1, time complexity

First, the algorithm is introduced

import time
start_time = time.time()
for a in range(0, 1001):
    for b in range(0, 1001):
        for c in range(0, 1001):
            if a**2 + b**2 == c**2 and a+b+c == 1000:
                print("a, b, c: %d, %d, %d" % (a, b, c))
end_time = time.time()
print("elapsed: %f" % (end_time - start_time))
print("complete!")

a, b, c: 0, 500, 500
a, b, c: 200, 375, 425
a, b, c: 375, 200, 425
a, b, c: 500, 0, 500
elapsed: 214.583347
complete!

依据三个变量之间的关系，减少一个变量，降低维度

import time
start_time = time.time()
for a in range(0, 1001):
    for b in range(0, 1001):
        c = 1000-a-b
        if a**2 + b**2 == c**2:
            print("a, b, c:%d, %d, %d" % (a, b, c))
end_time = time.time()
print("times:%d" % (end_time - start_time))

a, b, c:0, 500, 500
a, b, c:200, 375, 425
a, b, c:375, 200, 425
a, b, c:500, 0, 500
times:1

Second, the time complexity

1. Why do we need the complexity of

事后统计法：把代码跑一遍，通过统计、监控，就能得到算法执行的时间和占用的内存大小。

• The test results are very dependent test environment

• Test results are affected by large-scale data
  needs without a specific test data to test the method on the efficiency of the algorithm can be roughly estimated.

  2, a large degree of complexity O notation

  All code execution time T (n) n is proportional to the frequency of each line of code.
  Big O time complexity of a code indicating the execution time with the scale of growth trends in data, also known as asymptotic time complexity, short time complexity.

• Time complexity analysis
  only concerned with the number of cycles up to a code
  addition rule: the complexity of the overall complexity equal to the magnitude of the largest part of the code of
  the multiplication rule: the complexity of nested code equal to the complexity of the inner and outer nested code

  3, several common time complexity of the case study

• (1) O
  In general, as long as the loop, a recursive algorithm statement does not exist, even if there are thousands of lines of code, the time complexity is O (1).
• O (logn), O (nlogn )
  when using large tag complexity O, the coefficient can be neglected, i.e. O (Cf (n)) = O (f (n))

• O (m + n), O (m * n)
  code complexity is determined by the size of two data

  4, space complexity analysis

  Asymptotic space complexity: that the relationship between the growth of storage algorithms and data scale.

  5, the time complexity of the possibilities:

  Algorithm to complete the work requires a minimum number of basic operations, that is the optimal time complexity of
  how much the basic arithmetic operations take up the job done, that is the worst time complexity of
  the algorithm to complete the work required average number of basic operations, namely the average time complexity