👩‍❤️‍👩 🏭 💕 Algorithms for the exam in the SHAD 🤾🏽 ♻️ 🎥

Hello! My name is Alexander Kurilkin, and I am teaching a course on algorithms at ShAD Helper. In this post, I will analyze several tasks from the entrance exams of past years, so that you can see what awaits you and understand what we can teach you in our course. I hope that you share my love for interesting tasks on algorithms and will receive sincere pleasure from reading this post! So, let's get started ...

05/28/2016, No. 4

Are given $n$ segments $[a_i; b_i]$ . We call the segment nesting index $[a_i; b_i]$ the number of segments that contain it. Suggest an algorithm that determines whether there is a segment in the set with a nesting index exceeding 1000. The time limit is $O(n \log n)$ , for additional memory - $O(n)$ .

Decision

, "". , - , , - , . " ", $a_i$ , " ", $b_i$ . . , 1. , 1000, , — 1000 . — - , , . , (std::multiset C++), . , — 1000 . , , , , *set.begin(), ( 1000 ) . , ! , , . , . , , !

: 1000 ? , - 1000 , 1000 , - , ? , - 1000 , , , , , .

, . $O(n \log n)$ , $O(n \log 1000) = O(n)$ O(n) . , : $O(n \log n)$ . $O(n)$ .

05/25/2019, No.4

An array of real numbers is given $A$ . Suggest an algorithm that finds for each element $A$ index of the element closest to the right, at least twice its size. If there is no such element, then the value should be returned. $None$ . Time limit $O(n \log n)$ , for additional memory - $O(n)$ .

Decision

$(, )$ . ( $None$ ), $(A[n - 1], n - 1)$ , . $i$ . , , , ? , . , , , ? . , . , $(A[i], i)$ . , , ? , ( ) $i$ . (, , std::vector) , $A[i] \cdot 2$ . , ( ), , $None$ .

, , ? , O(n), $n$ , $O(n \log n)$ , . , , , $O(n)$ .

06/10/12, No. 5
In the set of $n$ each person may or may not know the other (if $A$ knows $B$ , it does not follow that $B$ knows $A$ ) All acquaintances are given by a Boolean matrix $n×n$ . — , , . , , . — $O(n)$ , — $O(1)$ .

$K_{ij} = 1$ , $i$ - $j$ - , 0 .

, $K_{ij} = 1 ~ (i \neq j)$ , $i$ - , - , $j$ - , $K_{ij} = 0$ , $j$ - , $i$ - .

: — . , 1, , $l$ . , ( 0), , , , , - ( , , $l$ ). $l$ - — , $(l, l + 1)$ . 1, , , . , . , , , ( ), , , , . , , , $O(n)$ , , $O(n)$ . : $O(n)$ . , , (, ), $O(1)$ . !

, , !

2019, -, D

, 2
: 2
: 256Mb

$n$ $m$ . $q$ « $i$ 1». .

— , . , .

. , , .

$n$ $m$ — ( $1 \le n, m \le 10^5, 2 \le n)$ . $m$ $u$ , $v$ , $w$ . , $w$ , $u$ $v$ ( $1 \le u, v \le n, 1 \le w \le 10$ ).

$q$ — ( $1 \le q \le 10^5$ ). $q$ $id$ ( $1 \le id \le m$ ). , $id$ . .

$q$ numbers separated by spaces or line breaks - the weight of the minimum spanning tree after each request.

Decision

, , 99% . .

" $u$ $v$ ", $k$ . $u$ $v$ $k$ ( . $k+1$ , , $u$ $v$ $> k + 1$ , $k + 1$ , , ́ ), , 1. . , , . $O(n)$ , , , $O(\log n)$ link-cut tree, , ( ). .

1 10, 10 ( , ). $k$ - $\le k$ . . : , ( ) , , . , , , , .

$(u, v)$ 1 $k$ . $k$ - . , $k$ - $u$ $v$ . , $k+1$ , , . , $u$ $v$ $k$ - , 1. $u$ $v$ $k$ - , $\le k$ , .

, , ? $O(\alpha(n))$ , $O(10 \cdot m \log n) = O(m \log n)$ . , Accepted :)

Algorithms for the exam in the SHAD

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