Domain-aware mpr election method based on multi-dimensional identification

Abstract

The application belongs to the field of wireless communication, and discloses a multi-dimensional identification-based domain-aware MPR election method, which comprises the following steps: 1, in an NID domain, a node broadcasts and receives a HELLO message to discover the same domain neighbor nodes; 2, a part of the same domain neighbor nodes are elected as an MS node set for MPR node election; 3, after a node is elected as an MPR node by neighbor nodes, the neighbor nodes that elect the node are identified through the received HELLO message, and the MS node set is maintained; 4, link information between the MPR node and each node in the MS node set is encapsulated into TC information; and 5, the TC message is flooded to the entire NID domain where the TC message is located. The application eliminates invalid redundant nodes, can improve the success rate of data transmission, ensures that each selected MPR node has stable connection in a high mobility environment, and improves network efficiency.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of wireless communication, and particularly relates to a multi-dimensional identifier-based domain-aware MPR election method. BACKGROUND

[0002] In a traditional link state routing protocol, the propagation of network topology information usually depends on each node broadcasting a message to all its neighbor nodes, which can cause broadcast redundancy, bandwidth waste and significant decline in network performance in a large-scale network. With the increase in the number of nodes, repeated propagation of broadcast messages not only increases the network burden, but also causes an increase in delay and a decrease in throughput, which is more obvious in bandwidth-limited environments such as wireless self-organizing networks.

[0003] Based on a network identifier (NID), the prior art has designed a node and link state-based routing protocol (Zone Link State Routing, ZLSR). The protocol reduces the number of times of forwarding broadcast messages by selecting a small number of key nodes as relays.

[0004] The traditional MPR (Multi Point Relays) selection method has limitations in handling large-scale or complex network topologies due to the use of a greedy strategy, so the selected MPR set is often non-optimal and can easily generate additional network overhead. The energy consumption of MPR and the stability of nodes are not fully considered, and further optimization is needed to improve its performance in practical applications. SUMMARY

[0005] To solve the above technical problems, the application provides a multi-dimensional identifier-based domain-aware MPR election method, which limits the optimized MPR broadcast within a specific dynamic network domain, solves the limitations caused by the greediness of the traditional MPR method, and solves the deficiencies of the traditional method in resource utilization and connection reliability.

[0006] To achieve the above purpose, the application is implemented by the following technical solutions:

[0007] The application is a multi-dimensional identifier-based domain-aware MPR election method, which includes the following steps:

[0008] Step 1: In a specific multi-dimensional identifier NID domain, nodes discover the same domain neighbor nodes by periodically broadcasting and receiving HELLO packets, the HELLO packets contain the geographic coordinate information and multi-dimensional identifier NID information of one-hop neighbor nodes and the geographic coordinate information and multi-dimensional identifier NID information of two-hop neighbor nodes, so that the neighbor node discovery process has identity and domain awareness capability;

[0009] Step 2, the same domain neighbor nodes are discovered based on the HELLO packet, and a part of the MS node set, i.e. the multi-point relay selection node set, is elected as the MPR node set;

[0010] Step 3, after the neighbor nodes are elected as the MPR nodes, the nodes elected by themselves are identified through the received HELLO packet, and the MS node set is maintained;

[0011] Step 4, the elected MPR nodes encapsulate the link information between the MPR nodes and each node in the MS node set into the TC (Topology Control) information;

[0012] Step 5, the MPR nodes flood the TC message encapsulated in step 4 to the entire NID domain where the TC message is located by using the MPR forwarding method, so as to effectively spread the key topology information.

[0013] The further improvement of the application is that in step 1, the multi-dimensional identification NID information includes the access network node, the core network node, the vehicle-mounted network node and the terminal network node.

[0014] The further improvement of the application is that in step 2, the metrics for the MPR node election include the coverage degree of the node, the node connection and the distribution stability.

