Output-Sensitive Tools for Range Searching in Higher Dimensions

Let $P$ be a set of $n$ points in ${\mathbb R}^{d}$. A point $p \in P$ is $k$\emph{-shallow} if it lies in a halfspace which contains at most $k$ points of $P$ (including $p$). We show that if all points of $P$ are $k$-shallow, then $P$ can be partitioned into $\Theta(n/k)$ subsets, so that any hyperplane crosses at most $O((n/k)^{1-1/(d-1)} \log^{2/(d-1)}(n/k))$ subsets. Given such a partition, we can apply the standard construction of a spanning tree with small crossing number within each subset, to obtain a spanning tree for the point set $P$, with crossing number $O(n^{1-1/(d-1)}k^{1/d(d-1)} \log^{2/(d-1)}(n/k))$. This allows us to extend the construction of Har-Peled and Sharir \cite{hs11} to three and higher dimensions, to obtain, for any set of $n$ points in ${\mathbb R}^{d}$ (without the shallowness assumption), a spanning tree $T$ with {\em small relative crossing number}. That is, any hyperplane which contains $w \leq n/2$ points of $P$ on one side, crosses $O(n^{1-1/(d-1)}w^{1/d(d-1)} \log^{2/(d-1)}(n/w))$ edges of $T$. Using a similar mechanism, we also obtain a data structure for halfspace range counting, which uses $O(n \log \log n)$ space (and somewhat higher preprocessing cost), and answers a query in time $O(n^{1-1/(d-1)}k^{1/d(d-1)} (\log (n/k))^{O(1)})$, where $k$ is the output size

A novel steganography approach for audio files

We present a novel robust and secure steganography technique to hide images into audio files aiming at increasing the carrier medium capacity. The audio files are in the standard WAV format, which is based on the LSB algorithm while images are compressed by the GMPR technique which is based on the Discrete Cosine Transform (DCT) and high frequency minimization encoding algorithm. The method involves compression-encryption of an image file by the GMPR technique followed by hiding it into audio data by appropriate bit substitution. The maximum number of bits without significant effect on audio signal for LSB audio steganography is 6 LSBs. The encrypted image bits are hidden into variable and multiple LSB layers in the proposed method. Experimental results from observed listening tests show that there is no significant difference between the stego audio reconstructed from the novel technique and the original signal. A performance evaluation has been carried out according to quality measurement criteria of Signal-to-Noise Ratio (SNR) and Peak Signal-to-Noise Ratio (PSNR)