix. Each time the edge closest to the fix—with the least edge weight to any of the vertices in the set—is added. After the prepare contains all the vertices, the edges in the set grade a minimum spanning tree.

The pseudocode of Prim's algorithm is given in Algorithm 4.14. The role PrimMST uses a priority queue minQ to store those vertices not yet included in the fractional MST. Every vertex in minQ is keyed with its minimum border weight to the partial MST. In line vii, the vertex with the minimum primal is extracted from minQ, and the keys of its next vertices are updated accordingly, as shown in lines eight through 11. The parameter predecessor refers to MST edges.

Algorithm iv.14

Prim's MST algorithm

Similar Dijkstra's algorithm, the data structure of minQ determines the runtime of Prim's algorithm. PrimMST has a time complication of O(V ² + E) if minQ is implemented with a linear array. However, less time complication can be achieved by apply of a more sophisticated data structure.

Figure 4.18 shows an instance in which Prim's MST algorithm selects the vertex five ₀ as the starting vertex. In fact, an MST can exist built from any starting vertex. Moreover, an MST is non necessarily unique. For example, if the edge (v ₇, v ₈) replaces the border (v _iii, v _eight), as shown in Figure 4.eighteen, the new set of edges still forms an MST.

The strategy used past Prim's algorithm is actually very similar to that of Dijkstra'south shortest-path algorithm. Dijkstra'south algorithm implicitly keeps a ready of processed vertices and chooses an unprocessed vertex that has a minimum shortest-path estimate at the moment to be the side by side target of relaxation. This strategy follows the principle of a greedy algorithm. This concept volition be explained in Subsection 4.4.1.

Read total chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780123743640500114

Graph theory

Mary Attenborough , in Mathematics for Electrical Technology and Calculating, 2003

19.iv Trees

A tree is a connected graph with no cycles. A forest is a graph with no cycles but may or may not be continued (i.e. a forest is a graph whose components are trees). Figure 19.13(a) shows a tree, while Figure xix.12(b) shows a forest.

Figure xix.13. (a) A tree is a continued graph with no cycles. (b) A woods has no cycles, but may or may not be connected.

If T is a tree with at least 2 vertices, then it has the three properties (T1) (T2), and (T3):

(T1) There is exactly one path from any vertex v _i in T to any other vertex v _j.

(T2) The graph obtained from T by removing any edge has 2 components, each of which is a tree.

Trees have many applications, particularly rooted trees. Decision trees are used to stand for the possible decisions at each stage of a problem or algorithm. Probability trees can be used to analyse conditional probabilities, which we shall see in Chapter 21. Another application is to parsing of a sentence. The tree in Figure 19.14 represents the judgement 'Fortune favours the dauntless'. The vertices, other than the terminal vertices, represent grammatical categories and the terminal vertices stand for the words of the sentence. The same sort of idea can be used to analyse allowed constructs of statements in programming languages and the syntax of arithmetics expressions. We wait at this application of copse in the next chapter on language theory.

Effigy 19.fourteen. A parsing tree for the sentence 'Fortune favours the dauntless'.

Spanning trees

A spanning tree of a graph G is a tree T which is a spanning subgraph of Thousand. That is, T has the same vertex gear up as K. Examples of graphs with spanning trees marked are given in Figure 19.15.

Figure 19.15. Graphs with spanning trees shaded.

How to abound a spanning tree

Take any vertex v of One thousand as an initial partial tree. Add edges one by one so each new edge joins a new vertex to the fractional tree. If there are n vertices in the graph Chiliad then the spanning tree volition have n vertices and north – i edges.

Minimum spanning tree

Supposing nosotros have a group of offices which need to be connected by a network of communication lines. The offices may communicate with each other directly or through another office. In order to decide on which offices to build links between we firstly work out the cost of all possible connections. This will so requite united states of america a weighted complete graph every bit shown in Figure 19.16.

Figure 19.16. A weighted complete graph. The vertices stand for offices and the edges represent possible communication links. The weights on the edges correspond the toll of structure of the link.

The minimum spanning tree is and then the spanning tree that represents the minimum price.

The greedy algorithm for the minimum spanning tree

1

Cull whatsoever start vertex to grade the initial partial tree (v _i ).

2

Add the cheapest edge, east _i, to a new vertex to grade a new fractional tree.

3

Repeat Step 2 until all vertices take been included in the tree.

