4-7

a) 必要性显然成立
充分性：
∵∑k−i−1l=0(A[i+l,j]+A[i+l+1,j+1])≤∑k−i−1l=0(A[i+l,j+1]+A[i+l+1,j])
∴A[i,j]+A[k,j+1]≤A[i,j+1]+A[k,j]

∵∑t−l−1l=0(A[i,j+l]+A[k,j+l+1])≤∑t−l−1l=0(A[i,j+l+1]+A[k,j+l])
∴A[i,j]+A[k,t]≤A[i,t]+A[k,j]
故得证
b) 把22改成24
c) 若存在f(i)>f(i+1)，
则A[i,f(i+1)]+A[i+1,f(i)]≥A[i,f(i)]+A[i+1,f(i+1)]
与Monge性质矛盾，故f(i)≤f(i+1)
d) 记第i个偶数行的f值为gi，第i个奇数行的f值为hi，则
gi≤hi≤gi+1,只需检查gi+1−gi+1次，总次数为n+m/2=O(n+m)
e) T(m)=T(m/2)+cm+dn,T=O(nlgm+m)