[0015] The further improvement of the application is that in step 2, the MS node set election method specifically includes the following steps:

[0016] Step 2.1, multi-dimensional screening is performed in all one-hop neighbor nodes: high-quality nodes with high coverage and stable distribution are screened out based on the geographic coordinate information of the one-hop neighbor nodes and the multi-dimensional identification NID information, and a high-quality node candidate pool is formed;

[0017] Step 2.2, according to the number of two-hop neighbor nodes that can be connected by the high-quality nodes in the high-quality node candidate pool in step 2.1, the high-quality nodes are sorted: the one-hop neighbor nodes in the high-quality node candidate pool that can cover the isolated two-hop neighbor nodes are selected as the required nodes of the high-quality node candidate pool, and the isolated two-hop neighbor nodes refer to the two-hop neighbor nodes connected to the target node through only one neighbor node;

[0018] Step 2.3, after the required nodes are screened out, the selection of the MS node set is performed in the remaining one-hop neighbor nodes until all the two-hop neighbor nodes are covered.

[0019] The further improvement of the application is that step 2.1 specifically includes the following steps:

[0020] Step 2.1.1, starting from any node , the node Discovering one-hop neighbor set and two-hop neighbor set of the node through HELLO packet , , and obtaining and storing geographic coordinate information and multi-dimensional identification NID information of all one-hop neighbor nodes through HELLO packet

[0021] Step 2.1.2, traversing all one-hop neighbor nodes and calculating quality of neighbor nodes of each one-hop neighbor set :

[0022]

[0023]

[0024]

[0025] For each one-hop neighbor node , define neighbors in four regions determined by angles :

[0026] : where is the neighbor of ;

[0027] : where is the neighbor of ;

[0028] : where is the neighbor of ;

[0029] : where is the neighbor of ;

[0030] wherein is a weight factor, which is dynamically adjusted according to network strategy, , , and are the four regions, is the weight assigned to each region, , is the neighbor coverage of the neighbor node, is the variance of the number of neighbors in the four regions, i.e. distribution stability, nodes with higher value and lower variance will be given priority, is the node connection score, which is used to determine whether to forward according to multi-dimensional identification NID information, and nodes with high quality value are included in the high-quality node candidate pool, i.e. HQ-POOL set.

[0031] The further improvement of the present application is that in the step 2.2, the following steps are specifically included:

[0032] In step 2.2.1, for the nodes of the MPR node set to be calculated, all one-hop neighbors in the high-quality node candidate pool are regarded as a set F, and the one-hop neighbor nodes connected to all two-hop neighbor nodes are elements of the set F;

[0033] In step 2.2.2, if the set F is empty, waiting until the topology changes, and if the set F is not empty, using the letter …numbering the elements in the set F with the numbers 1, 2, 3, 4…, numbering the elements in the set S with the numbers 1, 2, 3, 4…, and setting the initial value of the MPR node set to be empty;

[0034] In step 2.2.3, the set F is sorted from large to small according to the number of two-hop neighbor nodes of the set S connected by the one-hop neighbor nodes in the set F;

[0035] In step 2.2.4, in the sorted set F, a two-hop neighbor node belonging to only one one-hop neighbor node is first found, then the one-hop neighbor node connected to the two-hop neighbor node is added from the set F to the MPR node set, and the two-hop neighbor nodes connected by the one-hop neighbor node are marked in the set S, if not found, the smallest element in the set F, i.e. the one-hop neighbor node, is first deleted, then it is judged whether the union set of the two-hop neighbor nodes corresponding to the one-hop neighbor node in the set F is equal to the two-hop neighbor nodes not marked in the set S, if the union set of the two-hop neighbor nodes corresponding to the one-hop neighbor node is equal to the two-hop neighbor nodes not marked in the set S, the one-hop neighbor node not meeting the judgment condition is directly deleted from the set F, and the two-hop neighbor nodes connected by the deleted node are marked in the set S, and each time a set F stage is moved to the MPR node set, a union set operation of the MPR node is needed, and then it is judged whether the two-hop neighbor nodes connected by the MPR node after the union set operation are completely the same as the two-hop neighbor nodes of the set S, if the judgment result is the same, the process is ended;

[0036] In step 2.2.5, step 2.2.4 is repeated until all elements in the MPR node set, i.e. nodes, are completely traversed.

[0037] The further improvement of the present application is that in step 5, the MPR node uses the local broadcast and MPR forwarding method to flood the TC message to the multi-dimensional identification network, the node completes the grasp of the topology of the multi-dimensional identification network by receiving the TC message, and the MPR node encapsulates the link information between the MPR node and the MS node into the TC message and sends it to a specific NID domain.

