Question

参考下面的qn，增长的升序如下： G2，G3，G4，G1，G5？ g2 = 4n，g3 = 18nlgn，g4 = 12n ^ 2lg（n-1），g1 = 25n ^ 2lgn，g5 = 15n ^ 3

Answer 1

这个问题要求你使用一些代数来转换函数，以便你可以比较它们。请记住，主导常数对于渐近比较并不重要。指数/对数基数有时候做很重要，所以要小心。

g1(n) = n^2 lg n，使用approximation lg n! ~ n lg n。
g2(n) = 4^(lg n) = n^(lg 4) << n^2，假设lg是基础e。
g3(n) = n^2 lg n使用formula作为算术级数的总和。
g4(n) = n^2 (lg lg n)^2
g5(n) = n^3

此时，您可以将lg n ^ k << n用于任何固定k。这意味着n的更高功率支配lg n的更高功率。同样(lg lg n)^k << lg n适用于任何k。

这会产生排序g2 << g4 << g1 = g3 << g5。

Answer 2

您可能需要仔细检查，因为我相信结果是：g2 g4 g3 g5 g1

这是一个测试功能的C程序。您应该仔细检查程序[主要是g*函数]和g4。所以，这是：

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

#define FLT long double

struct rtn {
    const char *name;
    FLT val;
};
typedef struct rtn rtn_t;

#define DO(_alg) \
    do { \
        ptr = &list[_alg - 1]; \
        g##_alg(ptr,n); \
        printf("%s: %Lg\n",ptr->name,ptr->val); \
    } while (0)

FLT
fact(unsigned long long n)
{
    unsigned long long i;
    FLT f;

    f = 1.0;

    for (i = 1;  i <= n;  ++i) {
        f *= n;
    }

    return f;
}

void
g1(rtn_t *rtn,unsigned long long n)
{
    FLT val;

    rtn->name = "g1";

    val = 25.0;
    val *= n;
    val *= logl(fact(n));

    rtn->val = val;
}

void
g2(rtn_t *rtn,unsigned long long n)
{
    FLT val;

    rtn->name = "g2";

    val = logl(n);
    val += val;
    val = powl(2,val);

    rtn->val = val;
}

void
g3(rtn_t *rtn,unsigned long long n)
{
    FLT val;
    FLT lgn;
    unsigned long long k;

    rtn->name = "g3";

    lgn = logl(n);

    val = 0;

    for (k = 1;  k <= n;  ++k)
        val += lgn * k;

    val *= 18;

    rtn->val = val;
}

void
g4(rtn_t *rtn,unsigned long long n)
{
    FLT val;
    FLT lgn;

    rtn->name = "g4";

    val = n;
    val *= val;
    val *= 12;

    lgn = logl(n);
    lgn = logl(lgn);
    lgn *= lgn;

    val *= lgn;

    rtn->val = val;
}

void
g5(rtn_t *rtn,unsigned long long n)
{
    FLT val;

    rtn->name = "g5";

    val = n;
    val = val * val * val;

    val *= 15;

    rtn->val = val;
}

int
cmp(const void *lhs,const void *rhs)
{
    const rtn_t *lp;
    const rtn_t *rp;
    int cmpflg;

    lp = lhs;
    rp = rhs;

    do {
        cmpflg = 0;

        if (lp->val < rp->val) {
            cmpflg = -1;
            break;
        }

        if (lp->val > rp->val) {
            cmpflg = 1;
            break;
        }
    } while (0);

    return cmpflg;
}

void
order(unsigned long long n)
{
    rtn_t list[5];
    rtn_t *ptr;

    printf("\n");
    printf("n: %llu\n",n);

    DO(1);
    DO(2);
    DO(3);
    DO(4);
    DO(5);

    qsort(list,5,sizeof(rtn_t),cmp);

    for (ptr = list;  ptr < &list[5];  ++ptr)
        printf(" %s",ptr->name);
    printf("\n");

    fflush(stdout);
}

int
main(void)
{
    unsigned long long n;

    n = 1;
    for (n = 1;  n < (1LL << 62);  n *= 2)
        order(n);

    return 0;
}

以下是该计划的输出：

n: 1
g1: 0
g2: 1
g3: 0
g4: inf
g5: 15
 g1 g3 g2 g5 g4

n: 2
g1: 69.3147
g2: 2.61406
g3: 37.4299
g4: 6.44792
g5: 120
 g2 g4 g3 g1 g5

n: 4
g1: 554.518
g2: 6.83333
g3: 249.533
g4: 20.4845
g5: 960
 g2 g4 g3 g1 g5

n: 8
g1: 3327.11
g2: 17.8628
g3: 1347.48
g4: 411.625
g5: 7680
 g2 g4 g3 g1 g5

n: 16
g1: 17744.6
g2: 46.6944
g3: 6787.3
g4: 3194.74
g5: 61440
 g2 g4 g3 g1 g5

n: 32
g1: 88722.8
g2: 122.062
g3: 32938.4
g4: 18983.3
g5: 491520
 g2 g4 g3 g1 g5

n: 64
g1: 425870
g2: 319.078
g3: 155709
g4: 99843.8
g5: 3.93216e+06
 g2 g4 g3 g1 g5

