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‎LICENSE

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MIT License
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Copyright (c) 2017 Jonas Raoni Soares da Silva
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Permission is hereby granted, free of charge, to any person obtaining a copy
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of this software and associated documentation files (the "Software"), to deal
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in the Software without restriction, including without limitation the rights
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to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
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copies of the Software, and to permit persons to whom the Software is
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furnished to do so, subject to the following conditions:
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The above copyright notice and this permission notice shall be included in all
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copies or substantial portions of the Software.
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THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
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OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
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SOFTWARE.

‎Lesson 1 - Iterations/BinaryGap.cs

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using System;
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class Solution {
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public int solution(int N) {
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bool started = false;
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int length = 0, max = 0;
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for (uint i = 1; i <= N; i <<= 1) {
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if ((i & N) != 0) {
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if (started) {
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if (length > max)
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max = length;
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length = 0;
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}
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started = true;
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}
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else if (started)
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++length;
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}
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return max;
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}
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}

‎Lesson 1 - Iterations/BinaryGap.md

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A binary gap within a positive integer N is any maximal sequence of consecutive zeros that is surrounded by ones at both ends in the binary representation of N.
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For example, number 9 has binary representation 1001 and contains a binary gap of length 2. The number 529 has binary representation 1000010001 and contains two binary gaps: one of length 4 and one of length 3. The number 20 has binary representation 10100 and contains one binary gap of length 1. The number 15 has binary representation 1111 and has no binary gaps.
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Write a function:
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class Solution { public int solution(int N); }
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that, given a positive integer N, returns the length of its longest binary gap. The function should return 0 if N doesn't contain a binary gap.
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For example, given N = 1041 the function should return 5, because N has binary representation 10000010001 and so its longest binary gap is of length 5.
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Assume that:
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N is an integer within the range [1..2,147,483,647].
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Complexity:
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expected worst-case time complexity is O(log(N));
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expected worst-case space complexity is O(1).

‎Lesson 10 - Prime and composite numbers/CountFactors

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A positive integer D is a factor of a positive integer N if there exists an integer M such that N = D * M.
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For example, 6 is a factor of 24, because M = 4 satisfies the above condition (24 = 6 * 4).
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Write a function:
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class Solution { public int solution(int N); }
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that, given a positive integer N, returns the number of its factors.
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For example, given N = 24, the function should return 8, because 24 has 8 factors, namely 1, 2, 3, 4, 6, 8, 12, 24. There are no other factors of 24.
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Assume that:
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N is an integer within the range [1..2,147,483,647].
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Complexity:
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expected worst-case time complexity is O(sqrt(N));
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expected worst-case space complexity is O(1).

‎Lesson 10 - Prime and composite numbers/CountFactors.cs

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using System;
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class Solution {
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public int solution(int N) {
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int end = (int)Math.Sqrt(N), count = 0, i = 1;
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while (++i <= end) {
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if (N % i == 0)
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++count;
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}
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return count * 2 + (end * end == N ? 1 : 2);
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}
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}

‎Lesson 10 - Prime and composite numbers/Flags

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A non-empty zero-indexed array A consisting of N integers is given.
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A peak is an array element which is larger than its neighbours. More precisely, it is an index P such that 0 < P < N − 1 and A[P − 1] < A[P] > A[P + 1].
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For example, the following array A:
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A[0] = 1
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A[1] = 5
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A[2] = 3
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A[3] = 4
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A[4] = 3
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A[5] = 4
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A[6] = 1
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A[7] = 2
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A[8] = 3
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A[9] = 4
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A[10] = 6
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A[11] = 2
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has exactly four peaks: elements 1, 3, 5 and 10.
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You are going on a trip to a range of mountains whose relative heights are represented by array A, as shown in a figure below. You have to choose how many flags you should take with you. The goal is to set the maximum number of flags on the peaks, according to certain rules.
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Flags can only be set on peaks. What's more, if you take K flags, then the distance between any two flags should be greater than or equal to K. The distance between indices P and Q is the absolute value |P − Q|.
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For example, given the mountain range represented by array A, above, with N = 12, if you take:
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two flags, you can set them on peaks 1 and 5;
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three flags, you can set them on peaks 1, 5 and 10;
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four flags, you can set only three flags, on peaks 1, 5 and 10.
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You can therefore set a maximum of three flags in this case.
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Write a function:
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class Solution { public int solution(int[] A); }
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that, given a non-empty zero-indexed array A of N integers, returns the maximum number of flags that can be set on the peaks of the array.
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For example, the following array A:
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A[0] = 1
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A[1] = 5
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A[2] = 3
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A[3] = 4
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A[4] = 3
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A[5] = 4
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A[6] = 1
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A[7] = 2
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A[8] = 3
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A[9] = 4
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A[10] = 6
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A[11] = 2
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the function should return 3, as explained above.
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Assume that:
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N is an integer within the range [1..400,000];
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each element of array A is an integer within the range [0..1,000,000,000].
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Complexity:
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expected worst-case time complexity is O(N);
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expected worst-case space complexity is O(N), beyond input storage (not counting the storage required for input arguments).

