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Internet Engineering Task Force                                  S. Chen
Internet-Draft                                                     T. Li
Intended status: Informational                           Arista Networks
Expires: 4 September 2020                                   3 March 2020


         An Algorithm for Computing Dynamic Flooding Topologies
              draft-chen-lsr-dynamic-flooding-algorithm-00

Abstract

   Link-state routing protocols suffer from excessive flooding in dense
   network topologies.  Dynamic flooding [I-D.ietf-lsr-dynamic-flooding]
   alleviates the problem by decoupling the flooding topology from the
   physical topology.  Link-state protocol updates are flooded only on
   the sparse flooding topology while data traffic is still forwarded on
   the physical topology.

   This document describes an algorithm to obtain a sparse subgraph from
   a dense graph.  The resulting subgraph has certain desirable
   properties and can be used as a flooding topology in dynamic
   flooding.

   This document discloses the algorithm that we have developed in order
   to make it easier for other developers to implement similar
   algorithms.  We do not claim that our algorithm is optimal, rather it
   is a pragmatic effort and we expect that further research and
   refinement can improve the results.

   We are not proposing that this algorithm be standardized, nor that
   the working group use this as a basis for further standardization
   work, however we have no objections if the working group chooses to
   do so.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
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   Drafts is at https://datatracker.ietf.org/drafts/current/.

   Internet-Drafts are draft documents valid for a maximum of six months
   and may be updated, replaced, or obsoleted by other documents at any
   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."



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   This Internet-Draft will expire on 4 September 2020.

Copyright Notice

   Copyright (c) 2020 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents (https://trustee.ietf.org/
   license-info) in effect on the date of publication of this document.
   Please review these documents carefully, as they describe your rights
   and restrictions with respect to this document.  Code Components
   extracted from this document must include Simplified BSD License text
   as described in Section 4.e of the Trust Legal Provisions and are
   provided without warranty as described in the Simplified BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
   2.  Problem Statement . . . . . . . . . . . . . . . . . . . . . .   3
   3.  Algorithm Outline . . . . . . . . . . . . . . . . . . . . . .   3
   4.  Algorithm Details . . . . . . . . . . . . . . . . . . . . . .   4
     4.1.  Initial Cycle Setup . . . . . . . . . . . . . . . . . . .   4
     4.2.  Arc Path Selection  . . . . . . . . . . . . . . . . . . .   5
     4.3.  Exceptions  . . . . . . . . . . . . . . . . . . . . . . .   6
     4.4.  LANs  . . . . . . . . . . . . . . . . . . . . . . . . . .   7
   5.  Example . . . . . . . . . . . . . . . . . . . . . . . . . . .   7
   6.  Security Considerations . . . . . . . . . . . . . . . . . . .   9
   7.  Informative References  . . . . . . . . . . . . . . . . . . .   9
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .   9

1.  Introduction

   In [I-D.ietf-lsr-dynamic-flooding], dynamic flooding is proposed to
   reduce the flooding of link-state protocol packets in the network.
   The basic idea is to find a sparse flooding topology from the
   physical topology and flood link-state protocol data units (LSPDUs or
   LSPs) only on the flooding topology.  The flooding topology should
   have the following properties:

   1.  It should include all nodes in the area.  This ensures that LSPs
       can reach all nodes.

   2.  It should be biconnected if possible.  This ensures that the LSP
       delivery is resilient to a single node or link failure.

   3.  It has a limited diameter.  Being a subgraph, the flooding
       topology often has a larger diameter than the physical topology.



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       A larger diameter indicates a longer convergence time.  The
       tradeoff between flooding reduction and convergence should be
       considered during the flooding topology computation.

   4.  It has a balanced degree of distribution.  The degree of a node
       on the flooding topology indicates its burden in flooding LSPs.
       It is desirable to balance this burden across multiple nodes.
       Hence, the degree of each node should also be considered during
       flooding topology computation.

   With the above properties in mind, we propose an iterative algorithm
   to compute the flooding topology.

2.  Problem Statement

   We model the physical topology as an undirected graph.  Each system
   or pseudonode in the area is represented by a node in the graph.  An
   edge connects two nodes who advertise each other as neighbors.  Given
   the set of the nodes and the set of edges, we propose an algorithm to
   compute a biconnected (if possible) subgraph that covers all nodes.
   The subgraph is computed with consideration of diameter and node
   degree.

3.  Algorithm Outline

   A simple cycle that covers all nodes is a biconnected subgraph with
   balanced node degrees.  While it has some desirable properties, a
   simple cycle is not suitable as a flooding topology at large scale.
   With N nodes in the area, a link-state update has to take N/2 hops to
   reach all nodes.  The undue propagation delay causes a long
   convergence time.

   The proposed algorithm constructs a subgraph composed of small
   overlapping cycles.  The base graph is denoted by G(V, E), where V is
   the set of all reachable nodes in this area, and E is the set of
   edges.  The subgraph to be computed is denoted by G'({}, {}), which
   starts with an empty set of nodes and an empty set of edges.

   1.  Select a subset of nodes V(0) from V and a subset of edges E(0)
       from E to form an initial cycle.  This cycle is added to the
       subgraph: G'(V(0), E(0)).

   2.  Select a subset of nodes V(i) and a subset of edges E(i), that
       are not included in the current subgraph.  That is, V(i) is
       selected from V - V(0) - ... - V(i-1) and E(i) is selected from E
       - E(0) - ... - E(i-1).  These nodes and the edges are selected to
       form an arc path whose two endpoints are included in the current




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       subgraph.  This arc path is added to the subgraph: G'( V(0) +
       V(1) + ... + V(i), E(0) + E(1) + ... + E(i) ).

   3.  Repeat step 2 until all nodes are included in the subgraph G'.

   The subgraph constructed by this algorithm has the following
   properties:

   1.  It covers all nodes in the area.

   2.  It is biconnected.  This can be easily proven by induction.
       Specifically, the initial cycle is biconnected.  Adding an arc
       path, whose two endpoints differ, to a biconnected subgraph
       maintains the biconnectivity of the subgraph.

   3.  It has a limited diameter.  By selecting small cycles, the
       subgraph will have a smaller diameter.  More specifically, the
       implementation can pick endpoints of each arc path to reduce the
       diameter of the subgraph.  The degree of a node in the subgraph
       is determined by the number of arc paths it is on.  By carefully
       selecting the arc endpoints, we may balance the node degrees.

   Together with the encoding scheme in [I-D.ietf-lsr-dynamic-flooding],
   this algorithm can be used to implement centralized dynamic flooding.
   The area leader can build the base graph from its link-state database
   (LSDB), apply this algorithm to compute the flooding topology, and
   then encode it into the Dynamic Flooding Path TLVs specified in
   [I-D.ietf-lsr-dynamic-flooding].  In a topology change event, the
   area leader can repeat the above process and send out the new
   flooding topology.

4.  Algorithm Details

   The outlined algorithm allows for different approaches to find the
   initial cycle and subsequent arc paths.  We do not intend to find the
   theoretically optimal solution.  Our aim is to find a practical
   approach that works for any connected base graph, and is also easy to
   implement.

4.1.  Initial Cycle Setup

   The initial cycle forms a base of the subgraph computation.
   Intuitively, we would like to place the initial cycle around the
   centroid of the base graph, and gradually expand it outwards.
   Complicated graph analysis can help, but is not desired.

   We propose to select a starting node and then search a path that ends
   at this node.  We suggest selecting the node with the highest degree



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   as the starting node.  The degree of each node can be easily
   determined when the base graph is constructed from the LSDB.
   Starting from this node, we perform a depth-first search (DFS) for a
   limited number of steps, and then a breadth-first search (BFS) to
   find the shortest path back to the starting node.  The restriction of
   the DFS depths and the use of BFS effectively help limit the diameter
   of the initial cycle.  Below is a summary of the procedure.  We omit
   the details of the well-known DFS and BFS algorithms.

   1.  Let V(0) = [] be the list of nodes on the initial cycle.

   2.  Find the starting node, denoted by n0.  V(0) = [n0].

   3.  Perform DFS starting from n0 until either the first leaf is
       reached or the depth exceeds a preset limit.  The visited nodes
       are denoted by n1, n2, ..., ni, where i < DFS depth limit, and
       added to the list V(0) in order.

   4.  Mark nodes in V(0) as visited to avoid BFS visiting these nodes.
       Then perform BFS to find the shortest path from ni to n0.  Append
       nodes on the shortest path to V(0).  Since this is a cycle, node
       n0 appears twice in V(0): the first and the last places.

