计算和存储超大功率

Question

I am finding pow(2,i) where i can range: 0<=i<=100000. Apart i have MOD=1000000007

powers[100000];
powers[0]=1;
for (i = 1; i <=100000; ++i)
{
  powers[i]=(powers[i-1]*2)%MOD;
}

for i=100000 won't power value become greater than MOD ?

How do I store the power correctly?

The operation doesn't look feasible to me. I am getting correct value up to i=70 max I guess.

I have to find sum+= ar[i]*power(2,i) and finally print sum%1000000007 where ar[i] is an additional array with some big numbers up to 10^5

Answer 1

As long as your modulus value is less than half the capacity of your data type, it will never be exceeded. That's because you take the previous value in the range 0..1000000006 , double it, then re-modulo it bringing it back to that same range.

However, I can't guarantee that higher values won't cause you troubles, it's more mathematical analysis than I'm prepared to invest given the simple alternative. You could spend a lot of time analysing, checking and debugging, but it's probably better just to not allow the problem to occur in the first place.

The alternative? I'd tend to use the pre-generation method (having a program do the gruntwork up front, inserting the pre-generated values into an array easily and speedily accessible from your real program).

With this method, you can use tools that are well tested and known to work with massive values. Since this data is not going to change, it's useless calculating it every time your program starts.

If you want an easy (and efficient) way to do this, the following bash script in conjunction with bc and awk can do this:

#!/usr/bin/bash

bc >nums.txt <<EOF
    i = 1;
    for (x = 0;x <= 10000; x++) {
        i % 1000000007;
        i = i * 2;
    }
EOF

awk 'BEGIN { printf "static int array[] = {" }
           { if (NR % 5 == 1) printf "\n    ";
             printf "%s, ",$0;
             next
           }
     END   { print "\n};" }' nums.txt

The bc part is the "meat" of the matter, it creates the large powers of two and outputs them modulo the number you provided. The awk part is simply to format them in C-style array elements, five per line.

Just take the output of that and put it into your code and, voila, there you have it, a compile-time-expensed array that you can use for fast lookup.

It takes only a second and a half on my box to generate the array and then you never need to do it again. You also won't have to concern yourself with the vagaries of modulo math :-)

static int array[] = {
    1,2,4,8,16,
    32,64,128,256,512,
    1024,2048,4096,8192,16384,
    32768,65536,131072,262144,524288,
    1048576,2097152,4194304,8388608,16777216,
    33554432,67108864,134217728,268435456,536870912,
    73741817,147483634,294967268,589934536,179869065,
    359738130,719476260,438952513,877905026,755810045,
    511620083,23240159,46480318,92960636,185921272,
    371842544,743685088,487370169,974740338,949480669,
    898961331,797922655,595845303,191690599,383381198,
    766762396,533524785,67049563,134099126,268198252,
    536396504,72793001,145586002,291172004,582344008,
    164688009,329376018,658752036,317504065,635008130,
    270016253,540032506,80065005,160130010,320260020,
    640520040,281040073,562080146,124160285,248320570,
    :
    861508356,723016705,446033403,892066806,784133605,
    568267203,136534399,273068798,546137596,92275185,
    184550370,369100740,738201480,476402953,952805906,
    905611805,
};

Answer 2

If you notice that your modulo can be stored in int. MOD=1000000007(decimal) is equivalent of 0b00111011100110101100101000000111 and can be stored in 32 bits.

 - i      pow(2,i)        bit representation 
 - 0            1         0b00000000000000000000000000000001
 - 1            2         0b00000000000000000000000000000010 
 - 2            4         0b00000000000000000000000000000100 
 - 3            8         0b00000000000000000000000000001000
 - ...
 - 29   536870912         0b00100000000000000000000000000000

Tricky part starts when pow(2,i) is grater than your MOD=1000000007, but if you know that current pow(2,i) will be greater than your MOD, you can actually see how bits look like after MOD

 - i      pow(2,i)  pow(2,i)%MOD      bit representation
 - 30   1073741824  73741817    0b000100011001010011000000000000
 - 31   2147483648  147483634   0b001000110010100110000000000000
 - 32   4294967296  294967268   0b010001100101001100000000000000
 - 33   8589934592  589934536   0b100011001010011000000000000000

So if you have pow(2,i-1)%MOD you can do *2 actually on pow(2,i-1)%MOD till you're next pow(2,i) will be greater than MOD.

In example for i=34 you will use (589934536*2) MOD 1000000007 instead of (8589934592*2) MOD 1000000007, because 8589934592 can't be stored in int.

Additional you can try bit operations instead of multiplication for pow(2,i). Bit operation same as multiplication for 2 is bit shift left .