1. Data Structures for Shaping and Scheduling
Heap Calendar Queue Timing Wheel
ECE 1545

2. Data Structures
Data structures used to perform scheduling and shaping:
Priority queues
Calendar queues
Timing wheel

3. Priority Queue
A priority queue is an abstract data type that stores a collection of data with a priority
Priority queue operations
insert: insert an element with a priority
deleteMin: remove the element with the lowest priority
Property of priority queue:
If two elements x and y are in the queue, if x has a lower priority than y, x will be removed before y

4. Implementing a Priority Queue: Candidates
Linked list
O(N) worst-case time on either insert() or deleteMin()
Binary Search Tree
O(log N) average time on insert() and deleteMin()
Overkill: all elements are sorted
However, we only need the minimum element
Heap
O(log N) worst case for both insertion and deleteMin operation

5. Heap A heap is a binary tree which satisfies the heap property:
“The key of every node is less than or equal to the key of any of its children.”
Properties:
The smallest element in a heap is the root
No conclusion can be drawn about the order of children
The tree is completely filled on all levels except possibly the lowest
root 3 4 7 5 6

6. Vector Representation
Use a vector A to store a heap
Store elements in level-order
Parent (k) : ⌊ k/2 ⌋
Left (k) : 2k
Right (k): 2k + 1
Heap property: A[Parent(k)] ≤ A[k]
root R l r ll lr rl rr 7 6 5 4 3 2 1 A: R l r ll lr rl rr

7. insert(x) To insert an element with value x: Create a new leaf node
If heap property is violated, swap node with parent node
Repeat until parent node is less than x

8. insert(14) 14 14 14
Example from slides by A. Srinivasan (Florida State)

9. deleteMin() Replace root with the last leaf (last vector element)
This maintains the complete binary tree property but may violate the partially ordered tree property
Start at root: If heap property is violated swap position with smaller child 31 31

10. deleteMin() 31 31
Example from slides by A. Srinivasan (Florida State) 31

11. Calendar Queue
A calendar queue is a priority queue with where the average enqueue and dequeue complexity is O(1)
Implemented as an array of sorted linked lists
Element of the array is called a ”bucket”
Each bucket is associated with a time interval (“width”)
Elements of linked list are sorted
buckets
linked lists
current time pointer …

12. Calendar Queue … Inspired by a desk calendar:
one page (bucket) per day of year
items of same day are listed in sorted order
listed items can be for subsequent years
items not in current year are ignored
buckets
linked lists
current time pointer
March 18
3/18/2019, 3pm
3/18/2019, 6:30pm
3/18/2020 8am
2019 …

13. Calendar Queue
Calendar queue with n buckets, where each bucket has width w
Value x is added to bucket x/w mod n
Important: Number and width of buckets must be adjusted
Buckets should not have many items in list
otherwise enqueue into sorted list is too long
Not too many buckets should be empty  otherwise dequeue must search too many buckets
Bucket width should not be too large  otherwise data is clustered in few buckets and most other buckets are empty
Bucket width should not be too small  otherwise most data is not in “current year”

14. Calendar Queue: Adjusting number of buckets
Target: One item for each bucket
Rules of thumb:
Double the number of buckets when number of stored entries is twice the number of buckets
Halve the number of buckets when number of stored entries is half the number of buckets
When bucket number is adjusted, copy items to a new calendar queue

15. Calendar Queue: Adjusting the bucket width
Target: Set Bucket width equal to the average separation of adjacent elements
 minimizes enqueue and dequeue times
Rules of thumb:
Recompute bucket width when copying to a new calendar queue
Select samples around place of current time pointer and estimate average separation of adjacent elements
Set bucket width to estimate of average separation

16. Calendar Queue: Tightly clustered items
1000 items, 512 buckets
500 items uniformly in [0, 0.1]
500 items uniformly in [5.6, 5.7]
After removing elements in [0,0.01], current time pointer has to cycle through 54 “years” for next items
Solution: After running through an “empty year”, make global search for smallest item
bucket width = 0.002
buckets
linked lists
current time pointer
[0, 0.002)
5.6
[0.002, 0.004)
0.002
5.6002
[0.004, 0.006)
0.004
5.6004
[0.006, 0.008)
0.006
5.6006 …
[1.022, 1.024)

17. Calendar Queue: Evaluation
Evaluation from 1988 (on 5 Mhz PC)
Exponential separation between adjacent elements
Comparison:
Linear linked list
Splay tree
Supports claim of constant enqueue and dequeue times

18. Calendar Queue: Evaluation
Different distributions of separation:

19. Calendar Queue: Applications
Main application for calendar queue is the management of events in discrete event simulations
Suitable to compute timers (for timeout values)
Suitable for shaping functions that compute the release time (eligibility time) of packets
Less suitable for a sorted scheduler queues (e.g., EDF, WFQ)
Due to potential search for smallest element

20. Timing Wheel O(1) data structure for managing timers
Inspired by timing wheel in logic simulators

21. Simple Timing Wheel
Assumes that timers are set to at most MAX value into the future (Here: MAX=8), i.e., within a rotation from the current position
A cursor in the timing wheel moves one location every time unit
A list for each location
If cursor is at position i and if timer is set for i+j, add timer to position (i+j) mod MAX 1 7 2 6 3 5 4
Enqueue and dequeue is O(1)
Drawback: required wheel size can be large

22. Hashed Timing Wheel Works for arbitrary values of the timer
Record the number of remaining rounds with the stored entry
If cursor is at position i and if timer is set for i+j,
add timer to position (i+j) mod MAX
with remaining rounds (i+j) div MAX
Lists for a position can be sorted or unsorted
# of rounds remaining 4 1 7 2 6 3 1 2 5 4 2 1
In the worst-case, enqueue is O(n) with sorted lists
If # entries is less than wheel size, average enqueue should be O(1)

23. Hierarchical Timing Wheel
10:25:30
Seconds pointer advanced per clock tick
If seconds pointer wraps, move the minutes pointer
If minutes pointer wraps, move the hour pointer
Example:
Current time: 10h 25m 30s
Timer value: 50m 50s
Timeout at: 11h 15m 20s
Add entry for 11h, also store minutes and seconds
hours wheel 1 23 2 … … 11 10 20 15 1
minutes wheel 59 2 … 3 25 … 1
seconds wheel 59 2 … 3 … 4

24. Hierarchical Timing Wheel
11:00:00
At 11h 0m 0s, remove all entries from 11h slot and move to minutes wheel, into position of minute value
In minutes wheel, store the seconds
hours wheel 1 23 2 … … 11 10 20 15 1
minutes wheel 59 2 … 3 15 … 20 1
seconds wheel 59 2 … 3 … 4

25. Hierarchical Timing Wheel
11:15:00
At 11h 15m 0s, remove all entries from 15m slot and move to seconds wheel, into position of seconds value
At 11h 15m 20s, the seconds wheel moves to 20s and finds the entry
hour wheel 1 23 2 … … 11 10 1
minute wheel 59 2 … 3 15 … 20 1
seconds wheel 59 2 … 3 20 4

26. Timing Wheels: Summary
Application areas for timing wheels similar to calendar queues:
Timers
Simulations
Shapers
Hashed timing wheel is essentially a calendar queue
Main difference of timing wheel and calendar queue:
Timing wheel does not resize