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Recently i did Leetcode Longest Common Subsequence problem. I used the basic dp approach and solved it in 18 ms using 12 mb space. After that i decided to check the fastest solutions for the problem and got this code

Fastest LCS code

class Solution {
public:
    int longestCommonSubsequence(string X, string Y) {
        if ( X.size() < Y.size() ) swap(X,Y) ;
        int m = X.size() , n = Y.size();
        if (m == 0 || n == 0) return 0;
        int w = (m + 31) >> 5;
        std::uint32_t S[256][w];
        std::memset(S, 0, sizeof(std::uint32_t) * 256 * w);
        std::int32_t set = 1;
        for (int i = 0, j = 0; i < m; ++i) {
            S[X[i]][j] |= set;
            set <<= 1;
            if (!set) set++,j++;
        }
        std::uint32_t L[w];
        std::memset(L, 0, sizeof(std::uint32_t) * w);
        for (int i = 0; i < n; ++i) {
            std::uint32_t b1 = 1;
            std::uint32_t b2 = 0;
            for (int j = 0; j < w; ++j) {
                std::uint32_t U  = L[j] | S[Y[i]][j];
                std::uint32_t c  = L[j] >> 31;
                std::uint32_t V  = U - (L[j] << 1 | b1+b2);
                b1 = c;
                b2 = (V > U);
                L[j] = U & (~V);
            }
        }      
        int res = 0;
        for (int i = 0; i < w; ++i)
            for (; L[i]; ++res)
                L[i] &= L[i] - 1;
        return res;
    }
};

I submitted it and it ran in 4 ms using 7.5 mb space. I tried to deconstruct the algorithm and understand how it was working. I cleaned the code a bit and got it to run in 3 ms, but I still could not understand it.

My optimized code

#include <cstring>

typedef unsigned int uint;

class Solution {
public:
    int lcs(string X, string Y){
        if ( X.size() < Y.size() ) swap(X,Y) ;
        int m = X.size() , n = Y.size();
        if (m == 0 || n == 0) return 0;
        int w = (m + 31) >> 5;
        uint S[256][w];
        memset(S, 0, sizeof(S));
        uint set = 1;
        for (int i = 0, j = 0; i < m; ++i) {
            S[X[i]][j] |= set;
            set <<= 1;
            if (!set) set++,j++;
        }
        uint L[w];
        memset(L, 0, sizeof(L));
        for (int i = 0; i < n; ++i) {
            uint b1 = 1;
            uint b2 = 0;
            for (int j = 0; j < w; ++j) {
                uint U  = L[j] | S[Y[i]][j];
                uint V  = U - (L[j] << 1 | b1+b2);
                b1 = L[j] >> 31;
                b2 = (V > U);
                L[j] = U & (~V);
            }
        }      
        int res = 0;
        for (int i = 0; i < w; ++i)
            res += __builtin_popcount(L[i]);
        return res;
    }

    int longestCommonSubsequence(string text1, string text2) {
        return lcs(text1, text2);
    }
};

I could not find any relevant blogs explaining it. Can anyone help me understand how this works or link any blogs. Thanks.

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