Find nth SET bit in an int
Instead of just the lowest set bit, I want to find the position of the nth lowest set bit. (I\'m NOT talking about value on the nt
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def bitN (l: Long, i: Int) : Long = { def bitI (l: Long, i: Int) : Long = if (i == 0) 1L else 2 * { if (l % 2 == 0) bitI (l / 2, i) else bitI (l /2, i-1) } bitI (l, i) / 2 }A recursive method (in scala). Decrement i, the position, if a modulo2 is 1. While returning, multiply by 2. Since the multiplication is invoced as last operation, it is not tail recursive, but since Longs are of known size in advance, the maximum stack is not too big.
scala> n.toBinaryString.replaceAll ("(.{8})", "$1 ") res117: java.lang.String = 10110011 11101110 01011110 01111110 00111101 11100101 11101011 011000 scala> bitN (n, 40) .toBinaryString.replaceAll ("(.{8})", "$1 ") res118: java.lang.String = 10000000 00000000 00000000 00000000 00000000 00000000 00000000 000000讨论(0) -
The version from bit-twiddling hacks adapted to this case is, for example,
unsigned int nth_bit_set(uint32_t value, unsigned int n) { const uint32_t pop2 = (value & 0x55555555u) + ((value >> 1) & 0x55555555u); const uint32_t pop4 = (pop2 & 0x33333333u) + ((pop2 >> 2) & 0x33333333u); const uint32_t pop8 = (pop4 & 0x0f0f0f0fu) + ((pop4 >> 4) & 0x0f0f0f0fu); const uint32_t pop16 = (pop8 & 0x00ff00ffu) + ((pop8 >> 8) & 0x00ff00ffu); const uint32_t pop32 = (pop16 & 0x000000ffu) + ((pop16 >>16) & 0x000000ffu); unsigned int rank = 0; unsigned int temp; if (n++ >= pop32) return 32; temp = pop16 & 0xffu; /* if (n > temp) { n -= temp; rank += 16; } */ rank += ((temp - n) & 256) >> 4; n -= temp & ((temp - n) >> 8); temp = (pop8 >> rank) & 0xffu; /* if (n > temp) { n -= temp; rank += 8; } */ rank += ((temp - n) & 256) >> 5; n -= temp & ((temp - n) >> 8); temp = (pop4 >> rank) & 0x0fu; /* if (n > temp) { n -= temp; rank += 4; } */ rank += ((temp - n) & 256) >> 6; n -= temp & ((temp - n) >> 8); temp = (pop2 >> rank) & 0x03u; /* if (n > temp) { n -= temp; rank += 2; } */ rank += ((temp - n) & 256) >> 7; n -= temp & ((temp - n) >> 8); temp = (value >> rank) & 0x01u; /* if (n > temp) rank += 1; */ rank += ((temp - n) & 256) >> 8; return rank; }which, when compiled in a separate compilation unit, on gcc-5.4.0 using
-Wall -O3 -march=native -mtune=nativeon Intel Core i5-4200u, yields00400a40 <nth_bit_set>: 400a40: 89 f9 mov %edi,%ecx 400a42: 89 f8 mov %edi,%eax 400a44: 55 push %rbp 400a45: 40 0f b6 f6 movzbl %sil,%esi 400a49: d1 e9 shr %ecx 400a4b: 25 55 55 55 55 and $0x55555555,%eax 400a50: 53 push %rbx 400a51: 81 e1 55 55 55 55 and $0x55555555,%ecx 400a57: 01 c1 add %eax,%ecx 400a59: 41 89 c8 mov %ecx,%r8d 400a5c: 89 c8 mov %ecx,%eax 400a5e: 41 c1 e8 02 shr $0x2,%r8d 400a62: 25 33 33 33 33 and $0x33333333,%eax 400a67: 