Example 19.1

Find the minimum spanning tree for the graph representing communication links between offices as shown in Figure 19.xvi.

Solution Nosotros first with whatsoever vertex and choose the 1 marked a. Add together the edge ab which is the cheapest edge of those incident to a.

Add a new edge in guild to form a partial tree and choose bc, which is one of the cheapest remaining edges incident either with a or b. Now nosotros add together edge advertising which is the cheapest remaining edge of those incident with a or b or c. Continuing in this manner we detect the minimum spanning tree, as shown in Figure 19.17. The total cost of the communication links in our solution is found to be ii + 3 + 3 + 2 + 4 = 14.

Figure 19.17. The graph of Figure 19.16 with its minimum spanning tree marked.

Read full chapter

URL:

https://www.sciencedirect.com/science/commodity/pii/B9780750658553500450

Troubleshooting Traffic for Network Optimization

Robert J. Shimonski , ... Yuri Gordienko , in Sniffer Pro Network Optimization and Troubleshooting Handbook, 2002

Spanning Tree Protocol

Spanning Tree Protocol (STP) is the de facto switch link management protocol you must principal as both a network engineer and/or a protocol analyst. STP offers one major benefit: Path back-up while preventing switch loops. STP will maintain a "tree" of all switches and paths in the network, and, if a link goes downward, it volition be able to reroute traffic through the redundant links that be. The problem that would occur if STP weren't enabled would exist that if redundant links and Mac addresses are learned from two unlike locations, a loop may (or more than probable volition) occur, and traffic would exist circulated at a very high rate, which is known to terminate all network traffic within no time at all. The issues with a spanning tree is the excessive time information technology takes to "learn" what it needs to know most hosts connected to the switched network, and the excessive traffic that the Bridge Protocol Data Units (BPDUs) generate during normal operations.

One problem nosotros tin can find and eliminate with the use of Sniffer Pro is the excessive BPDU traffic generated if you cannot turn Spanning Tree off. There are some things to exist enlightened of when using the STP on your switched network. If all of your switches are using the default configuration and the other switches make up one's mind ii of them to take the same path cost, the switch that has the everyman Mac Accost will be selected as the root switch. Using Sniffer Pro, you tin monitor the traffic on your network and decide if the correct switch is acting equally the root switch. If not, raise the priority of the better selection and make that switch the new root switch. In that location are many ways to optimize circulate traffic with the use of Spanning Tree and the all-time way to piece of work with this traffic is to do one of 2 things:

▪

Turn Spanning Tree off. It's not needed unless y'all have redundant paths in your network.

▪

Get out Spanning Tree on and discover ways for it to not slow downwardly your LAN through optimization.

That being said, let'southward await at means to optimize it if you make up one's mind to leave STP on.

Spanning Tree Optimization

As this chapter states, you lot volition want to know how to troubleshoot and optimize traffic with Sniffer Pro. To practise so, all y'all demand to practice is monitor the network utilization for acceptable broadcast traffic. If the traffic is not within adequate ranges, optimize your network to get information technology inside acceptable limits. Spanning Tree Protocol has a major downside; it is dull to reach convergence in a very large environment that has a link failure. It is possible to optimize STP operation, simply before we practice so, let'southward expect at why STP causes network traffic:

The root bridge is selected co-ordinate to the bridge ID value. (This is also configurable and then you tin can have your core switches interim every bit your root bridge instead of a cupboard-based access layer switch.) On the root bridge, all interfaces are placed in the forwarding state. For each segment that has more than one bridge connected to it, a designated span is selected that volition be the one to forward frames to the root. Each bridge selects a root port that will be used to frontwards frames toward the root bridge. STP selects all the designated bridges and root ports necessary for switched LAN functionality and identifies a loop-gratis path between the root bridge and all LANs. STP so places the selected bridge interfaces in to a forwarding country and all the others in a blocked country. The root bridge transmitting BPDUs every ii seconds by default maintains the spanning tree (this is where your traffic continues after convergence). Upon receipt of a BPDU from the root bridge, the other bridges transmit their own BPDUs.

Note

If yous are nevertheless in a jam trying to understand how Spanning Tree works, you can visit Cisco's Web site, where there is a concise article on exactly how Switching and Spanning Tree works. Information technology is definitely worth a read if you are confused for any reason: www.cisco.com/warp/public/473/lan-switch-cisco.shtml

Some would say that this is acceptable traffic, but that's for you lot to decide. I believe that a network can be fine-tuned and operate ameliorate when traffic flow and application period is optimized. Now that you can run into that switches running STP chat with each other pretty frequently, permit's wait at a style to optimize this traffic without turning STP off.

