1.
Counting the number of bits set CH Gowri Kumar [email_address]

2.
Copyright Notice <ul><li>© 2008, www.gowrikumar.com </li></ul><ul><li>You are given unrestricted freedom to </li></ul><ul><li>use, modify, copy and redistribute in </li></ul><ul><li>any format/media , as long as you retain </li></ul><ul><li>this copyright statement. </li></ul>

3.
Problem Statement <ul><li>Given an unsigned integer n, count the number of bits set in it. </li></ul><ul><li>Ex : </li></ul><ul><ul><li>3  11  2 </li></ul></ul><ul><ul><li>8  1000  1 </li></ul></ul><ul><ul><li>23  10111  4 </li></ul></ul>

4.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1

5.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1

6.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1

7.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1

8.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1

9.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1

10.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0

11.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0

12.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0

13.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0

14.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 0

15.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 0

16.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0

17.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1

18.
0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1

19.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1 0 1

20.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1

21.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1

22.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1

23.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1

24.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1

25.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1

26.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 1

27.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 1

28.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 0 1

29.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 0 1

30.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 0 1

31.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1

32.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1

33.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1

34.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 1

35.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 1

36.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 1

37.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1

38.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1

39.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1

40.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1

41.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1

42.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1

43.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1

44.
+ + + + + + + + 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1

45.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1

46.
C code <ul><li>x = (x & 0x5555) + ((x>>1) & 0x5555) </li></ul>

47.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1

48.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1

49.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1

50.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1

51.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1

52.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1

53.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1

54.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1

55.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1

56.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1

57.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1

58.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1

59.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0

60.
+ + + + 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1

61.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0

62.
C code <ul><li>x = (x & 0x5555) + ((x>>1) & 0x5555) </li></ul><ul><li>x = (x & 0x3333) + ((x>>2) & 0x3333) </li></ul>

63.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0

64.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0

65.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0

66.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0

67.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0

68.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0

69.
+ + 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0

70.
0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0

71.
C code <ul><li>x = (x & 0x5555) + ((x>>1) & 0x5555) </li></ul><ul><li>x = (x & 0x3333) + ((x>>2) & 0x3333) </li></ul><ul><li>x = (x & 0x0F0F) + ((x>>4) & 0x0F0F) </li></ul>

72.
+ 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0

73.
+ 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0

74.
0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0

75.
C code <ul><li>x = (x & 0x5555) + ((x>>1) & 0x5555) </li></ul><ul><li>x = (x & 0x3333) + ((x>>2) & 0x3333) </li></ul><ul><li>x = (x & 0x0F0F) + ((x>>4) & 0x0F0F) </li></ul><ul><li>x = (x & 0x00FF) + (x>>16) & 0x00FF) </li></ul>

76.
Code for 32-bit integers <ul><li>x = (x&0x55555555) + ((x>>1)&0x55555555) </li></ul><ul><li>x = (x&0x33333333) + ((x>>2)&0x33333333) </li></ul><ul><li>x = (x&0x0F0F0F0F) + ((x>>4)&0x0F0F0F0F) </li></ul><ul><li>x = (x&0x00FF00FF) + ((x>>8)&0x00FF00FF) </li></ul><ul><li>x = (x&0x0000FFFF) + ((x>>16)&0x0000FFFF) </li></ul><ul><li>return x; </li></ul>

77.
Reference <ul><li>Hackers Delight </li></ul>

78.
The End