[0038] The further improvement of the present application is that the HELLO packet includes a plurality of Link control messages, and the links between the neighbor nodes are grouped according to the link state and the neighbor type, and a group of links is included in a Link control message.

[0039] The beneficial effects of the present application are:

[0040] The present application improves the success rate of data transmission: the present application uses the cycle and set operation to optimize the selection of MPR set, eliminates invalid redundant nodes, and can improve the success rate of data transmission.

[0041] The present application reduces network overhead: the present application reduces network overhead by reducing unnecessary message forwarding. This is because the new method can more accurately select the nodes that need to be forwarded, thereby reducing redundant forwarding operations.

[0042] The present application optimizes the size of the MPR set: when selecting MPR nodes, the present application considers the connectivity of the nodes, so as to select the optimal MPR set, which helps to reduce the size of the MPR set and further improve the network efficiency.

[0043] The present application improves the stability of the network: the present application ensures that each selected MPR node has stable connection in a high mobility environment by screening the spatial distribution of the candidate nodes. BRIEF DESCRIPTION OF DRAWINGS

[0044] Figure 1 is a flowchart of the present application.

[0045] Figure 2 is a flowchart of the MPR set election method of the present application.

[0046] Figure 3 is an example topology of the MPR election method of the present application.

[0047] Figure 4 is a diagram of the MPR set size of the MPR election method of the present application and the traditional MPR node election method.

[0048] Figure 5 is a diagram of the data transmission success rate of the MPR election method of the present application and the traditional MPR node election method. DETAILED DESCRIPTION

[0049] The embodiments of the present application will be described below with reference to the drawings. For the purpose of clear description, many practical details will be described in the following description. However, it should be understood that these practical details are not used to limit the present application. That is, in some embodiments of the present application, these practical details are unnecessary.

[0050] As Figure 1As shown, the application is a multi-dimensional identification based domain-aware MPR election method, which specifically includes the following steps:

[0051] Step 1, in a specific multi-dimensional identification NID domain, nodes discover the same domain neighbor nodes by periodically broadcasting and receiving HELLO packets, the HELLO packets contain one-hop neighbor node geographic coordinate information and multi-dimensional identification NID information and two-hop neighbor node geographic coordinate information and multi-dimensional identification NID information, so that the neighbor node discovery process has identity and domain awareness, and the multi-dimensional identification NID information includes access network nodes, core network nodes, vehicle network nodes and terminal network nodes.

[0052] Step 2, a part of the same domain neighbor nodes discovered based on the HELLO packet is elected as the MS node set of the MPR node election, the metric of the MPR node election includes the coverage of the node, the node connection degree and the distribution stability, i.e. the neighbor number variance. For example, Figure 2 As shown, the MS node set election method specifically includes the following steps:

[0053] Step 2.1, multi-dimensional screening is performed in all one-hop neighbor nodes: based on the geographic coordinate information and multi-dimensional identification NID information of the one-hop neighbor nodes, high-quality nodes with high coverage and stable distribution are screened out to form a high-quality node candidate pool.

[0054] Specifically, starting from any node , the node discovered the one-hop neighbor set and two-hop neighbor set of the node through the HELLO packet, and obtains and stores the geographic coordinate information and multi-dimensional identification NID information of all one-hop neighbor nodes through the HELLO packet; all one-hop neighbor nodes are traversed, and the quality of the neighbor nodes of each one-hop neighbor set is calculated :

[0055]

[0056]

[0057]

[0058] For each one-hop neighbor node , the neighbors located in the four regions determined by the angle are defined as follows:

[0059] : wherein ​neighbors of

[0060] wherein neighbors of

[0061] wherein neighbors of

[0062] wherein neighbors of

[0063] wherein, is a weight factor, dynamically adjusted according to network policy, , , and is the weight assigned to each region, is the weight assigned to each region, , is the neighbor coverage of the neighbor node, is the variance of the number of neighbors in the four regions, i.e. the distribution stability, and the node with a higher value and a lower variance will be given priority, is the node connection score, which is used to determine whether to forward according to the multi-dimensional identifier NID information, and the node with a high quality value is included in the high-quality node candidate pool, i.e. the HQ-POOL set.