41 81 e0 33 33 33 33 and $0x33333333,%r8d 400a6e: 41 01 c0 add %eax,%r8d 400a71: 45 89 c1 mov %r8d,%r9d 400a74: 44 89 c0 mov %r8d,%eax 400a77: 41 c1 e9 04 shr $0x4,%r9d 400a7b: 25 0f 0f 0f 0f and $0xf0f0f0f,%eax 400a80: 41 81 e1 0f 0f 0f 0f and $0xf0f0f0f,%r9d 400a87: 41 01 c1 add %eax,%r9d 400a8a: 44 89 c8 mov %r9d,%eax 400a8d: 44 89 ca mov %r9d,%edx 400a90: c1 e8 08 shr $0x8,%eax 400a93: 81 e2 ff 00 ff 00 and $0xff00ff,%edx 400a99: 25 ff 00 ff 00 and $0xff00ff,%eax 400a9e: 01 d0 add %edx,%eax 400aa0: 0f b6 d8 movzbl %al,%ebx 400aa3: c1 e8 10 shr $0x10,%eax 400aa6: 0f b6 d0 movzbl %al,%edx 400aa9: b8 20 00 00 00 mov $0x20,%eax 400aae: 01 da add %ebx,%edx 400ab0: 39 f2 cmp %esi,%edx 400ab2: 77 0c ja 400ac0 <nth_bit_set+0x80> 400ab4: 5b pop %rbx 400ab5: 5d pop %rbp 400ab6: c3 retq 400ac0: 83 c6 01 add $0x1,%esi 400ac3: 89 dd mov %ebx,%ebp 400ac5: 29 f5 sub %esi,%ebp 400ac7: 41 89 ea mov %ebp,%r10d 400aca: c1 ed 08 shr $0x8,%ebp 400acd: 41 81 e2 00 01 00 00 and $0x100,%r10d 400ad4: 21 eb and %ebp,%ebx 400ad6: 41 c1 ea 04 shr $0x4,%r10d 400ada: 29 de sub %ebx,%esi 400adc: c4 42 2b f7 c9 shrx %r10d,%r9d,%r9d 400ae1: 41 0f b6 d9 movzbl %r9b,%ebx 400ae5: 89 dd mov %ebx,%ebp 400ae7: 29 f5 sub %esi,%ebp 400ae9: 41 89 e9 mov %ebp,%r9d 400aec: 41 81 e1 00 01 00 00 and $0x100,%r9d 400af3: 41 c1 e9 05 shr $0x5,%r9d 400af7: 47 8d 14 11 lea (%r9,%r10,1),%r10d 400afb: 41 89 e9 mov %ebp,%r9d 400afe: 41 c1 e9 08 shr $0x8,%r9d 400b02: c4 42 2b f7 c0 shrx %r10d,%r8d,%r8d 400b07: 41 83 e0 0f and $0xf,%r8d 400b0b: 44 21 cb and %r9d,%ebx 400b0e: 45 89 c3 mov %r8d,%r11d 400b11: 29 de sub %ebx,%esi 400b13: 5b pop %rbx 400b14: 41 29 f3 sub %esi,%r11d 400b17: 5d pop %rbp 400b18: 44 89 da mov %r11d,%edx 400b1b: 41 c1 eb 08 shr $0x8,%r11d 400b1f: 81 e2 00 01 00 00 and $0x100,%edx 400b25: 45 21 d8 and %r11d,%r8d 400b28: c1 ea 06 shr $0x6,%edx 400b2b: 44 29 c6 sub %r8d,%esi 400b2e: 46 8d 0c 12 lea (%rdx,%r10,1),%r9d 400b32: c4 e2 33 f7 c9 shrx %r9d,%ecx,%ecx 400b37: 83 e1 03 and $0x3,%ecx 400b3a: 41 89 c8 mov %ecx,%r8d 400b3d: 41 29 f0 sub %esi,%r8d 400b40: 44 89 c0 mov %r8d,%eax 400b43: 41 c1 e8 08 shr $0x8,%r8d 400b47: 25 00 01 00 00 and $0x100,%eax 400b4c: 44 21 c1 and %r8d,%ecx 400b4f: c1 e8 07 shr $0x7,%eax 400b52: 29 ce sub %ecx,%esi 400b54: 42 8d 14 08 lea (%rax,%r9,1),%edx 400b58: c4 e2 6b f7 c7 shrx %edx,%edi,%eax 400b5d: 83 e0 01 and $0x1,%eax 400b60: 29 f0 sub %esi,%eax 400b62: 25 00 01 00 00 and $0x100,%eax 400b67: c1 e8 08 shr $0x8,%eax 400b6a: 01 d0 add %edx,%eax 400b6c: c3 retqWhen compiled as a separate compilation unit, timing on this machine is difficult, because the actual operation is as fast as calling a do-nothing function (also compiled in a separate compilation unit); essentially, the calculation is done during the latencies associated with the function call.
