排序不等式(排序原理)
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设数组aaa, bbb和ccc
a[ ]为a1<=a2<=a3....<=an a[ ]数组递增a[~]为a_1<=a_2<=a_3....<=a_n~~a[~]数组递增a[ ]为a1<=a2<=a3....<=an a[ ]数组递增
b[ ]为b1<=b2<=b3....<=bn b[ ]数组递增b[~]为b_1<=b_2<=b_3....<=b_n~~b[~]数组递增b[ ]为b1<=b2<=b3....<=bn b[ ]数组递增
c[ ]为b[ ]数组的一个排列c[~]为b[~]数组的一个排列c[ ]为b[ ]数组的一个排列
记
S1=a1∗bn+a2∗bn−1+a3∗bn−2+...+an∗b1S_1=a_1*b_n + a_2*b_{n-1} + a_3*b_{n-2} + ...+a_n*b_1S1=a1∗bn+a2∗bn−1+a3∗bn−2+...+an∗b1 (反序和)
S=a1∗c1+a2∗c2+a3∗c3+.....an∗cnS=a_1*c_1+a_2*c_2+a_3*c_3+.....a_n*c_nS=a1∗c1+a2∗c2+a3∗c3+.....an∗cn (乱序和)
S2=a1∗b1+a2∗b2+a3∗b3+......+an∗bnS_2=a_1*b_1+a_2*b_2+a_3*b_3+......+a_n*b_nS2=a1∗b1+a2∗b2+a3∗b3+......+an∗bn (顺序和)
则 S1<=S<=S2S_1<=S<=S2S1<=S<=S2
即 : 反序和 <= 乱序和 <= 顺序和
可用排序原理证明
- a2+b2>=2aba^2+b^2>=2aba2+b2>=2ab (均值不等式)
- a2+b2+c2>=ab+bc+caa^2+b^2+c^2>=ab+bc+caa2+b2+c2>=ab+bc+ca
打水问题,nnn个人排队打水,第iii个人要等a[i]a[i]a[i]的单位时间,如何才能使得所有人的等待时间最小
很明显让最快的人先打水更优