Algorithm

I’m “studying” this algorithmic technique: Backtracking. This is an algorithmic approach to problem solution that I trust every good Computer Scientist already uses; but taking a good theo-practical look at it is something better. A Backtracking Tree I believe you can find enough informations about it online (the 2 links I provided are more than enough), so I’ll go straight to the problem. Problem definition Given an integer n, and the set S = { 1 … n }, calculate (print) all the possible subsets of S. For example, for n = 1, all the possible subsets are { 1 } and { } (empty set). For n = 2, all the possible subsets are: { 1 2 } { 1 } { 2 } { }. In general, for the set made of the first n Integers, the number of possible subsets is 2n. ...

This time it’s not something I make myself. Indeed, I still can’t “see” it 100%: I got it, but it’s a bit complex. A cute little lady counting (bits? ;-) ) It’s a method to count the number of bits in a number in O(1), in just 5 lines of code. INHUMAN. The “human” solutions Of course, there are methods that look way more easy and, given that the size of a number in memory is “fixed”, the O(1) still stands. For example: 0. Based on the “evenness/oddness” of the number 1 2 3 4 5 6 7 8 9 10 unsigned int bits_counter_v0(unsigned int x) { unsigned int count = 0; while ( x != 0 ) { // If odd, add 1 count += (x % 2 == 0) ? 0 : 1; x >>= 1; } return count; } 1. Counting one bit at a time (always the least significant one) 1 2 3 4 5 6 7 8 9 10 unsigned int bits_counter_v1(unsigned int x) { unsigned int count = 0; while ( x != 0 ) { // If least-significant bit is 1, add 1 count += (x & 1) ? 1 : 0; x >>= 1; } return count; } 2. Counting 4 bit at a time with max 8 shifts, using an “hashmap” with precalculated results The fact that it can count the bits in “max 8 shifts” has the trade off of the memory used by the hashmap. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 unsigned int bits_counter_v2(unsigned int x) { unsigned int count = 0; // "Hashmap" of the values for the least significant 4 bits unsigned int int_to_bits_count[16] = { 0, // 0 00 1, // 1 01 1, // 2 10 2, // 3 11 1, // 4 100 2, // 5 101 2, // 6 110 3, // 7 111 1, // 8 1000 2, // 9 1001 2, // 10 1010 3, // 11 1011 2, // 12 1100 3, // 13 1101 3, // 14 1110 4 // 15 1111 }; while ( x != 0 ) { // Add the bits count of the least significant 4 bits count += int_to_bits_count[ x & 15 ]; x >>= 4; } return count; } Let’s see what some insane people made. ...

My colleague and friend Luca (@lucabox) described a better solution to the problem of "Finding the first non repearing char in a string in O(n) time and O(1) space". It uses smartly the space, making the solution nicer and slicker. Or we are just 2 geeks that need to give more attention to their girlfriends :P Luca’s solution description The logic of this solution is based on the usage of an array of unsigned chars. Every char (assumed to be lowecase) has an associated small byte (1 char = 8 bits), where the bits 0x1 and 0x2 (the 2 least significant) represents, respectively, “present once in the input string” and “present multiple times in the input string”. After the input is “scanned” once, and every letter is marked with the correspondent “presence bit” (once, multiple or none), it get’s scanned a second time to find the first char of the input which has the bit “present once” set to “1”. ...

This guy, Jeff Moser, is mental! He made a loooong comic strip to make AES explain itself: from a very high level, non technical explanation, deep down to mathematical details. Worth a read for sure! ;)