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Correction of quizzes
Correction of quizzes
Correction of quizzes
Correction of quizzes
ADTs and implementations
Hash tables
Graphs
ADTs and implementationADTs and implementation
Correction of quizzes
ADTs: what can they do?Which ADTs have we seeen?
List Dictionary PriorityQueue Queue Stack
Each of these ADTs declares a set of functions that an implementation must be able to do.
add
getMax
removeMax
addLast
getFirst
removeFirst
addLast
getLast
removeLast
add(key)
get(key)
remove(key)
add(i)
get(i)
remove(i)
If there is no way to perform one function given the ones that are declared, then it’s N/A.
Correction of quizzes
ADTs: what can they do?Which ADTs have we seeen?
List Dictionary PriorityQueue Queue Stack
add
getMax
removeMax
addLast
getFirst
removeFirst
addLast
getLast
removeLast
add(key)
get(key)
remove(key)
add(i)
get(i)
remove(i)
ExampleExample: Can we perform addLast using a List?
Yes. In a list you can indicate where you want the element to be added and you have access to the size for all data structures, so for a list L call L.add(o,L.size());
Correction of quizzes
ADTs: what can they do?Which ADTs have we seeen?
List Dictionary PriorityQueue Queue Stack
add
getMax
removeMax
addLast
getFirst
removeFirst
addLast
getLast
removeLast
add(key)
get(key)
remove(key)
add(i)
get(i)
remove(i)
ExampleExample: Can we perform addLast using a Dictionary?
No. There is no sense of « first » or « last » in a dictionary because it is not a linear structure using an index i, but a structure that uses a key.
Correction of quizzes
ADTs: what can they do?Which ADTs have we seeen?
List Dictionary PriorityQueue Queue Stack
add
getMax
removeMax
addLast
getFirst
removeFirst
addLast
getLast
removeLast
add(key)
get(key)
remove(key)
add(i)
get(i)
remove(i)
ExampleExample: Can we perform removeMax using a List?
No. You can only use the index if you want to remove a specific object, but you do not know which element is the max.
Yes you could always have a loop that finds where is the max, keeps its index and ask to remove it. But this is not something you can do directly.
Correction of quizzes
ADTs: what can they do?Which ADTs have we seeen?
Stack
addLast
getLast
removeLast
Lets consider a stack using pointers:
• Add
• AddFirst
• AddLast
• Remove
• RemoveMax
• Get
• GetMax
N/A
N/A
N/A
N/A
N/A
N/A
O(1)
all 3 in O(1)
What if we used an array?
addLast: O(n) due to resize
getLast: O(1) because the index tells us directly where the last is
removeLast: O(1) if at the end
Correction of quizzes
ADTs: what can they do?Which ADTs have we seeen?
Queue
addLast
getFirst
removeFirst
Lets consider a queue using pointers:
• Add
• AddFirst
• AddLast
• Remove
• RemoveMax
• Get
• GetMax
N/A
N/A
N/A
N/A
N/A
N/A
O(1)
O(1)
What if we used an array?
addLast: O(n) due to resize
getFirst: O(1) thanks to index
removeFirst: O(n) due to shifting (if first is in the first cell)
Correction of quizzes
ADTs: what can they do?Which ADTs have we seeen?
List
add(i)
get(i)
remove(i)
Lets consider a List using pointers, with a shortcut to the tail:
• Add
• AddFirst
• AddLast
• Remove
• RemoveMax
• Get
• GetMax
O(n)
O(1)
O(1)
O(n)
N/A
O(n)
N/A
And implemented via an array?
O(n)
O(n)
O(n)
O(n)
O(1)
N/A
N/A
Correction of quizzes
ADTs: what can they do?Which ADTs have we seeen?
Dictionary
Dictionaries have many implementations.
Binary Search Tree (BST) Hash tables
Basic (not balanced)
AVL (balanced)
Red black (balanced)
Using probing Using chaining
Double hashing
Linear probing
Quadratic probing
LinkedList
Balanced BST
Hashtable
arraypointers
Correction of quizzes
ADTs: what can they do?Which ADTs have we seeen?
Dictionary
Dictionaries have many implementations.
Hash tables
Using probing
Double hashing
array• Add
• AddFirst
• AddLast
• Remove
• RemoveMax
• Get
• GetMax
O(n)
N/A
N/A
O(1)
N/A
O(1)
N/A
Using chaining
Balanced BST
Correction of quizzes
ADTs: what can they do?Which ADTs have we seeen?
Dictionary
Dictionaries have many implementations.
Hash tables
array• Add
• AddFirst
• AddLast
• Remove
• RemoveMax
• Get
• GetMax
O(log m)
N/A
N/A
O(log m)
N/A
O(log m)
N/A
Using chaining
Balanced BST
LinkedListWhere m is the
maximum number of elements for any
given cell.
Correction of quizzes
ADTs: what can they do?Which ADTs have we seeen?
Dictionary
Dictionaries have many implementations.
Hash tables
array• Add
• AddFirst
• AddLast
• Remove
• RemoveMax
• Get
• GetMax
O(1)
N/A
N/A
O(m)
N/A
O(m)
N/A
Using chaining
LinkedListWhere m is the
maximum number of elements for any
given cell.
Binary Search Tree (BST)
AVL (balanced)
pointers
Correction of quizzes
ADTs: what can they do?Which ADTs have we seeen?
Dictionary
Dictionaries have many implementations.
• Add
• AddFirst
• AddLast
• Remove
• RemoveMax
• Get
• GetMax
Binary Search Tree (BST)
AVL (balanced)
pointers O(log n)
N/A
N/A
O(log n)
N/A
O(log n)
N/A
Correction of quizzes
ADTs: what can they do?Which ADTs have we seeen?
PriorityQueue
add
getMax
removeMax
We have seen only one implementation for priority queue: the family of heaps, out of which
we only looked at the binary heap.
• Add
• AddFirst
• AddLast
• Remove
• RemoveMax
• Get
• GetMax
O(log n)
N/A
N/A
N/A
O(log n)
N/A
O(1)
Hash tablesHash tables
Correction of quizzes
1) Explain what is improved when we use quadratic probing instead of linear.
Linear probing tends to create clusters (i.e. groups of contiguous occupied cells) and thus the distribution is not very uniform. In quadratic probing, collisions are solved by going to further cells instead of the next one, which yields a more uniform distribution. As the efficiency depends on the distribution (i.e. the likeliness of collisions), it is improved.
• Linear probing: h (x) = (h(x) + i) mod ni
• Quadratic probing: h (x) = (h(x) + i²) mod ni
Correction of quizzes
2) Give a secondary hash function such that the probing will behave like a linear probing.
• Linear probing: h (x) = (h(x) + i) mod n
• Double hashing: h (x) = (h(x) + i.s(x)) mod n
where s(x) is a secondary hashing function.
If we want both functions h (x) to be the same, what should s(x) be?
i
i
i
s(x) = 1
Note that a hashing function takes a key and returns a position. That’s all it does. It does not look for collisions and how to resolve them: that’s
what the probing does.
Correction of quizzes
3) If the structure used for chaining is also a hash table, what is the time complexity of a lookup using O notation?
• You get your key, you hash it: you’ll get the cell where the structure is, since there are no collisions. So everything depends on the complexity of lookup in the structure used for chaining.
• What is the complexity of a lookup in a hash table?
∙ We assume that we have the best type of probing (double hashing).
∙ We proved that the complexity of lookup is O(1) in a hash table if the probing behaves like random (almost the case for double hashing).
• Thus the result is O(1).
GraphsGraphs
Correction of quizzes
A
I
C
F B
N
H G
L
D K
E J
M
1) Write the names of the nodes in the order in which you visit them in a breadth-first search from A to M. (i.e. the name of each node when you poll it from the queue)
A N H C B L K G I F D M J E
Correction of quizzes
A
I
C
F B
N
H G
L
D K
E J
M
2) If there are |E| edges and |V| nodes, explain what is the worst case time complexity of finding a path using breadth-first search.
Which situation leads to the worst case? The target node is not reachable.
How many nodes do you visit? |V|How many edges do you visit? |E|
O(|V| + |E|)
Correction of quizzes
3) What does it change to a breadth-first search when the collection of nodes is stored in an ArrayList instead of a LinkedList?
Nothing.
• You access the starting node from the main collection: the access is O(n) for a LinkedList or an ArrayList.
• You will never need to access the main collection again: you move locally by following the edges.
