Post on 19-Jan-2016
Data Structures
Priority Queue
Recall Queues
• FIFO: First-In, First-Out
• Some contexts where this seems right?
• Some contexts where some things should be allowed to skip ahead in the line?
Queues that Allow Line Jumping
• Need a new ADT• Operations: Insert an Item,
Remove the “Best” Item
insert deleteMin
2615 23
12 1845 3 7
Priority Queue ADT1. PQueue data : collection of data with priority
2. PQueue operations– insert– deleteMin
3. PQueue property: for two elements in the queue, x and y, if x has a lower priority value than y, x will be deleted before y
Applications of the Priority Queue
• Select print jobs in order of decreasing length• Forward packets on routers in order of
urgency• Select most frequent symbols for compression• Sort numbers, picking minimum first• Anything greedy
Potential Implementations
insert deleteMin
Unsorted list (Array)
Unsorted list (Linked-List)
Sorted list (Array)
Sorted list (Linked-List)
Binary Search Tree
AVL Trees
Potential Implementations
insert deleteMin
Unsorted list (Array) O(1) O(n)
Unsorted list (Linked-List) O(1) O(n)
Sorted list (Array) O(n) O(1)
Sorted list (Linked-List) O(n) O(1)
Binary Search Tree O(n) worst O(n) worst
AVL Trees O(log n) O(log n)
Recall From Lists, Queues, Stacks
• Use an ADT that corresponds to your needs
• The right ADT is efficient, while an overly general ADT provides functionality you aren’t using, but are paying for anyways
• Heaps provide O(log n) worst case for both insert and deleteMin, O(1) average insert
Heap Structure Property• A binary heap is a complete binary tree – binary
tree that is completely filled, with the possible exception of the bottom level, which is filled left to right.
Representing Complete Binary Trees in an Array
GED
CB
A
F
From node i:
7
1
2 3
4 5 6
J KH I L
left child: right child:parent:
98 10 11 12
0 1 2 3 4 5 6 7 8 9 10 11 1213
implicit (array) implementation:A B C D E F G H I J K L
Heap Order Property
Heap order property: For every non-root node X, the value in the parent of X is less than (or equal to) the value in X.
8020
10
996040
20
10
50 700
80
853015
not a heap
Heap Operations• findMin• insert(val)• deleteMin
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700 65
Heap – Deletemin
Basic Idea:1. Remove root (that is always the min!)2. Put “last” leaf node at root3. Find smallest child of node4. Swap node with its smallest child if needed.5. Repeat steps 3 & 4 until no swaps needed.
Building a Heap
• At every point, the new item may need to percolate all the way through the heap
• Adding the items one at a time is O(n log n) in the worst case.
• Can we do better?
BuildHeap: Floyd’s Method
8 1 7 212 5 11 3 106 9 4
Add elements arbitrarily to form a complete tree. Pretend it’s a heap and fix the heap-order property!
4 8 1 7 2
96103
115
12
BuildHeap: Floyd’s Method
115
12
4 8 1 7 2
96103
92103
115
12
4 8 1 7 6
92103
115
12
9213
115
12
4 8 1 7 6 4 8 10 7 6
92103
115
12
9213
115
12
4 8 1 7 6 4 8 10 7 6
8 10 7
9613
25
12
4 11
92103
115
12
9213
115
12
4 8 1 7 6 4 8 10 7 6
8 10 7
9613
25
12
8 10 711
9653
21
12
4 411
Finally…
23
1
12 8 10 711
9654
Note they’re not the same
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O(N log N)
1
O(N)
1
9654
12 8 10 711
9645
12 8 10 7 11
But that doesn’t matter, they’re both heaps
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1
7
Extension: d-Heaps
How does height compare to binary heap?
• Each node has d children• Still representable by array• Good choices for d:– choose a power of two– fit one set of children in a
cache line/memory page/disk block
4 8 5 12 11 10 6 9
1 2 7 3 4 8 5 12 11 10 69
Operations on d-Heap
• Insert: runtime = O(logd n) worst
• deleteMin: runtime =
Does this help insert or deleteMin more?
O(d logd n), worst/ave
Analysis of Priority Queue
• The run time for updating an item or adding a new item to a priority queue that has N items
Implementation Insert Delete Min Find MinUnordered array 1 N NOrdered array N 1 1AVL tree (RB tree) log N log N log NBinary heap log N log N 1d-ary heap logd N d logd N 1Fibonacci heap 1 log N 1
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Fibonacci Heaps: Structure
Fibonacci heap.Set of heap-ordered trees.Maintain pointer to minimum element.Set of marked nodes.
roots heap-ordered tree
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Fibonacci Heaps: Structure
Fibonacci heap.Set of heap-ordered trees.Maintain pointer to minimum element.Set of marked nodes.
min
find-min takes O(1) time
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Fibonacci Heaps: Insert
Insert.Create a new singleton tree.Add to root list; update min pointer (if necessary).
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21
insert 21
min
Heap H
Fibonacci Heaps: Insert
Insert.Create a new singleton tree.Add to root list; update min pointer (if necessary).
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21
min
insert 21
Fibonacci Heaps: Insert Analysis
Actual cost. O(1)
Change in potential. +1
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2123
min
Linking Operation
Linking operation. Make larger root be a child of smaller root.
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3
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56 24
15
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3
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56 24
15
smaller rootlarger root still heap-ordered
Fibonacci Heaps: Delete Min
Delete min.Delete min; meld its children into root list; update min.Consolidate trees so that no two roots have same rank.
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3
44
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min
Fibonacci Heaps: Delete Min
Delete min.Delete min; meld its children into root list; update min.Consolidate trees so that no two roots have same rank.
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min
Fibonacci Heaps: Delete Min
Delete min.Delete min; meld its children into root list; update min.Consolidate trees so that no two roots have same rank.
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min current
Fibonacci Heaps: Delete Min
Delete min.
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0 1 2 3
currentmin
rank
Fibonacci Heaps: Delete Min
Delete min.
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411723 18 52
30
7
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26 46
24
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0 1 2 3
min current
rank
Fibonacci Heaps: Delete Min
Delete min.
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411723 18 52
30
7
35
26 46
24
44
0 1 2 3
min
current
rank
Fibonacci Heaps: Delete Min
Delete min.
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30
7
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26 46
24
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0 1 2 3
min
current
rank
link 23 into 17
Fibonacci Heaps: Delete Min
Delete min.
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18 52
30
7
35
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0 1 2 3
min
current
rank
link 17 into 7
Fibonacci Heaps: Delete Min
Delete min.
39
417
30
18 52
17
35
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24
44
0 1 2 3
23
current
min
rank
link 24 into 7
Fibonacci Heaps: Delete Min
Delete min.
39
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18 52
23
17
35
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24 44
0 1 2 3
min
current
rank
Fibonacci Heaps: Delete Min
Delete min.
39
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30
18 52
23
17
35
26 46
24 44
0 1 2 3
min
current
rank
Fibonacci Heaps: Delete Min
Delete min.
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30
18 52
23
17
35
26 46
24 44
0 1 2 3
min
current
rank
Fibonacci Heaps: Delete Min
Delete min.
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30
18 52
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17
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24 44
0 1 2 3
min
current
rank
link 41 into 18
Fibonacci Heaps: Delete Min
Delete min.
3941
7
30
1852
23
17
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0 1 2 3
min
current
rank
Fibonacci Heaps: Delete Min
Delete min.
7
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23
17
35
26 46
24
0 1 2 3
min
rank
3941
18
44
current
Fibonacci Heaps: Delete Min
Delete min.Delete min; meld its children into root list; update min.Consolidate trees so that no two roots have same rank.
7
30
52
23
17
35
26 46
24
min
3941
18
44
stop