It seems to be slightly faster than my suggestion of a binary search,
unsigned int nth_bit_set(uint32_t value, unsigned int n) { uint32_t mask = 0x0000FFFFu; unsigned int size = 16u; unsigned int base = 0u; if (n++ >= __builtin_popcount(value)) return 32; while (size > 0) { const unsigned int count = __builtin_popcount(value & mask); if (n > count) { base += size; size >>= 1; mask |= mask << size; } else { size >>= 1; mask >>= size; } } return base; }where the loop is executed exactly five times, compiling to
00400ba0 <nth_bit_set>: 400ba0: 83 c6 01 add $0x1,%esi 400ba3: 31 c0 xor %eax,%eax 400ba5: b9 10 00 00 00 mov $0x10,%ecx 400baa: ba ff ff 00 00 mov $0xffff,%edx 400baf: 45 31 db xor %r11d,%r11d 400bb2: 66 0f 1f 44 00 00 nopw 0x0(%rax,%rax,1) 400bb8: 41 89 c9 mov %ecx,%r9d 400bbb: 41 89 f8 mov %edi,%r8d 400bbe: 41 d0 e9 shr %r9b 400bc1: 41 21 d0 and %edx,%r8d 400bc4: c4 62 31 f7 d2 shlx %r9d,%edx,%r10d 400bc9: f3 45 0f b8 c0 popcnt %r8d,%r8d 400bce: 41 09 d2 or %edx,%r10d 400bd1: 44 38 c6 cmp %r8b,%sil 400bd4: 41 0f 46 cb cmovbe %r11d,%ecx 400bd8: c4 e2 33 f7 d2 shrx %r9d,%edx,%edx 400bdd: 41 0f 47 d2 cmova %r10d,%edx 400be1: 01 c8 add %ecx,%eax 400be3: 44 89 c9 mov %r9d,%ecx 400be6: 45 84 c9 test %r9b,%r9b 400be9: 75 cd jne 400bb8 <nth_bit_set+0x18> 400beb: c3 retqas in, not more than 31 cycles in 95% of calls to the binary search version, compared to not more than 28 cycles in 95% of calls to the bit-hack version; both run within 28 cycles in 50% of the cases. (The loop version takes up to 56 cycles in 95% of calls, up to 37 cycles median.)
To determine which one is better in actual real-world code, one would have to do a proper benchmark within the real-world task; at least with current x86-64 architecture processors, the work done is easily hidden in latencies incurred elsewhere (like function calls).
讨论(0) -
My approach is to calculate the population count for each 8-bit quarters of the 32-bit integer in parallel, then find which quarter contains the nth bit. The population count of quarters that are lower than the found one can be summarized as the initial value of later calculation.