   Note that since DFS and BFS do not process a node more than once, we
   are ensured to obtain a cycle (if one exists) from the above
   procedure.

4.2.  Arc Path Selection

   After obtaining an initial cycle, we recursively add arc paths to the
   subgraph until all nodes are included.  Each arc path's two endpoints
   are chosen from the current subgraph.  This ensures that the
   resulting subgraph remains biconnected.  To limit the diameter of the
   resulting subgraph, we select an arc path with limited length and
   attach it closer to the initial cycle.

   In each iteration, we have a starting subgraph, which includes the
   initial cycle and the arc paths obtained in earlier iterations.  We
   first select a node from the starting subgraph, that has at least one
   neighbor that is not in the starting subgraph.  To balance the degree
   distribution, we prefer to select a node that has the least degree in
   the starting subgraph.  Meanwhile, we intend to place the new arc
   path closer to the initial cycle, which will help reduce the diameter
   of the resulting subgraph.  As the number of iterations increases, it
   becomes hard to find such nodes that meet both conditions.  A
   tradeoff between the node degree and the node distance to the initial
   cycle has to be made.




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   Starting from the selected node, we perform a DFS for a limited
   number of steps in the base graph to include nodes and edges that do
   not belong to the starting subgraph.  Then BFS is performed to find
   the shortest path back to any node in the starting subgraph except
   the starting node of this iteration.  The resulting path is combined
   with the starting subgraph to generate a new subgraph, which serves
   as the starting subgraph in the next iteration.  The iteration is
   repeated until all nodes in the base graph are included in the
   subgraph.

   The procedure in each iteration is very similar to the one used to
   find the initial cycle, except that the two endpoints of the new path
   do not match:

   1.  Let V(i) = [] be the list of nodes found in the i-th iteration.
       The starting subgraph includes nodes in V(0) + ... + V(i-1), and
       edges between each pair of adjacent nodes in each node list V(j).

   2.  Select a starting node n0 for this iteration from the starting
       subgraph.  V(i) = [n0].

   3.  Mark all nodes in V(0) + ... + V(i-1) as visited.  Then perform
       DFS from n0 until either the first leaf is reached or the depth
       exceeds a preset limit.  Nodes visited in this iteration are
       denoted by n1, n2, ..., ni in order, and appended to the list
       V(i).

   4.  Mark all nodes in V(0) + ... + V(i-1) + V(i) as visited.  Then
       perform BFS to find the shortest path from ni to any node in V(0)
       + ... V(i-1) - [n0], where n0 is the starting node in this
       iteration.  If a path is found, append its nodes to V(i) and
       repeat the iteration from step 1.

   By correctly marking the visited nodes before DFS and BFS, we ensure
   that the obtained arc path (if it exists) has two endpoints and only
   these two points on the starting subgraph.

4.3.  Exceptions

   If the base graph is biconnected, there exists a simple cycle between
   any two nodes.  We are thus ensured to find one arc path in each
   iteration and the algorithm described above will yield a biconnected
   subgraph that covers all nodes in the base graph.  Otherwise,
   however, we may not be able to find an arc path with both endpoints
   belonging to the starting subgraph in that iteration.  When this
   happens, we know that the edge between the last node found by DFS and
   its parent is a cut edge in the base graph.  For connectivity, this
   edge must be included in the resulting subgraph.  Hence, when step 4



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   (the BFS stage in earlier procedures) fails, we should amend it with
   the following:

   5.  If BFS does not find a shortest path from ni, and V(i) contains
       only two nodes, i.e., the cut edge case, then go back to step 1
       to continue the iteration.

   6.  If BFS does not find a shortest path from ni, and V(i) contains
       more than two nodes, then remove the last node from V(i), and go
       back to step 4.

   Similarly, we might face the same problem when selecting the initial
   cycle.  We can apply step 6 until we find a cycle.  However, if we
   happen to find a cut edge, we can change the first neighbor of the
   starting node that is visited by the DFS and repeat the procedure.
   If all edges connecting to the starting node are cut edges, we can
   change the starting node.  If an initial cycle is not found after all
   the above efforts, indicating that the base graph does not have a
   cycle, then we will return the base graph as the result.