[0064] Step 2.2, according to the number of two-hop neighbor nodes that can be connected by the high-quality nodes in the high-quality node candidate pool in step 2.1, sort: select the one-hop neighbor node that can cover the isolated two-hop neighbor node in the high-quality node candidate pool as the required node of the high-quality node candidate pool, the isolated two-hop neighbor node refers to the two-hop neighbor node connected to the target node through only one neighbor node. Specifically: step 2.2.1, for the node set of MPR to be calculated, all one-hop neighbors in the high-quality node candidate pool are regarded as set F, and the elements of set F, i.e. the one-hop neighbor nodes, connect all two-hop neighbor nodes, which are the elements of set S;

[0065] Step 2.2.2, if set F is empty, wait until the topology changes, if F is not empty, use the letter …to number the elements in set F, and use the numbers 1, 2, 3, 4…to number the elements in set S, and set the initial value of the MPR node set to empty;

[0066] Step 2.2.3, according to the number of two-hop neighbor nodes in set S connected by the one-hop neighbor nodes in set F, sort set F from large to small;

[0067] Step 2.2.4, in the ordered set F, first find a two-hop neighbor node which only belongs to one one-hop neighbor node, then add the one-hop neighbor node connected to the two-hop neighbor node from set F to the MPR node set, and mark the two-hop neighbor node connected to the one-hop neighbor node in set S, if not found, first assume to delete the minimum element in set F, i.e. the one-hop neighbor node, then judge whether the union set of the two-hop neighbor nodes corresponding to the one-hop neighbor nodes in set F is equal to the unmarked two-hop neighbor nodes in set S, if the union set of the two-hop neighbor nodes corresponding to the one-hop neighbor nodes is equal to the unmarked two-hop neighbor nodes in set S, then directly delete the one-hop neighbor node which does not meet the judgment condition from set F, and mark the two-hop neighbor nodes connected to the deleted node in set S, and each time the set F is moved to the MPR node set, the union set of the MPR nodes needs to be operated, and then it is judged whether the two-hop neighbor nodes connected by the MPR nodes after the union set operation are completely the same as the two-hop neighbor nodes in set S, if the judgment result is the same, the process ends;

[0068] Step 2.2.5, repeat step 2.2.4 until all elements in the MPR node set, i.e. nodes, are completely traversed.

[0069] Step 2.3, after the selection of the mandatory nodes, select the MS node set in the remaining one-hop neighbor nodes until all two-hop neighbor nodes are covered.

[0070] Step 3, after the neighbor nodes are elected as MPR nodes, identify the nodes elected by the received HELLO packet, and maintain the MS node set;

[0071] Step 4, the elected MPR node encapsulates the link information between the MPR node and each node in the MS node set into TC (Topology Control) information;

[0072] Step 5, the MPR node uses the MPR forwarding method to flood the TC message encapsulated in step 4 to the entire NID domain where the TC message is located. The key topology information is effectively diffused, the MPR node uses the local broadcast and MPR forwarding method to flood the TC message to the multi-dimensional identification network, and the node completes the grasp of the multi-dimensional identification network topology by receiving the TC message. The MPR node encapsulates the link information between the MPR node and the MS node into the TC message and sends it to the specific NID domain.

[0073] Embodiment

[0074] The one-hop neighbor nodes in the HQ-POOL set are used as 、 , 、 、 、 The one-hop neighbors are numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively, and the two-hop neighbors are numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively. The MPR set is defined as an empty set. F is the set of all one-hop neighbors, and S is the set of all two-hop neighbors.

[0075] Count the number of elements according to the number of elements 、 、 、 、 and , and sort them in descending order.

[0076]

[0077] First, traverse the one-hop neighbor set to find whether there is a node whose corresponding two-hop neighbor node can only be covered by the node; if the condition is met, add the node to the MPR set and merge it with the existing elements in the MPR. Subsequently, mark the covered nodes in the two-hop neighbor set S as covered, and remove the node from the one-hop neighbor set F to ensure that it no longer participates in the subsequent steps.

[0078]

[0079]

[0080]

[0081] After ignoring the marked two-hop neighbors, check whether there is a two-hop neighbor node that can only be covered by one one-hop neighbor node.