After that count set bits one-by-one until the
nis reached. Without branches and using an incomplete implementation of population count algorithm, my example is the following:#include <stdio.h> #include <stdint.h> int main() { uint32_t n = 10, test = 3124375902u; /* 10111010001110100011000101011110 */ uint32_t index, popcnt, quarter = 0, q_popcnt; /* count set bits of each quarter of 32-bit integer in parallel */ q_popcnt = test - ((test >> 1) & 0x55555555); q_popcnt = (q_popcnt & 0x33333333) + ((q_popcnt >> 2) & 0x33333333); q_popcnt = (q_popcnt + (q_popcnt >> 4)) & 0x0F0F0F0F; popcnt = q_popcnt; /* find which quarters can be summarized and summarize them */ quarter += (n + 1 >= (q_popcnt & 0xff)); quarter += (n + 1 >= ((q_popcnt += q_popcnt >> 8) & 0xff)); quarter += (n + 1 >= ((q_popcnt += q_popcnt >> 16) & 0xff)); quarter += (n + 1 >= ((q_popcnt += q_popcnt >> 24) & 0xff)); popcnt &= (UINT32_MAX >> (8 * quarter)); popcnt = (popcnt * 0x01010101) >> 24; /* find the index of nth bit in quarter where it should be */ index = 8 * quarter; index += ((popcnt += (test >> index) & 1) <= n); index += ((popcnt += (test >> index) & 1) <= n); index += ((popcnt += (test >> index) & 1) <= n); index += ((popcnt += (test >> index) & 1) <= n); index += ((popcnt += (test >> index) & 1) <= n); index += ((popcnt += (test >> index) & 1) <= n); index += ((popcnt += (test >> index) & 1) <= n); index += ((popcnt += (test >> index) & 1) <= n); printf("index = %u\n", index); return 0; }
A simple approach which uses loops and conditionals can be the following as well:
#include <stdio.h> #include <stdint.h> int main() { uint32_t n = 11, test = 3124375902u; /* 10111010001110100011000101011110 */ uint32_t popcnt = 0, index = 0; while(popcnt += ((test >> index) & 1), popcnt <= n && ++index < 32); printf("index = %u\n", index); return 0; }讨论(0) -
Based on a method by Juha Järvi published in the famous Bit Twiddling Hacks, I tested this implementation where
nandiare used as in the question:a = i - (i >> 1 & 0x55555555); b = (a & 0x33333333) + (a >> 2 & 0x33333333); c = b + (b >> 4) & 0x0f0f0f0f; r = n + 1; s = 0; t = c + (c >> 8) & 0xff; if (r > t) { s += 16; r -= t; } t = c >> s & 0xf; if (r > t) { s += 8; r -= t; } t = b >> s & 0x7; if (r > t) { s += 4; r -= t; } t = a >> s & 0x3; if (r > t) { s += 2; r -= t; } t = i >> s & 0x1; if (r > t) s++; return (s);Based on my own tests, this is about as fast as the loop on x86, whereas it is 20% faster on arm64 and probably a lot faster on arm due to the fast conditional instructions, but I can't test this right now.
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Edit
After giving it some thought and using the
__builtin_popcountfunction, I figured it might be better to decide on the relevant byte and then compute the whole result instead of incrementally adding/subtracting numbers. Here is an updated version:int GetBitAtPosition(unsigned i, unsigned n) { unsigned bitCount; bitCount = __builtin_popcount(i & 0x00ffffff); if (bitCount <= n) { return (24 + LUT_BitPosition[i >> 24][n - bitCount]); } bitCount = __builtin_popcount(i & 0x0000ffff); if (bitCount <= n) { return (16 + LUT_BitPosition[(i >> 16) & 0xff][n - bitCount]); } bitCount = __builtin_popcount(i & 0x000000ff); if (bitCount <= n) { return (8 + LUT_BitPosition[(i >> 8) & 0xff][n - bitCount]); } return LUT_BitPosition[i & 0xff][n]; }
I felt like creating a LUT based solution where the number is inspected in byte-chunks, however, the LUT for the n-th bit position grew quite large (256*8) and the LUT-free version that was discussed in the comments might be better.
Generally the algorithm would look like this:
unsigned i = 0x000006B5; unsigned n = 4; unsigned result = 0; unsigned bitCount; while (i) { bitCount = LUT_BitCount[i & 0xff]; if (n < bitCount) { result += LUT_BitPosition[i & 0xff][n]; break; // found } else { n -= bitCount; result += 8; i >>= 8; } }Might be worth to unroll the loop into its up to 4 iterations to get the best performance on 32 bit numbers.
