Proposal of new steiner tree algorithm applied to P2MP traffic engineering

Abstract

There are many polynomial-time heuristic Steiner tree algorithms since seeking the minimum point to multi point (P2MP) tree in a network, which is known as the Steiner problem in networks (SPN), is nondeterministic polynomial time complete. Takahashi and Matsuyama's minimum-cost path heuristic algorithm (MPH) is widely applied to various multicast services. MPH has to run Dijkstra's algorithm m times in its algorithm process, where m is the number of end nodes of the multicast tree. + n log n)). By using Fibonacci heaps (F-heaps), the time complexity of MPH is O(m(l + n log n)), where l is the number of links and n is the number of nodes on the network. A new Steiner tree algorithm called branch-based multi-cast (BBMC) is proposed for this study, which produces exactly the same P2MP tree as MPH does. BBMC shortens MPH's average time complexity to O((log m)(l In addition, the algorithm speed of BBMC is much faster than that of MPH because BBMC does not use Dijkstra's algorithm in its algorithm process and drastically reduces the number of accesses to the F-heaps. MPH's processing time increasesproportionally to m, especially when m is large, but BBMC's processing time is almost independent from m. The speed of BBMC is about the same as that of the destination-driven multi-cast algorithm (DDMC), which has a time complexity of O(l + n log n), though BBMC produces a P2MP tree with a tree cost much smaller than that of DDMC.

Improvement of Steiner Tree Algorithm: Branch-Based Multi-Cast

“…In the experimental results, BBMC's Steiner tree costs were 10 to 21% smaller than DDMC's, though BBMC was as fast as DDMC in terms of algorithm speed [17]. This average case time complexity equality between BBMC and DDMC authenticates these results.…”

“…For example, BBMC's Steiner tree costs were 10 to 21% smaller than those produced by DDMC, which is another Steiner tree algorithm, in the experimental result [17]. DDMC is similar to BBMC except that DDMC only initializes a branch candidate from a new branch destination, namely bn2 in Fig.…”

“…The author recently proposed a Steiner tree algorithm named branch-based multi-cast (BBMC), which produces exactly the same multicast tree of MPH in a constant processing time irrespective of the number of multicast end nodes m in the experimental result [17]. Therefore, when m became larger, the processing time of BBMC was less than one tenth that of MPH.…”

“…In addition, the processing time of BBMC was almost the same as that of another Steiner tree algorithm named destination-driven multi-cast (DDMC) [8], whose average case time complexity is O(l + n log n). However, the author only proved that BBMC has the average case time complexity of O(log m(l + n log n)) [17], which is still dependent on m and does not reflect the experimental result of m-independency in processing time. Thus, this paper proves that BBMC has the average case time complexity of O(l + n log n) on the condition there is an upper limit of the node degree, which is the number of links connected to a node.…”

“…MPH calls Dijkstra's algorithm as many times as there are multicast end nodes because MPH determines the shortest branch from each current tree, which means the multicast tree created by the algorithm to date, to the multicast end nodes whose "reach" is still "no". Dijkstra's algorithm uses path queue (PQ) to register a new branch candidate by using the "insert" or "replace" function and to remove the shortest candidate from PQ by using the "delete-min" function [17]. PQ has two different forms to hold branch candidates: one is F-heaps and the other is a binary search tree, so either one is chosen.…”