[0082]

[0083] In this check, no two-hop neighbor node is found that can only be covered by one one-hop neighbor node. Therefore, remove the element with the lowest priority from the sorted one-hop neighbor set F. After removal, the set F is updated to { , , , }, that is, the union of , , , .

[0084] ​

[0085] So can be removed directly.

[0086] S6: Ignore the marked two-hop neighbor nodes, and determine whether there is a two-hop neighbor node that only belongs to a unique one-hop neighbor.

[0087]

[0088] The one-hop neighbor node The two-hop neighbor nodes 3 and 4 of the one-hop neighbor node can only be covered. Therefore, first, add to the MPR set, and remove from the one-hop neighbor set F. Mark the two-hop neighbor nodes 3, 4, 5, 6, and 7 related to as covered. Finally, find the union of the two-hop neighbor nodes covered by all nodes in the MPR set, and determine whether the union is equal to the two-hop neighbor set S, to verify whether all two-hop neighbor nodes have been completely covered.

[0089]

[0090] Because continue to calculate.

[0091] After ignoring the marked two-hop neighbor nodes, check whether there is a two-hop neighbor node that is not covered and can only be covered by a unique one-hop neighbor node.

[0092]

[0093] Currently, no two-hop neighbor node that can only be covered by one one-hop neighbor node is found. According to the sorting result, remove the element with the lowest priority from the one-hop neighbor set F. After removal, the set F is updated to , , that is, the union of and .

[0094]

[0095] Because , so can be removed.

[0096] After ignoring the marked two-hop neighbor nodes, check whether there is a two-hop neighbor node that is not covered and can only be covered by a unique one-hop neighbor node.

[0097]

[0098] One-hop neighbor node The two-hop neighbor nodes 11 and 12 can only be Coverage. Therefore, Add to the MPR set and remove from the one-hop neighbor set F. Next, the markers and... The relevant two-hop neighbor nodes 8, 9, 10, 11, and 12 are in a covered state. Finally, the union of the two-hop neighbor nodes covered by all nodes in the MPR set is calculated, and it is determined whether the union is equal to the two-hop neighbor set S to verify whether all two-hop neighbor nodes have been completely covered.

[0099]

[0100] because

[0101] Therefore, it covers all two-hop neighbor nodes, and the method ends.

[0102] This application compares its performance with that of the traditional MPR node election method, and the comparison results are as follows: Figure 4 and Figure 5 As shown. Compared to traditional algorithms, this application achieves a smaller final set of elected MPR nodes and a shorter algorithm execution time under the same topology.

[0103] The above description is merely an embodiment of this application and is not intended to limit this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principle of this application should be included within the scope of the claims of this application.

