By Michael R. Fellows (auth.), Susanne Albers, Tomasz Radzik (eds.)
This booklet constitutes the refereed court cases of the twelfth Annual eu Symposium on Algorithms, ESA 2004, held in Bergen, Norway, in September 2004.
The 70 revised complete papers awarded have been conscientiously reviewed from 208 submissions. The scope of the papers spans the total diversity of algorithmics from layout and mathematical concerns to real-world functions in a number of fields, and engineering and research of algorithms.
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Extra info for Algorithms – ESA 2004: 12th Annual European Symposium, Bergen, Norway, September 14-17, 2004. Proceedings
Hence, if Max Thru and Greedy Swap have the same conﬁguration for node u, Opt has the same conﬁguration at u as well. We ﬁnally examine each node u that Max Thru and Greedy Swap conﬁgure diﬀerently. This node u can only exist on a closed loop L created by Max Thru and each closed loop can only have one such node. For every node v ∈ L and v = u, Max Thru and Greedy Swap conﬁgure v in the same way by the construction of Greedy Swap. Hence, by our argument above, Opt conﬁgures v in the same way as well.
The overlap is of odd-length. The conditions of the above lemma are also useful for our problem. Lemma 2. The number of mismatches that are not part of a swap, is exactly the number of the overlaps that implement condition 1. and 2. of lemma 1. Proof: We will examine all possibilities: 1. Condition 1. of the lemma does not hold. Then there is no misalignment of the text. Indeed it matches the pattern. 2. Condition 1. holds but condition 2. does not. According to lemma 1 there is a swap-match. 3.
The Cut Paren algorithm deﬁnes a set of proper line systems. Lemma 3. Given simple routing paths, each transparent section deﬁned by Max Thru is cut into at most 2 pieces by Cut Paren. By Lemma 2 Cut Paren deﬁnes a set of proper line systems. By Lemmas 1 and 3 we know that Cut Paren is at most twice as expensive as any optimal solution. Hence, Theorem 1. Given simple routing paths, Cut Paren is a 2-approximation algorithm. Using the lower bound given by Lemma 1, no algorithm can beat the approximation ratio of 2 since the optimal solution can cost twice as much as the infeasible solution generated by Max Thru.