n: 128
g1: 1.98739e+06
g2: 834.091
g3: 721051
g4: 490438
g5: 3.14573e+07
 g2 g4 g3 g1 g5

n: 256
g1: 9.08522e+06
g2: 2180.37
g3: 3.28345e+06
g4: 2.30749e+06
g5: 2.51658e+08
 g2 g4 g3 g1 g5

n: 512
g1: 4.08835e+07
g2: 5699.62
g3: 1.47468e+07
g4: 1.05429e+07
g5: 2.01327e+09
 g2 g4 g3 g1 g5

n: 1024
g1: 1.81704e+08
g2: 14899.2
g3: 6.54775e+07
g4: 4.71655e+07
g5: 1.61061e+10
 g2 g4 g3 g1 g5

n: 2048
g1: inf
g2: 38947.4
g3: 2.8796e+08
g4: 2.07694e+08
g5: 1.28849e+11
 g2 g4 g3 g5 g1

n: 4096
g1: inf
g2: 101811
g3: 1.25625e+09
g4: 9.03472e+08
g5: 1.03079e+12
 g2 g4 g3 g5 g1

n: 8192
g1: inf
g2: 266140
g3: 5.44307e+09
g4: 3.89214e+09
g5: 8.24634e+12
 g2 g4 g3 g5 g1

n: 16384
g1: inf
g2: 695707
g3: 2.34457e+10
g4: 1.66359e+10
g5: 6.59707e+13
 g2 g4 g3 g5 g1

n: 32768
g1: inf
g2: 1.81862e+06
g3: 1.00478e+11
g4: 7.06453e+10
g5: 5.27766e+14
 g2 g4 g3 g5 g1

n: 65536
g1: inf
g2: 4.754e+06
g3: 4.28701e+11
g4: 2.98373e+11
g5: 4.22212e+15
 g2 g4 g3 g5 g1

n: 131072
g1: inf
g2: 1.24273e+07
g3: 1.82197e+12
g4: 1.25439e+12
g5: 3.3777e+16
 g2 g4 g3 g5 g1

n: 262144
g1: inf
g2: 3.24856e+07
g3: 7.71653e+12
g4: 5.2528e+12
g5: 2.70216e+17
 g2 g4 g3 g5 g1

n: 524288
g1: inf
g2: 8.49195e+07
g3: 3.25808e+13
g4: 2.19211e+13
g5: 2.16173e+18
 g2 g4 g3 g5 g1

n: 1048576
g1: inf
g2: 2.21985e+08
g3: 1.37182e+14
g4: 9.12084e+13
g5: 1.72938e+19
 g2 g4 g3 g5 g1

n: 2097152
g1: inf
g2: 5.80283e+08
g3: 5.76166e+14
g4: 3.78499e+14
g5: 1.38351e+20
 g2 g4 g3 g5 g1

n: 4194304
g1: inf
g2: 1.5169e+09
g3: 2.41441e+15
g4: 1.56705e+15
g5: 1.1068e+21
 g2 g4 g3 g5 g1

n: 8388608
g1: inf
g2: 3.96527e+09
g3: 1.00966e+16
g4: 6.47442e+15
g5: 8.85444e+21
 g2 g4 g3 g5 g1

n: 16777216
g1: inf
g2: 1.03655e+10
g3: 4.21424e+16
g4: 2.66999e+16
g5: 7.08355e+22
 g2 g4 g3 g5 g1

n: 33554432
g1: inf
g2: 2.7096e+10
g3: 1.75593e+17
g4: 1.09924e+17
g5: 5.66684e+23
 g2 g4 g3 g5 g1

n: 67108864
g1: inf
g2: 7.08306e+10
g3: 7.30468e+17
g4: 4.51869e+17
g5: 4.53347e+24
 g2 g4 g3 g5 g1

n: 134217728
g1: inf
g2: 1.85156e+11
g3: 3.03425e+18
g4: 1.85497e+18
g5: 3.62678e+25
 g2 g4 g3 g5 g1

n: 268435456
g1: inf
g2: 4.84009e+11
g3: 1.25865e+19
g4: 7.60524e+18
g5: 2.90142e+26
 g2 g4 g3 g5 g1

n: 536870912
g1: inf
g2: 1.26523e+12
g3: 5.21442e+19
g4: 3.11451e+19
g5: 2.32114e+27
 g2 g4 g3 g5 g1

增长顺序如下：

2 个答案: