1. Computing Spans Given an an array X, the span S[i] of X[i] is
the maximum number of consecutive elements X[j] immediately preceding X[i]
such that X[j]  X[i]
Spans have applications to financial analysis
E.g., stock at 52-week high X 6 3 4 5 2 1 S
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Stacks

2. Algorithm 1 For each index i Array size is N
Scan backward to find a higher value
Array size is N
Worst-case time complexity?
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Stacks

3. Quadratic Algorithm Algorithm spans1(X, n) Input array X of n integers
Output array S of spans of X #
S  new array of n integers n
for i  0 to n  1 do n
s  n
while s  i  X[i - s]  X[i] …+ (n  1)
s  s …+ (n  1)
S[i]  s n
return S
Algorithm spans1 runs in O(n2) time
© 2014 Goodrich, Tamassia, Goldwasser
Stacks

4. Computing Spans with a Stack
We keep in a stack the indices of the last element that is taller when “looking back”
We scan the array from left to right
Let i be the current index
We pop indices from the stack until we find index j such that X[i]  X[j]
We set S[i]  i - j
We push i onto the stack
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Stacks

5. Algorithm 2
Keep on the stack the last building that is taller than the current one
Root top that can’t be seen
Ie. While a building is shorter on the stack
Pop the building
© 2014 Goodrich, Tamassia, Goldwasser
Stacks

6. Algorithm 2 While shorter building on the stack pop the building
Push current building
Stack (top on the right)
[push 0]
© 2014 Goodrich, Tamassia, Goldwasser
Stacks

7. Algorithm 2 While shorter building on the stack pop the building
Push current building
Stack (top on the right)
[push 0]
0, [push 1]
© 2014 Goodrich, Tamassia, Goldwasser
Stacks

8. Algorithm 2 While shorter building on the stack pop the building
Push current building
Stack (top on the right)
[push 0]
0, [push 1]
0, [pop 1, push 2]
© 2014 Goodrich, Tamassia, Goldwasser
Stacks

9. Algorithm 2 While shorter building on the stack pop the building
Push current building
Stack (top on the right)
[push 0]
0, [push 1]
0, [pop 1, push 2]
0,2, [push 3]
© 2014 Goodrich, Tamassia, Goldwasser
Stacks

10. Algorithm 2 While shorter building on the stack pop the building
Push current building
Stack (top on the right)
[push 0]
0, [push 1]
0, [pop 1, push 2]
0,2, [push 3]
0,2, [pop 3, push 4]
© 2014 Goodrich, Tamassia, Goldwasser
Stacks

11. Algorithm 2 While shorter building on the stack pop the building
Push current building
Stack (top on the right)
[push 0]
0, [push 1]
0, [pop 1, push 2]
0,2, [push 3]
0,2, [pop 3, push 4]
0,2, [pop 4, push 5]
© 2014 Goodrich, Tamassia, Goldwasser
Stacks

12. Algorithm 2 While shorter building on the stack pop the building
Push current building
Stack (top on the right)
[push 0]
0, [push 1]
0, [pop 1, push 2]
0,2, [push 3]
0,2, [pop 3, push 4]
0,2, [pop 4, push 5]
0, [pop 5,2; push 6]
© 2014 Goodrich, Tamassia, Goldwasser
Stacks

13. Algorithm 2 While shorter building on the stack pop the building
Push current building
Stack (top on the right)
[push 0]
0, [push 1]
0, [pop 1, push 2]
0,2, [push 3]
0,2, [pop 3, push 4]
0,2, [pop 4, push 5]
0, [pop 5,2; push 6]
0,6, [push 7]
© 2014 Goodrich, Tamassia, Goldwasser
Stacks

14. Algorithm 2 Stack (top on the right) Span 0 [push 0] 1 0,1 [push 1] 1
0, [pop 1, push 2]
0,2, [push 3]
0,2, [pop 3, push 4]
0,2, [pop 4, push 5]
0, [pop 5,2; push 6]
0,6, [push 7]
How to get the span?
© 2014 Goodrich, Tamassia, Goldwasser
Stacks

15. Algorithm 2 Stack (top on the right) Span 0 [push 0] 1 0,1 [push 1] 1
0, [pop 1, push 2]
0,2, [push 3]
0,2, [pop 3, push 4]
0,2, [pop 4, push 5]
0, [pop 5,2; push 6]
0,6, [push 7]
How to get the span?
© 2014 Goodrich, Tamassia, Goldwasser
Stacks

16. Algorithm 2 Stack (top on the right) Span 0 [push 1] 1 0,1 [push 1] 1
0, [pop 1, push 2]
0,2, [push 3]
0,2, [pop 3, push 4]
0,2, [pop 4, push 5]
0, [pop 5,2; push 6]
0,6, [push 7]
What is the time complexity (stack operations)?
© 2014 Goodrich, Tamassia, Goldwasser
Stacks

17. Linear Time Algorithm Each index of the array
Is pushed into the stack exactly one
Is popped from the stack at most once
The statements in the while-loop are executed at most n times
Algorithm spans2 runs in O(n) time
Algorithm spans2(X, n) #
S  new array of n integers n
A  new empty stack
for i  0 to n  1 do n
while (A.isEmpty() 
X[A.top()]  X[i] ) do n
A.pop() n
if A.isEmpty() then n
S[i]  i n
else
S[i]  i - A.top() n
A.push(i) n
return S
© 2014 Goodrich, Tamassia, Goldwasser
Stacks

18. Worst-case Analysis N data items Algorithm 1 Algorithm 2
Time Complexity
Space Complexity
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Stacks

19. Worst-case Analysis N data items Algorithm 1 Algorithm 2
Time Complexity
O(N2)
O(N)
Space Complexity
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Stacks

20. Which Data Structure?
Suppose that intList is a variable of type Collection<Integer>, i.e., intList is a list of integer elements. Write a code segment that uses an iterator to compute the sum of all the integer values in the collection. Write a second code segment that does the same thing using a for-each loop.
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Stacks

21. Which Data Structure?
For each of the following scenarios choose the “best” data structure from the following list or a combination of data structures: an unsorted array, linked list, Doubly Linked List, circular Linked List, stack, queue. In each case, justify your answer briefly.
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22. Which Data Structure?
Suppose that a grocery store decided that customers who come first will be served first
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23. Which Data Structure?
A list must be maintained so that any element can be accessed randomly
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24. Which Data Structure?
A program needs to remember operations it performed in opposite order
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25. Which Data Structure?
The size of a file is unknown. The entries need to be entered as they come in. Entries must be deleted when they are no longer needed. It is important that structure has flexible memory management
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Stacks

26. Which Data Structure?
A list must be maintained so that elements can be added to the beginning or end in O(1)
© 2014 Goodrich, Tamassia, Goldwasser
Stacks