Amortized analysis
10/28/21
Administrivia
• Read Section 17.3 for tomorrow (10/29)
Where we left off: Subset sum
Given a set of n (positive) values v 1 , v 2 , ..., v n , is there a subset of them summing to S?
• T[i][j] = whether can make i using v 1 , v 2 , ..., v j
• T[0][j] = true for all j
• T[i][0] = false for i ≠ 0
• T[i][j] = T[i][j-1] || T[i-v
j][j-1]
Recall: Adding to an ArrayList
Doubling the array void add(T value) {
if(num == vals.length) {
T[] temp = (T[]) new Object[num*2];
for(int i=0; i<num; i++) temp[i] = vals[i];
vals = temp;
}
vals[num] = value;
num++;
}
Growing the array by 1 void add(T value) {
if(num == vals.length) {
T[] temp = (T[]) new Object[num+1];
for(int i=0; i<num; i++) temp[i] = vals[i];
vals = temp;
}
vals[num] = value;
num++;
}
Recall: Adding to an ArrayList
Doubling the array void add(T value) {
if(num == vals.length) {
T[] temp = (T[]) new Object[num*2];
for(int i=0; i<num; i++) temp[i] = vals[i];
vals = temp;
}
vals[num] = value;
num++;
}
Growing the array by 1 void add(T value) {
if(num == vals.length) {
T[] temp = (T[]) new Object[num+1];
for(int i=0; i<num; i++) temp[i] = vals[i];
vals = temp;
}
vals[num] = value;
num++;
}
What are the worst-case running times of these methods?
A. O(1) and O(1) B. O(1) and O(n) C. O(n) and O(1)
D. O(n) and O(n) E. None of the above
Recall: Adding to an ArrayList
Doubling the array void add(T value) {
if(num == vals.length) {
T[] temp = (T[]) new Object[num*2];
for(int i=0; i<num; i++) temp[i] = vals[i];
vals = temp;
}
vals[num] = value;
num++;
}
Growing the array by 1 void add(T value) {
if(num == vals.length) {
T[] temp = (T[]) new Object[num+1];
for(int i=0; i<num; i++) temp[i] = vals[i];
vals = temp;
}
vals[num] = value;
num++;
}
What are the worst-case running times of these methods?
A. O(1) and O(1) B. O(1) and O(n) C. O(n) and O(1)
D. O(n) and O(n) E. None of the above
Both have O(n) worst case time per operation, but…
• Times (in sec) for adding chars one at a time to a list of chars:
Final length doubling incrementing