Optimizing STP Timers

If you are looking to optimize STP traffic, you lot should focus your efforts on the timers that send BPDUs. The timers you lot can optimize are those that ship BPDUs at default intervals across the tree and those that determine when a missing BPDU is indicative of a link failure. The primal timer values are ready at the root bridge and are the hello time, max age, and forward delay. Let'due south wait at the tweaks you lot can put in for optimization:

▪

Configure the hello fourth dimension, max age, and forward delay timers on your switch in a test lab, so you lot can make sure y'all inquiry your switch type for its tunable parameter range. Each switch is unlike, and then you will accept to enquiry each configuration on each switch separately.

▪

You can utilise portfast to eliminate the wait time for nodes to exist learned by the switch and so they can transmit information on the network segment they are attached to more quickly.

▪

Eliminate STP where it is not needed, or it will never exist used.

Note

Take extreme circumspection when working with and tweaking your infrastructure. Make sure you programme everything out and have a good backout plan. Some Cisco switches write immediately to retention, and a error can exist costly. Spanning Tree loops and circulate storms can cripple a network in only a few minutes.

We have looked at i manner to perform assay using Sniffer Pro to optimize traffic on your network. Let's look at another way to utilize Sniffer Pro. In the next instance nosotros will connect directly to a switch to analyze information technology in hopes of improving network traffic.

Read total chapter

URL:

https://www.sciencedirect.com/science/commodity/pii/B9781931836579500162

DSP System Blueprint

Lars Wanhammar , in DSP Integrated Circuits, 1999

vii.6.7 Maximum Spanning Tree Method

The maximum spanning tree method, which was developed by Renfors and Neuvo [ 20], can be used to achieve rate optimal schedules. The method is based on graph-theoretical concepts. The starting indicate is the fully specified SFG. The SFG corresponds to a ciphering graph Due north, which is formed by inserting the functioning delays into the SFG. Further, a new graph North' is formed from Due north by replacing the delay elements by negative filibuster elements (− T). In this graph we find the maximum distance spanning tree—i.e., a spanning tree where the distance from the input to each node is maximal. Next, shimming delays are inserted in the link branches (i.east., the remaining branches) so that the total delay in the loops becomes zero. Finally, remove the negative delays. The remaining delays, apart from the operation delays, are the shimming delays. These concepts are described by an instance.

Instance 7.7

Consider the second-society department as in Example 7.vi and derive a rate optimal schedule using the maximum spanning tree method.

Figure 7.47 shows the ciphering graph North'—i.e., the ciphering graph with the performance delays inserted and T replaced by   − T. Note that the graphs are of the blazon activity-on-arcs and that the delay of adders is assigned to the border immediately later the addition. The maximal spanning tree is shown in Effigy 7.48 where edges belonging to the tree are drawn with thick lines and the link branches with sparse lines. Next, insert link branches i by one and add shimming delays so that the total delay in the cardinal loops that are formed becomes null. Finally, we remove the negative delays and arrive at the scheduled ciphering graph shown in Effigy 7.49.

Effigy 7.47. Modified computation graph N'

Figure seven.48. Scheduled computation graph

Figure 7.49. The maximum spanning tree of Due north′

Note that there are no shimming delays in the critical loops. All operations are scheduled as early equally possible, which in general will be suboptimal from a resources point of view. The resulting schedule is shown in Figure 7.50. Thus, this approach does non contain a minimization pace—for example, minimization of the amount of resources. It but finds a feasible charge per unit optimal schedule. Withal, it may be possible in a subsequent design pace to movement operations in time so that the required number of processors is minimized.

Effigy 7.50. Rate optimal operation schedule

Read total chapter

URL:

https://www.sciencedirect.com/scientific discipline/article/pii/B9780127345307500076

Spanning Trees and Spanners

David Eppstein , in Handbook of Computational Geometry, 2000

two.5 Minimum diameter spanning trees

The previous spanning tree problems have all been based on the weight of the tree synthetic. Nosotros now consider other criteria for the quality of a tree. The diameter of a tree is just the length of its longest path. Since geometric spanning copse are often used in applications such as VLSI, in which the time to propagate a signal through the tree is proportional to its diameter, it makes sense to look for a spanning tree of minimum diameter. Ho et al. [82] give an algorithm for this problem, based on the following fact:

Lemma 14 (Ho et al.)