4.4.  LANs

   We model a pseudonode as a node in the base graph.  The proposed
   algorithm can be applied as-is.  There are, however, possible
   optimizations for the multi-access LAN case.  First, a pseudonode is
   not required to be on the flooding topology.  The algorithm can thus
   be terminated as soon as all real nodes are included in the subgraph.
   Second, if a pseudonode is included on the flooding topology, all
   nodes connecting to this LAN will have to flood their LSPs to this
   LAN (see [I-D.ietf-lsr-dynamic-flooding] Section 6.6).  Hence, if a
   pseudonode is included in the subgraph, then it will automatically
   provide uni-connectivity to all its neighbors that are not yet
   included.  The algorithm can take advantage of this LAN property to
   reduce the edges in the subgraph.

5.  Example

   The proposed algorithm can be applied to any connected base graph.
   For ease of explanation, we consider a complete graph of 10 nodes and
   45 edges.  To limit the diameter of the resulting subgraph, we pre-
   set the maximum steps in the DFS to 3.

   1.  Let ni, i = 0, 1, ... 9, denote nodes in the base graph.

   2.  Find an initial cycle.

       a.  Without loss of generality, we select node n0 as the starting
           point.



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       b.  Perform DFS from n0 for 3 steps.  We obtain a path n0 - n1 -
           n2 - n3.

       c.  Perform BFS from n3 to n0.  Since this is a complete graph,
           where every node is directly connected to any other node, the
           shortest path is only one hop away.

       d.  The initial cycle is found n0 - n1 - n2 - n3 - n0.

   3.  Find the first arc path.

       a.  Select a node on the initial cycle, say n0.

       b.  Perform DFS from n0 for 3 steps.  We obtain a path n0 - n4 -
           n5 - n6.

       c.  Perform BFS from n6 to any node on the initial cycle except
           n0, i.e., {n1, n2, n3}. These three nodes have the same
           degree.  We may select any one of them as the endpoint.
           Suppose that n1 is selected.

       d.  The first arc path is found n0 - n4 - n5 - n6 - n1.

   4.  Find the second arc path.

       a.  When selecting the starting node for this path, we may
           consider the current node degree as well as the node distance
           to n0 (the starting point of the initial cycle).  We notice
           that both n0 and n1 have a degree of 3 while other nodes have
           a degree of 2.  Nodes n1, n3, n4 are the closest to n0.  We
           select a node with a lower degree and closer to n0.  Suppose
           n3 is selected.

       b.  Perform DFS from n3 for 3 steps.  We obtain a path n3 - n7
           -n8 - n9.

       c.  Perform BFS from n9 to any node except n3.  Using the same
           criteria in a. to select the endpoint, we select n4.

       d.  The second arc path is found n3 - n7 -n8 - n9 - n4.

   5.  The subgraph has included all nodes.  The iteration ends.

   The subgraph found by the proposed algorithm can be represented by
   three paths:

      n0 - n1 - n2 - n3 - n0




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      n0 - n4 - n5 - n6 - n1

      n3 - n7 -n8 - n9 - n4

   The subgraph has 12 edges, significantly reduced from 45 in the base
   graph.  The highest node degree is 3 and the lowest node degree is 2.
   The diameter of the subgraph is 4, increased, as expected, from that
   of the base graph.

6.  Security Considerations

   This document introduces no new security issues.  Security issues
   within dynamic flooding are already discussed in
   [I-D.ietf-lsr-dynamic-flooding].

7.  Informative References

   [I-D.ietf-lsr-dynamic-flooding]
              Li, T., Psenak, P., Ginsberg, L., Chen, H., Przygienda,
              T., Cooper, D., Jalil, L., and S. Dontula, "Dynamic
              Flooding on Dense Graphs", Work in Progress, Internet-
              Draft, draft-ietf-lsr-dynamic-flooding-04, 26 November
              2019, <https://tools.ietf.org/html/draft-ietf-lsr-dynamic-
              flooding-04>.

Authors' Addresses

   Sarah Chen
   Arista Networks
   5453 Great America Parkway
   Santa Clara, California 95054
   United States of America

   Email: sarahchen@arista.com


   Tony Li
   Arista Networks
   5453 Great America Parkway
   Santa Clara, California 95054
   United States of America

   Email: tony.li@tony.li








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