‎Lesson 10 - Prime and composite numbers/MinPerimeterRectangle

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An integer N is given, representing the area of some rectangle.
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The area of a rectangle whose sides are of length A and B is A * B, and the perimeter is 2 * (A + B).
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The goal is to find the minimal perimeter of any rectangle whose area equals N. The sides of this rectangle should be only integers.
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For example, given integer N = 30, rectangles of area 30 are:
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(1, 30), with a perimeter of 62,
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(2, 15), with a perimeter of 34,
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(3, 10), with a perimeter of 26,
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(5, 6), with a perimeter of 22.
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Write a function:
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class Solution { public int solution(int N); }
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that, given an integer N, returns the minimal perimeter of any rectangle whose area is exactly equal to N.
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For example, given an integer N = 30, the function should return 22, as explained above.
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Assume that:
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N is an integer within the range [1..1,000,000,000].
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Complexity:
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expected worst-case time complexity is O(sqrt(N));
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expected worst-case space complexity is O(1).

‎Lesson 10 - Prime and composite numbers/MinPerimeterRectangle.cs

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using System;
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class Solution {
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public int solution(int N) {
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int end = (int)Math.Sqrt(N), min = N + 1, i = 1;
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while (++i <= end) {
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int d = N / i;
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if (d * i == N && min > d + i)
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min = d + i;
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}
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return min * 2;
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}
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}

‎Lesson 10 - Prime and composite numbers/Peaks

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A non-empty zero-indexed array A consisting of N integers is given.
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A peak is an array element which is larger than its neighbors. More precisely, it is an index P such that 0 < P < N − 1, A[P − 1] < A[P] and A[P] > A[P + 1].
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For example, the following array A:
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A[0] = 1
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A[1] = 2
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A[2] = 3
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A[3] = 4
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A[4] = 3
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A[5] = 4
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A[6] = 1
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A[7] = 2
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A[8] = 3
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A[9] = 4
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A[10] = 6
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A[11] = 2
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has exactly three peaks: 3, 5, 10.
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We want to divide this array into blocks containing the same number of elements. More precisely, we want to choose a number K that will yield the following blocks:
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A[0], A[1], ..., A[K − 1],
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A[K], A[K + 1], ..., A[2K − 1],
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...
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A[N − K], A[N − K + 1], ..., A[N − 1].
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What's more, every block should contain at least one peak. Notice that extreme elements of the blocks (for example A[K − 1] or A[K]) can also be peaks, but only if they have both neighbors (including one in an adjacent blocks).
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The goal is to find the maximum number of blocks into which the array A can be divided.
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Array A can be divided into blocks as follows:
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one block (1, 2, 3, 4, 3, 4, 1, 2, 3, 4, 6, 2). This block contains three peaks.
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two blocks (1, 2, 3, 4, 3, 4) and (1, 2, 3, 4, 6, 2). Every block has a peak.
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three blocks (1, 2, 3, 4), (3, 4, 1, 2), (3, 4, 6, 2). Every block has a peak. Notice in particular that the first block (1, 2, 3, 4) has a peak at A[3], because A[2] < A[3] > A[4], even though A[4] is in the adjacent block.
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However, array A cannot be divided into four blocks, (1, 2, 3), (4, 3, 4), (1, 2, 3) and (4, 6, 2), because the (1, 2, 3) blocks do not contain a peak. Notice in particular that the (4, 3, 4) block contains two peaks: A[3] and A[5].
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The maximum number of blocks that array A can be divided into is three.
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Write a function:
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class Solution { public int solution(int[] A); }
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that, given a non-empty zero-indexed array A consisting of N integers, returns the maximum number of blocks into which A can be divided.
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If A cannot be divided into some number of blocks, the function should return 0.
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For example, given:
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A[0] = 1
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A[1] = 2
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A[2] = 3
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A[3] = 4
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A[4] = 3
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A[5] = 4
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A[6] = 1
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A[7] = 2
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A[8] = 3
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A[9] = 4
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A[10] = 6
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A[11] = 2
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the function should return 3, as explained above.
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Assume that:
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N is an integer within the range [1..100,000];
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each element of array A is an integer within the range [0..1,000,000,000].
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Complexity:
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expected worst-case time complexity is O(N*log(log(N)));
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expected worst-case space complexity is O(N), beyond input storage (not counting the storage required for input arguments).

