算法課_算法分析_
O(n^2),
O(nlgn),
O(n)
t=an^b+c
given two groups of t and n, the b can be calculucated by lg(t2-t1)=blgn+lga, assuming that t and n are large enough.
空間複雜度
Question 1
3-SUM in quadratic time. Design an algorithm for the 3-SUM problem that takes time proportional to N2 in
the worst case. You may assume that you can sort the N integers
in time proportional to N2 or
better.
| Your Answer | Score | Explanation | |
|---|---|---|---|
| Total | 0.00 / 0.00 |
Question Explanation
Hint: given an integer x and a sorted array a[] of N distinct integers, design a linear-time algorithm to determine if there exists two distinct indicesi and j such that a[i] + a[j] == x.
Hint: given an integer x and a sorted array a[] of N distinct integers, design a linear-time algorithm to determine if there exists two distinct indicesi and j such that a[i] + a[j] == x.
即：sort 數組a，對a裏面的每一個小於等於0的數a[j]，找出-a[j]是否在數組中，用lgn的時間，找出最接近-a[j]的位置i，滿足a[i]+a[j]<=0，然後再在i同時向左右看，l=i，r=i+1，a[j]+a[l]+a[r]，如果<0，則向右移r=r+1；>0，則向左移，l=l-1；=0，則左右都移。
Question 2
Search in a bitonic array. An array is bitonic if it is comprised of an increasing sequence of integers followed immediately by a decreasing sequence of integers. Write a program that, given a bitonic array of N distinct
integer values, determines whether a given integer is in the array.
Standard version: Use ∼3lgN compares
in the worst case.
Signing bonus: Use ∼2lgN compares
in the worst case (and prove that no algorithm can guarantee to perform fewer than ∼2lgNcompares
in the worst case).
| Your Answer | Score | Explanation | |
|---|---|---|---|
| Total | 0.00 / 0.00 |
Question Explanation
Hints: Standard version. First, find the maximum integer using ∼1lgN compares—this divides the array into the increasing and decreasing pieces.
Hints: Standard version. First, find the maximum integer using ∼1lgN compares—this divides the array into the increasing and decreasing pieces.
找最大值時只需 less(a[i-1],a[i]) == true 向右 false 向左，直到最後
Signing bonus. Do it without finding the maximum integer.這個太複雜，實現很麻煩。考慮單調性和大小兩個邏輯。而且我認爲2lgn這個有錯，應該是3lgn。worsest case 3lgn： 51...,100,49,...1，找50是否存在，a[50]=100，a[49]=99，則左邊單調增，朝下找50，用1*lg50的時間，右邊a[75]=25，a[74]=26,單調減，要判斷單調性和a[75]與50的大小，2lg50。綜合起來3lg50，worse case，吃力不討好。上面一段是我一開始的想法，在論壇裏看到了別人的2lnN的解法。在另一篇文檔裏Question 3
Egg drop. Suppose that you have an N-story
building and plenty of eggs. An egg breaks if it is dropped from floor T or
higher and does not break otherwise. Your goal is to devise a strategy to determine the value of T given
the following limitations on the number of eggs and tosses:
Version 0: 1 egg, ≤T tosses.
Version 1: ∼1lgN eggs
and ∼1lgN tosses.
Version 2: ∼lgT eggs
and ∼2lgT tosses.
Version 3: 2 eggs
and ∼2N−−√ tosses.
Version 4: 2 eggs
and ≤cT−−√ tosses
for some fixed constant c.
| Your Answer | Score | Explanation | |
|---|---|---|---|
| Total | 0.00 / 0.00 |
Question Explanation
Hints: Version 0: sequential search. 一個一個對，1，2，3，4，5的找 Version 1: binary search. 二分 Version 2: find an interval containing T of size ≤2T, then do binary search. 1，2，4，8，..，2^n search，找到sizeT的區間containingT，然後做二分。 對於T<N/2比較好。比較好是次數比較少。Version 3: find an interval of size N−−√, then do sequential search. Note: can be improved to ∼2N−−−√ tosses. 1，sqrt(N)，2*sqrt(N)，，，sqrt(N)*sqrt(N)這樣的序列找T所在的區間 Version 4: 1+2+3+…+t∼12t2. Aim for c=22√. 對於 T>N/2 更好點
Hints: Version 0: sequential search. 一個一個對，1，2，3，4，5的找 Version 1: binary search. 二分 Version 2: find an interval containing T of size ≤2T, then do binary search. 1，2，4，8，..，2^n search，找到sizeT的區間containingT，然後做二分。 對於T<N/2比較好。比較好是次數比較少。Version 3: find an interval of size N−−√, then do sequential search. Note: can be improved to ∼2N−−−√ tosses. 1，sqrt(N)，2*sqrt(N)，，，sqrt(N)*sqrt(N)這樣的序列找T所在的區間 Version 4: 1+2+3+…+t∼12t2. Aim for c=22√. 對於 T>N/2 更好點