By Pankaj K. Agarwal (auth.), Takeshi Tokuyama (eds.)

ISBN-10: 3540771182

ISBN-13: 9783540771180

This publication constitutes the refereed lawsuits of the 18th foreign Symposium on Algorithms and Computation, ISAAC 2007, held in Sendai, Japan, in December 2007.

The seventy seven revised complete papers provided including 2 invited talks have been rigorously reviewed and chosen from 220 submissions. The papers are prepared in topical sections on graph algorithms, computational geometry, complexity, graph drawing, disbursed algorithms, optimization, info constitution, online game concept, database functions, on-line algorithms, I/O algorithms, networks, geometric functions, and string.

**Read Online or Download Algorithms and Computation: 18th International Symposium, ISAAC 2007, Sendai, Japan, December 17-19, 2007. Proceedings PDF**

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**Additional info for Algorithms and Computation: 18th International Symposium, ISAAC 2007, Sendai, Japan, December 17-19, 2007. Proceedings**

**Sample text**

Otherwise select a minimal deﬁcient set W ⊆ V − (S0 −{vj }) wrt. vj , and let W0 := W0 ∪ {W }. (Step 4) If j < n, then j := j + 1 and go to Step 3. Otherwise output S0 as a solution. Note that in the case where S0 − {vj } does not satisfy (1) in Step 3, there exists a minimal deﬁcient set W ⊆ V − (S0 − {vj }) wrt. vj . Before deleting vj from S0 , S0 is feasible and hence by Lemma 2, every deﬁcient set contains a source in S0 . On the other hand, S0 − {vj } is infeasible. Again by Lemma 2, there is a deﬁcient set W with W ∩ S0 = {vj } such that W − {vj } is not deﬁcient.

Surprisingly the following fact holds: the last two vertices in an MD ordering gives a ﬂat pair. We prove this, and then show that all extreme subsets of a graph (G, w) can be computed by using ﬂat pairs in O(mn + n2 log n) time. It is already known [11] that all extreme subsets in a graph can be computed in O(mn + n2 log n) time by applying an MA ordering after augmenting the given graph with a new vertex and edges. However, the augmenting process is rather artiﬁcial, and no direct extension of this algorithm to the case of submodular set functions has been successful.

Our conjecture on the lower bound for the increment operation is Ω(n). By exhaustive searching we can show that the lower bound holds for a small n. Future goal is to ﬁnd a lower bound of Ω(n) bit probes per operation of a counting sequence of dimension n using no extra space. The addition of a moderate amount of extra space speeds up the operations. We present several data structures that uses little extra space for eﬃcient increment/decrement and addition/subtraction operations. Our ﬁrst solution uses log n + 3 extra bits that requires at most 2 log n + 4 bit inspections and at most 4 bit changes for the increment/decrement operation.

### Algorithms and Computation: 18th International Symposium, ISAAC 2007, Sendai, Japan, December 17-19, 2007. Proceedings by Pankaj K. Agarwal (auth.), Takeshi Tokuyama (eds.)

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