两个数字x和y来自两个不同的数组。 查找是否存在总和z使得z = x + y

Question

I need to add that there are n integers in each array, and each integer is between 0 and n^5. Is there any way to solve this problem in linear-time algorithm?

Answer 1

Yes it's possible in linear time under these assumptions:

• Your inputs are arrays of (for example) 32-bit integers.
• Adding two integers is an O(1) operation.
• Your machine has unlimited memory and reading a byte anywhere in memory is an O(1) operation.

1) Convert one of the arrays into a hash set with approximately O(1) time complexity for lookups. Construction of the hash set takes approximately linear time.

2) Iterate over the other array and for each element i, check if x - i is in the hash set. If there is a match then (i, x - i) is a solution. This step requires linear time.

Answer 2

I can not think of a linear time algorithm, but I can think of a O (m log n) solution where m is the length of the larger list and n is the length of the smaller list.

let n be the length of the smaller list and m be the length of the larger list.

Step 1: sort the smaller list (merge sort for example): O(n log n) Step 2: for each item in the larger list, try to find (target number - item) in the sorted list. If you find a match, you have found the two numbers you are looking for. O (m log n).

The complexity is O (n log n) + O (m log n) which is O (m log n) because m is the larger list.