‎Lesson 11 - Sieve of Eratosthenes/CountNonDivisible

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You are given an array A consisting of N integers.
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For each number A[i] such that 0 ≤ i < N, we want to count the number of elements of the array that are not the divisors of A[i]. We say that these elements are non-divisors.
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For example, consider integer N = 5 and array A such that:
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A[0] = 3
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A[1] = 1
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A[2] = 2
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A[3] = 3
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A[4] = 6
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For the following elements:
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A[0] = 3, the non-divisors are: 2, 6,
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A[1] = 1, the non-divisors are: 3, 2, 3, 6,
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A[2] = 2, the non-divisors are: 3, 3, 6,
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A[3] = 3, the non-divisors are: 2, 6,
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A[4] = 6, there aren't any non-divisors.
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Write a function:
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class Solution { public int[] solution(int[] A); }
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that, given an array A consisting of N integers, returns a sequence of integers representing the amount of non-divisors.
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The sequence should be returned as:
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a structure Results (in C), or
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a vector of integers (in C++), or
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a record Results (in Pascal), or
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an array of integers (in any other programming language).
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For example, given:
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A[0] = 3
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A[1] = 1
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A[2] = 2
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A[3] = 3
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A[4] = 6
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the function should return [2, 4, 3, 2, 0], as explained above.
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Assume that:
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N is an integer within the range [1..50,000];
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each element of array A is an integer within the range [1..2 * N].
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Complexity:
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expected worst-case time complexity is O(N*log(N));
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expected worst-case space complexity is O(N), beyond input storage (not counting the storage required for input arguments).

‎Lesson 11 - Sieve of Eratosthenes/CountSemiprimes

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A prime is a positive integer X that has exactly two distinct divisors: 1 and X. The first few prime integers are 2, 3, 5, 7, 11 and 13.
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A semiprime is a natural number that is the product of two (not necessarily distinct) prime numbers. The first few semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26.
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You are given two non-empty zero-indexed arrays P and Q, each consisting of M integers. These arrays represent queries about the number of semiprimes within specified ranges.
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Query K requires you to find the number of semiprimes within the range (P[K], Q[K]), where 1 ≤ P[K] ≤ Q[K] ≤ N.
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For example, consider an integer N = 26 and arrays P, Q such that:
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P[0] = 1 Q[0] = 26
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P[1] = 4 Q[1] = 10
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P[2] = 16 Q[2] = 20
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The number of semiprimes within each of these ranges is as follows:
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(1, 26) is 10,
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(4, 10) is 4,
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(16, 20) is 0.
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Write a function:
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class Solution { public int[] solution(int N, int[] P, int[] Q); }
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that, given an integer N and two non-empty zero-indexed arrays P and Q consisting of M integers, returns an array consisting of M elements specifying the consecutive answers to all the queries.
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For example, given an integer N = 26 and arrays P, Q such that:
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P[0] = 1 Q[0] = 26
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P[1] = 4 Q[1] = 10
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P[2] = 16 Q[2] = 20
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the function should return the values [10, 4, 0], as explained above.
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Assume that:
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N is an integer within the range [1..50,000];
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M is an integer within the range [1..30,000];
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each element of arrays P, Q is an integer within the range [1..N];
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P[i] ≤ Q[i].
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Complexity:
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expected worst-case time complexity is O(N*log(log(N))+M);
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expected worst-case space complexity is O(N+M), beyond input storage (not counting the storage required for input arguments).

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