Leetcode algorithm series _0891_ The sum of the width of the subsequence

Leetcode algorithm series _0891_ The sum of the width of the subsequence

Title Description

Given an integer array A, consider all non-empty sequences of a.

For any sequence s, the width of S is the difference between the largest and smallest elements of S.

Returns the sum of the widths of all the sub-sequences of a.

Since the answer can be very large, please return to the answer modulo 10^9+7.

Example 1:

输入：[2,1,3]输出：6解释：子序列为 [1]，[2]，[3]，[2,1]，[2,3]，[1,3]，[2,1,3] 。相应的宽度是 0，0，0，1，1，2，2 。这些宽度之和是 6 。

Tips:

• 1 <= a.length <= 20000
• 1 <= a[i] <= 20000

Algorithm

const mod = 1e9 + 7func sumSubseqWidths(a []int) int {    //[3,2,4,1] 和 排序后的 [1,2,3,4] 宽度之和相同    sort.Ints(a)    n := len(a)    res := 0    /**    作为最大值出现的次数    a[0] a[1] a[2]    1    2    4    [2,1,3],3作为最大值进行排列组合    [2,3],[1,3][,2,1,3],[3]    */    times := 1    for i := 0; i < n; i++ {        //a[i]作为最大值出现的次数==a[n-1-i]作为最小值出现的次数        res += (a[i] - a[n-1-i]) * times        //res可能非常大,所以取模        res %= mod        //times可能非常大,取模        times = (times << 1) % mod    }    return res}

Personal ideas

1. A subsequence is a subset of array elements that are arranged to form several arrays, the maximum value in the array-the minimum value that is the subarray width
2. [3,2,4,1] and [1,2,3,4] have different sub-sequences, but the sum of the width of the subsequence is the same
3. Sort an array, and the width of a subsequence is the difference between the tail and the elements
4. Example: Array [1,2,3,4], each element may be the maximum value of a subsequence, the minimum value, n is the length of the array
5. A[i] As the maximum value, there are I elements smaller than it, can form a 2^i sub-sequence
6. A[i] As a minimum, there are n-1-i elements larger than it, which can form a 2^ (n-1-i) sub-sequence
7. Rule: a[0] as the maximum number of subsequence ==a[n-1] as the minimum number of sub-series, the same a[1] and a[n-1-1] ..., that is, a[i] the minimum number of times equal to A[n-1-i]
8. Sum of the subsequence width = (max a[i]2^i+ ...) -(minimum value a[n-1-i]2^i ...)
9. The sum of the subsequence width = (a[i]-a[n-1-i]*2^i) + ...
10. The value of 2^i, which in turn is 1,2,4,8 ...., it will be large, so you need to take a model of 10^9+7

Summarize

• Solve the problem, always find out the mathematical law behind the problem

GitHub

• Project Source is here
• The author will always maintain the project, solve the algorithm problem in Leetcode, and write down his own ideas and opinions, and devote to the algorithm that everyone can understand.

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