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You have just described the order. First you include the middle of the list. Then the middles of the two halves. And so on. That's the same storage convention as a heap: the children of node number $x …

There is no accepted formal definition of the divide and conquer paradigm (see this question for some suggestions), and so we must regard this paradigm as an informal concept. The main idea in divide …

The idea is that you can check whether the answer is at most some value $m$ in time $O(n)$. Applying binary search on $m$, you obtain an $O(n\log n)$ solution. How do you check whether the answer is …

Let's denote the indices by $l,h,m$. If $l \leq h$ and $h-l$ is even then $m = \frac{l+h}{2}$ and so in the following iteration, the new values $l',h'$ will either be $\frac{l+h}{2}+1,h$ or $l,\frac{l …

Your interviewer is wrong. Suppose that you had an algorithm which gets as input a sorted integer array $A_1,\ldots,A_n$, outputs Yes iff $A_i + A_j = 0$ for some $i \neq j$, and always accesses at mo …

First, of all, the worst-case running time of binary search is $\Theta(\log n)$. What you are looking for is the average running time of binary search under some reasonable random model, for example t …

Here is a solution in time $O((Q+n)\log^2 n)$ and space $O(n\log n)$. For simplicity, assume that $n$ is a power of $2$, and that the array is zero-based. A dyadic interval is an interval of the form …

Your problem is known as group testing. An essentially optimal algorithm for your problem is Generalized binary splitting. There are also pretty good non-adaptive algorithms (such algorithms fix all q …

Your question was completely solved by Bentley and Yao, An almost optimal algorithm for unbounded sorting, and Beigel, Unbounded searching algorithms. They showed that if $f\colon \mathbb{N} \to \math …

Let me start with the first question. Since $\lfloor x \rfloor \leq x$, $$ \left\lfloor \frac{\mathit{high}+\mathit{low}}{2} \right\rfloor - \mathit{low} \leq \frac{\mathit{high}+\mathit{low}}{2} - \m …

Use your web browser. You can get the first 2 million or so digits in this link, and then use your browser's searching facilities. I was able to locate 351814 within the first 2 million digits (though …

Hint: Use FFT to multiply the polynomials $$ \sum_a x^{a^2 \pmod{n}}, \qquad \sum_b x^{b^3 \pmod{n}}. $$

For the first question, suppose that you have an algorithm that supports one error, and uses $m$ questions. For each element $i \in \{1,\ldots,n\}$, let $x_{i0}$ be the vector containing the answers t …

If you knew the distribution of $n^*$, you could find an optimal binary decision tree (there are several $O(n\log n)$ algorithms) in terms of expected number of queries (you can't beat binary search i …

Let me start by recalling the question: Given an unsorted array of $n$ elements in the range $1,\ldots,m$, find the $k$th smallest element in $O(n\log m)$ time and $O(1)$ case. Calling the array …