1 Efficient Trie Braiding in Scalable Virtual Routers Author: Haoyu Song, Murali Kodialam, Fang Hao, T.V. Lakshman Publisher: IEEE/ACM TRANSACTIONS ON NETWORKING Presenter: Zi-Yang Ou Date: 2012/10/17

Introduction 2 This paper proposes a mechanism called trie braiding that can be used to combine tries from different virtual routers into just one trie. Trie braiding enables each trie node to swap its left child node and right child node freely. The changed shape is memorized by a single bit at each trie node. Two optimal dynamic programming algorithms and a faster heuristic algorithm are presented. Trie braiding leads to significant savings in high-speed memory, and hence improves scalability.

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Trie Braiding In Fig. b, we swap node a’s child nodes and node c’s child nodes. 4 a cb d ef a cb d ef

Problem Formulation Definition 1: A mapping M of T 2 to T 1 Definition 2: A permissible mapping 5

Definition 3: Definition 4: Distance 6

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Braid : A DP Algorithm Input : T 1 and T 2 Output : 8

Step1: Compute Leaf Weights Step2: Distance Computation Step3: It starts from r 2 and uses S(, ) to obtain the optimal mapping and braiding bit for each node v 2. 9

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FAST-BRAID: Braiding With Isomorphism Detection The motivation is the result in Lemma 1. If we can identify the fact that, then we need to compute the value of only once. Using the technique developed by [26] in linear time algorithm for tree isomorphism to keep track of the nodes with isomorphic subtrees. We first process each tree separately. Two nodes will be given the same label if and only if the subtrees rooted at those two nodes are isomorphic. In the FAST-BRAID algorithm, we have to compute the value of for two labels, each from a different tree. 12

k-Braid: A k-Step Lookahead Heuristic for Braiding This heuristic algorithm determines the mapping from root to leaves. 13

Combining Multiple Trees So far, we have dealt with the problem of combining two trees. If we want to combine more than two trees, then the running time of the optimal algorithm grows exponentially with the number of trees. We use an incremental approach where we first merge T 1 and T 2 and then merge T i (i >= 3) onto combined tree. 14

Algorithm Evaluation 15

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