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We provide proofs of the following theorems by considering the entropy of random walks. Theorem 1.(Alon, Hoory and Linial) Let G be an undirected simple graph with n vertices, girth g, minimum degree at least 2 and average degree d̄. Odd girth: If g = 2r + 1, then n ≥ 1 + d̄ r−1 ∑ i=0 (d̄− 1)i. Even girth: If g = 2r, then n ≥ 2 r−1 ∑ i=0 (d̄− 1)i. Theorem(More)
We study the complexity of the following problems in the streaming model. Membership testing for DLIN. We show that every language in DLIN can be recognised by a randomized one-passO(log n) space algorithm with inverse polynomial one-sided error, and by a deterministic p-pass O(n/p) space algorithm. We show that these algorithms are optimal. Membership(More)
In this paper, we give streaming algorithms for some problems which are known to be in deterministic log-space, when the number of passes made on the input is unbounded. If the input data is massive, the conventional deterministic log-space algorithms may not run efficiently. We study the complexity of the problems when the number of passes is bounded. The(More)
In this paper, we have described a numerical integration technique for solving singularly perturbed delay differential equations. The second order singularly perturbed boundary value problem is transformed into an asymptotically equivalent first order neutral differential equation. Then numerical integration and linear interpolation is used to get the(More)
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