Claims

1. A domain-aware MPR election method based on multidimensional identifiers, characterized in that: The domain-aware MPR election method specifically comprises the following steps: Step 1, in an NID domain, nodes discover same-domain neighbor nodes by periodically broadcasting and receiving HELLO packets, wherein the HELLO packets contain geographic coordinate information and multi-dimensional identification NID information of one-hop neighbor nodes and geographic coordinate information and multi-dimensional identification NID information of two-hop neighbor nodes; Step 2, a part of the same-domain neighbor nodes discovered based on the HELLO packets are elected as an MS node set, i.e., a multi-point relay selection node set; Step 3, after the neighbor nodes are elected as MPR nodes, the MPR nodes identify the elected nodes by received HELLO packets and maintain the MS node set; Step 4, the elected MPR nodes encapsulate link information between the MPR nodes and each node in the MS node set into TC information; Step 5, the MPR nodes flood the TC information encapsulated in step 4 to the entire NID domain where the TC information is located by using an MPR forwarding method, wherein: In step 2, the MS node set election method specifically comprises the following steps: Step 2.1, multi-dimensional screening is performed on all one-hop neighbor nodes: high-quality nodes are screened out based on geographic coordinate information and multi-dimensional identification NID information of the one-hop neighbor nodes to form a high-quality node candidate pool; Step 2.2, the high-quality nodes in the high-quality node candidate pool are sorted according to the number of two-hop neighbor nodes that can be connected by the high-quality nodes: one-hop neighbor nodes in the high-quality node candidate pool that can cover isolated two-hop neighbor nodes are selected as mandatory nodes of the high-quality node candidate pool, wherein the isolated two-hop neighbor nodes refer to two-hop neighbor nodes that are connected to a target node through only one neighbor node; Step 2.3, after the mandatory nodes are screened out, selection of the MS node set is performed on the remaining one-hop neighbor nodes until all two-hop neighbor nodes are covered. 2.The multi-dimensional identifier based domain-aware MPR election method according to claim 1, characterized in that: In step 1, the multi-dimensional identification NID information comprises access network nodes, core network nodes, vehicle-mounted network nodes and terminal network nodes. 3.The multi-dimensional identity based domain-aware MPR election method according to claim 1, characterized in that: In step 2, metrics for election of the MPR nodes include coverage of the nodes, node connection degree and distribution stability. 4.The multi-dimensional identifier based domain-aware MPR election method of claim 1, wherein: Step 2.1 specifically comprises the following steps: Step 2.1.1, starting from an arbitrary node N, the node N discovers a one-hop neighbor set N1 and a two-hop neighbor set N2 of the node N through HELLO packets and obtains and stores geographic coordinate information and multi-dimensional identification NID information of all one-hop neighbor nodes through the HELLO packets; Step 2.1.2, iterate over all one-hop neighbor nodes and calculate the quality Q(N of each one-hop neighbor set N1 i ): Q(N i ) = a · C(N i ) + β · (1 / D(N i )) + δ · G(N i ) C(N i ) = w1|V i1 | + w2|V i2 | + w3|V i3 | + w4|V i4 | D(N i ) = Var(|V i1 |, |V i2 |, |V i3 |, |V i4 |) For each one-hop neighbor node X i , define the neighbors V ij : located in the four regions determined by the angle θ V i1 : a neighbor where 0° < Θ < 90°. V i2 : a neighbor where 90° < θ < 180°. V i3 : a neighbor where 180° < θ < 270°; V i4 : a neighbor with 270° < Θ < 360°. wherein a, b, g, d are weight factors, V i1 , V i2 , V i3 and V i4 are four areas, w j is the weight assigned to each area, j e [1, 2, 3, 4], C(N i ) is the neighbor coverage of the neighbor nodes, D(N i ) is the variance of the number of neighbors in the four areas, i.e. the distribution stability, G(N i ) is the node connectivity degree, and the nodes with high values of quality Q(N i ) are included in the high-quality node candidate pool, i.e. the HQ-POOL set.

5. The method of claim 1, wherein: In step 2.2, specifically comprises the following steps: Step 2.2.1, for a node to be calculated in the MPR node set, all one-hop neighbors in the high-quality node candidate pool are regarded as a set F, and elements of the set F, i.e., one-hop neighbor nodes, connect all elements of a set S, i.e., two-hop neighbor nodes; Step 2.2.2, if the set F is empty, waiting until the topology changes, if the set F is not empty, elements in the set F are numbered by letters and elements in the set S are numbered by numbers, and an initial value of the MPR node set is set as empty; Step 2.2.3, sort the set F from large to small according to the number of two-hop neighbor nodes connected by one-hop neighbor nodes in set S in set F; Step 2.2.4, in the sorted set F, first find a two-hop neighbor node belonging to only one one-hop neighbor node, then add the one-hop neighbor node connected by the two-hop neighbor node to the MPR node set from set F, and mark the two-hop neighbor node connected by the one-hop neighbor node in set S, if not found, first assume to delete the minimum element in set F, that is, the one-hop neighbor node, then judge whether the union set of the two-hop neighbor nodes corresponding to the one-hop neighbor node in set F is equal to the two-hop neighbor nodes not marked in set S, if the union set of the two-hop neighbor nodes corresponding to the one-hop neighbor node is equal to the two-hop neighbor nodes not marked in set S, then directly delete the one-hop neighbor node not meeting the judgment condition from set F, and mark the two-hop neighbor nodes connected by the deleted node in set S, and each time the set F in the stage is moved to the MPR node set, the union set operation of the MPR node needs to be performed, then judge whether the two-hop neighbor nodes connected by the MPR node after the union set operation are completely the same as the two-hop neighbor nodes in set S, if the judgment result is the same, then end; Step 2.2.5, repeat step 2.2.4 until all elements in the MPR node set are completely traversed.

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