The LUT for bitcount (could be replaced by
__builtin_popcount):unsigned LUT_BitCount[] = { 0, 1, 1, 2, 1, 2, 2, 3, // 0-7 1, 2, 2, 3, 2, 3, 3, 4, // 8-15 1, 2, 2, 3, 2, 3, 3, 4, // 16-23 2, 3, 3, 4, 3, 4, 4, 5, // 24-31 1, 2, 2, 3, 2, 3, 3, 4, // 32-39 2, 3, 3, 4, 3, 4, 4, 5, // 40-47 2, 3, 3, 4, 3, 4, 4, 5, // 48-55 3, 4, 4, 5, 4, 5, 5, 6, // 56-63 1, 2, 2, 3, 2, 3, 3, 4, // 64-71 2, 3, 3, 4, 3, 4, 4, 5, // 72-79 2, 3, 3, 4, 3, 4, 4, 5, // 80-87 3, 4, 4, 5, 4, 5, 5, 6, // 88-95 2, 3, 3, 4, 3, 4, 4, 5, // 96-103 3, 4, 4, 5, 4, 5, 5, 6, // 104-111 3, 4, 4, 5, 4, 5, 5, 6, // 112-119 4, 5, 5, 6, 5, 6, 6, 7, // 120-127 1, 2, 2, 3, 2, 3, 3, 4, // 128 2, 3, 3, 4, 3, 4, 4, 5, // 136 2, 3, 3, 4, 3, 4, 4, 5, // 144 3, 4, 4, 5, 4, 5, 5, 6, // 152 2, 3, 3, 4, 3, 4, 4, 5, // 160 3, 4, 4, 5, 4, 5, 5, 6, // 168 3, 4, 4, 5, 4, 5, 5, 6, // 176 4, 5, 5, 6, 5, 6, 6, 7, // 184 2, 3, 3, 4, 3, 4, 4, 5, // 192 3, 4, 4, 5, 4, 5, 5, 6, // 200 3, 4, 4, 5, 4, 5, 5, 6, // 208 4, 5, 5, 6, 5, 6, 6, 7, // 216 3, 4, 4, 5, 4, 5, 5, 6, // 224 4, 5, 5, 6, 5, 6, 6, 7, // 232 4, 5, 5, 6, 5, 6, 6, 7, // 240 5, 6, 6, 7, 6, 7, 7, 8, // 248-255 };The LUT for bit position within a byte:
unsigned LUT_BitPosition[][8] = { // 0-7 {UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, // 8-15 {3,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,3,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,3,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,3,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,3,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,3,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,3,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, // 16-31 {4,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,4,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,4,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,4,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,4,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,4,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,4,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,4,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {3,4,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,4,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,3,4,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,3,4,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,3,4,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,3,4,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,3,4,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,3,4,UINT_MAX,UINT_MAX,UINT_MAX}, // 32-63 {5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {3,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,3,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,3,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,3,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,3,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,3,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,3,5,UINT_MAX,UINT_MAX,UINT_MAX}, {4,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,4,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,4,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,4,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,4,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,4,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,4,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,4,5,UINT_MAX,UINT_MAX,UINT_MAX}, {3,4,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,4,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,3,4,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,3,4,5,UINT_MAX,UINT_MAX,UINT_MAX}, {2,3,4,5,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,3,4,5,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,3,4,5,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,3,4,5,UINT_MAX,UINT_MAX}, // 64-127 {6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {3,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,3,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,3,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,3,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,3,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,3,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,3,6,UINT_MAX,UINT_MAX,UINT_MAX}, {4,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,4,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,4,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,4,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,4,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,4,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,4,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,4,6,UINT_MAX,UINT_MAX,UINT_MAX}, {3,4,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,4,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,3,4,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,3,4,6,UINT_MAX,UINT_MAX,UINT_MAX}, {2,3,4,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,3,4,6,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,3,4,6,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,3,4,6,UINT_MAX,UINT_MAX}, {5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,5,6,UINT_MAX,UINT_MAX,UINT_MAX}, {3,5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,3,5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,3,5,6,UINT_MAX,UINT_MAX,UINT_MAX}, {2,3,5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,3,5,6,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,3,5,6,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,3,5,6,UINT_MAX,UINT_MAX}, {4,5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,4,5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,4,5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,4,5,6,UINT_MAX,UINT_MAX,UINT_MAX}, {2,4,5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,4,5,6,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,4,5,6,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,4,5,6,UINT_MAX,UINT_MAX}, {3,4,5,6,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,4,5,6,UINT_MAX,UINT_MAX,UINT_MAX}, {1,3,4,5,6,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,3,4,5,6,UINT_MAX,UINT_MAX}, {2,3,4,5,6,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,3,4,5,6,UINT_MAX,UINT_MAX}, {1,2,3,4,5,6,UINT_MAX,UINT_MAX}, {0,1,2,3,4,5,6,UINT_MAX}, // 