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This is an interesting question. How do we define a family of binary trees is balanced? A generalized definition of balanced binary trees One of the most general definitions is that the height of a …

Here is a program in Java that prints all unordered unique m-combinations of N-sized subsets of numbers from 0 inclusive to m * N exclusive. Click the run button to see some result. The algorithm use …

TL;DR -- No, there is no better strategy than the simple strategy. Here is the main idea of the proof. When there are not enough balls, there will be a "ball path" from a $k$-full bin to a bin wit …

A simple algorithm that let all teams finish with same points Let $n$ teams sit around a circle evenly. If $n$ is odd, let each team win over the next $(n-1)/2$ teams clockwise. Otherwise $n$ is even …

First, there is one special case of $k$, the powers of 10, which are 1, 10, 100, etc. Their positions at a lexicographically ordered sequence of positive integers are fixed. 1 is always at the 1st po …

Here is the simplest algorithm, which is efficient when $k$ is much smaller than $n$ relatively. Input: two positive integers $n$ and $k$ with $k\le n$ Output: a random permutation of $k$ integers …

In fact, your result is correct and that reference solution is wrong! Here is where that reference solution goes wrong. Since we view the candidate ranking as reading an random permutation, this …

Given $n$ persons, this question is about a sequence of unordered pair of persons, called a two-person schedule(TPS). A TPS is good if any of its initial contiguous subsequences satisfies the followin …

One epic answer by David, one simplest answer by Yuval. However, there is no graphs. I mean, actual visual graphs. Here is the famous beautiful negative instances to 1-dimensional Weisfeiler-Lehman …

Here is an variant of the idea 1 in the question, where a factor of 2 is used to replace the original factor of 9. Let $v=[2^0, 2^1, 2^2, 2^3, \cdots, 2^{62}, 2^{63}, 0, 0, 0, \cdots]^T$. Obtain $b^ …

What we need to prove are the smallest period for $s_1$ is 1, which is trivially true. the smallest period for $s_i$ is $F_{k-1}$ for all $F_k-1\le i\le F_{k+1}-2$, where $k\ge3$. Let $k\ge3$. You …

Why 2 different edge min-cuts in an undirected multigraph must be completely disjoint? There is no "why" here since 2 different edge minimum-cuts in an undirected multigraph can have one edge in …