Any bespeak gear up has some minimum diameter spanning tree in which at that place are at nigh ii interior points.

Proof. We kickoff with any minimum diameter spanning tree T, and perform a sequence of diameter-preserving transformations until information technology is in the in a higher place form. Permit P be the longest path in the given tree, and number its vertices v _ane,v ₂,…,5_p .

Nosotros first form a forest by removing all edges of P from T, and for each vertex 5 of P, allow T_v announce the tree in this forest containing five. For any other vertex u, let P_u announce the vertex v such that u is in T_five . And so nosotros construct a new tree T′ by adding to P an edge from each vertex u to P_u . T′ has the same diameter equally the original tree, since the distance betwixt any ii vertices can not increase except when they are in the aforementioned tree T_v , and in that example (by the assumption that P is a diameter path) the altitude of each point to v is less than the distance from v to the endpoints of P.

At present suppose that P has four or more edges and the length of the path v ₁-v _ii-v ₃ is at most half the length of P. (If not, we can contrary P and consider the three vertices at its other end.) Form a tree T″ by removing every edge uv ₂ and reconnecting each such vertex u to v ₃. This can just decrease the lengths of paths already going through v ₃ in T′ so the only pairs of vertices with increased path lengths are those newly connected to 5 _iii. But the length of any such path is at most twice the length of the path 5 ₁-v _two-v ₃, so the diameter of T″ is no more than than that of T.

Each repetition of this transformation decreases the number of edges in P until it is at most 3, and preserves the holding that each vertex is within one edge of P, so nosotros will eventually achieve a tree of the form specified in the lemma.

Theorem nine (Ho et al.)

We can find a minimum diameter spanning tree of any bespeak set in time O(northward ³).

Proof. We but try all unmarried interior vertices and pairs of interior vertices. For the latter example, we still need to determine how to assign each remaining bespeak to ane of the 2 interior vertices. If the diameter path of the tree is five _ane-v ₂-five _three-v _iv, and we know the lengths of v _one-v ₂ and five ₃-v ₄, we can perform this consignment by drawing 2 circles, centered at the 2 interior vertices, with those lengths as radii (Figure 4(a)); these circles must together cover the whole signal ready and each betoken tin be assigned to the interior vertex corresponding to the circle covering information technology. Ho et al. prove that, if nosotros sort the points past their distances from 5 ₂ and v ₃, the space of all minimal pairs of covering circles can be searched in linear time. These sorted orders can be precomputed in a total of O(northⁱⁱ log n) time.

Fig. 4. (a) Minimum bore spanning tree corresponds to cover past two circles, (b) Point set with high diameter minimum spanning tree.

Ho et al. also consider optimization of combinations of bore and total weight; they show that it is NP-complete to find a tree satisfying given bounds on these two measures. Information technology is too of interest to combine these criteria with a spring on vertex degree. Of course for caste-two trees, minimum diameter and minimum weight are equivalent to the traveling salesman trouble. The minimum weight spanning tree itself can accept very high diameter (an $\Omega \left(\sqrt{n}\right)$ cistron away from optimal); for instance a betoken fix spaced nearly uniformly in a unit square can have a path of length k $\Omega \left(\sqrt{n}\right)$ equally its minimum spanning tree (Figure 4(b)). Conversely an upper spring of $\Omega \left(\sqrt{n}\right)$ follows from results on the worst case length of the minimum spanning tree of north points [24,126].

One can achieve a better diameter by relaxing the requirement of minimum weight; as we describe later, for 2- and iii-dimensional point sets, i tin can find a subgraph of the complete Euclidean graph with degree three and total weight O(ane) times that of the minimum spanning tree, for which shortest paths in the subgraph have length within O(ane) of the Euclidean distance [44]. A single-source shortest path tree in such a graph is a degree-three tree that has both weight and bore within a abiding of the minimum. Similar problems of constructing spanning trees combining diameter, weight, and bottleneck shortest path premises were considered by Salowe et al. [119] and Khuller et al. [91],

Read total chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444825377500103

longdeate1990.blogspot.com

Source: https://www.sciencedirect.com/topics/computer-science/spanning-tree