128-255 {7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {3,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,3,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,3,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,3,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,3,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,3,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,3,7,UINT_MAX,UINT_MAX,UINT_MAX}, {4,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,4,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,4,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,4,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,4,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,4,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,4,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,4,7,UINT_MAX,UINT_MAX,UINT_MAX}, {3,4,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,4,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,3,4,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,3,4,7,UINT_MAX,UINT_MAX,UINT_MAX}, {2,3,4,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,3,4,7,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,3,4,7,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,3,4,7,UINT_MAX,UINT_MAX}, {5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,5,7,UINT_MAX,UINT_MAX,UINT_MAX}, {3,5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,3,5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,3,5,7,UINT_MAX,UINT_MAX,UINT_MAX}, {2,3,5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,3,5,7,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,3,5,7,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,3,5,7,UINT_MAX,UINT_MAX}, {4,5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,4,5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,4,5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,4,5,7,UINT_MAX,UINT_MAX,UINT_MAX}, {2,4,5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,4,5,7,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,4,5,7,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,4,5,7,UINT_MAX,UINT_MAX}, {3,4,5,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,4,5,7,UINT_MAX,UINT_MAX,UINT_MAX}, {1,3,4,5,7,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,3,4,5,7,UINT_MAX,UINT_MAX}, {2,3,4,5,7,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,3,4,5,7,UINT_MAX,UINT_MAX}, {1,2,3,4,5,7,UINT_MAX,UINT_MAX}, {0,1,2,3,4,5,7,UINT_MAX}, {6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {2,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {3,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,3,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,3,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {2,3,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,3,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,3,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,3,6,7,UINT_MAX,UINT_MAX}, {4,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,4,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,4,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,4,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {2,4,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,4,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,4,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,4,6,7,UINT_MAX,UINT_MAX}, {3,4,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,4,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {1,3,4,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,3,4,6,7,UINT_MAX,UINT_MAX}, {2,3,4,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,3,4,6,7,UINT_MAX,UINT_MAX}, {1,2,3,4,6,7,UINT_MAX,UINT_MAX}, {0,1,2,3,4,6,7,UINT_MAX}, {5,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,5,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {1,5,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,5,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {2,5,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,5,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {1,2,5,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,2,5,6,7,UINT_MAX,UINT_MAX}, {3,5,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,5,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {1,3,5,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,3,5,6,7,UINT_MAX,UINT_MAX}, {2,3,5,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,3,5,6,7,UINT_MAX,UINT_MAX}, {1,2,3,5,6,7,UINT_MAX,UINT_MAX}, {0,1,2,3,5,6,7,UINT_MAX}, {4,5,6,7,UINT_MAX,UINT_MAX,UINT_MAX,UINT_MAX}, {0,4,5,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {1,4,5,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {0,1,4,5,6,7,UINT_MAX,UINT_MAX}, {2,4,5,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {0,2,4,5,6,7,UINT_MAX,UINT_MAX}, {1,2,4,5,6,7,UINT_MAX,UINT_MAX}, {0,1,2,4,5,6,7,UINT_MAX}, {3,4,5,6,7,UINT_MAX,UINT_MAX,UINT_MAX}, {0,3,4,5,6,7,UINT_MAX,UINT_MAX}, {1,3,4,5,6,7,UINT_MAX,UINT_MAX}, {0,1,3,4,5,6,7,UINT_MAX}, {2,3,4,5,6,7,UINT_MAX,UINT_MAX}, {0,2,3,4,5,6,7,UINT_MAX}, {1,2,3,4,5,6,7,UINT_MAX}, {0,1,2,3,4,5,6,7}, };讨论(0)
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