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Succinct Priority Indexing
Structures for the Management
of Large Priority Queues
Hao Wang and Bill Lin
University of California, San Diego
IEEE IWQoS 2009
Charleston, South Carolina
July 13-15, 2009
1
Introduction
• Priority queues used many network applications
– Per-flow advanced QoS scheduling
– Management of per-flow DRAM packet buffers
– Maintenance of per-flow statistics counters for realtime network measurements
• Items in priority queue sorted at all times
(e.g. smallest key first)
• Common operations: INSERT, FINDMIN, DELETE
• Challenge: Need to operate at high speeds
(e.g. 40+ Gb/s)
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Introduction
• Binary heap common structure for priority queues
– Has Q(log2n) time complexity, where n is # items
– e.g. in fine-grained per-flow scheduling, n can be very
large (e.g. 1 million)
– But Q(log2n) may be too slow for high line rates
• Pipelined heaps [Bhagwan, Lin 2000][Ioannou 2001][Wang,
Lin 2006]
– Reduced amortized time complexity to constant time
– At the expense of Q(log2n) pipeline stages
3
Introduction
• van Emde Boas (vEB) trees
– Instead of maintaining priority queue of sorted items,
maintain sorted dictionary of keys
– In many applications, since keys are represented by a
k-bit integer, possible keys can only be from a fixed
universe of U = 2k values
– Only Q(log2log2U) complexity vs. Q(log2n) for heaps
• Pipelined vEB trees [Wang, Lin 2007]
– Reduced amortized time complexity to constant time
– At the expense of Q(log2log2U) pipeline stages
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This Talk
• Propose 3 related Priority Indexing (PI) structures
that leverage built-in hardware optimized
instructions in modern 64-bit x86 processors (both
Intel and AMD)
• Specifically, given a W=64 bit word, the instructions
BSR (bit-scan-reverse) and BSF (bit-scan-forward)
return the positions of the most-significant and leastsignificant bits, respectively
0 0 1 0 1 1
BSR
…
0 0 1 0 0 0
BSF
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This Talk
• Most-significant (least-significant) bit positions can
also be easily implemented using efficient priority
encoder designs in custom hardware
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Basic Priority Indexing Structure
• Essentially a W-way tree. Maintains sorted subset S of N
elements from a fixed universe of size U = Wh, where N ≤ U.
• Each element i of the universe is associated with a binary bit bi
• Leaf node contains W bits of bi: bi = 1 if element i is in the set
• Non-leaf node serves as summary of child nodes: bit in non-leaf
node set to 1 if its child node has at least one non-zero bit
Data Structure of PI with h = 3
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Example Operations
• TEST(i): Just check bi
• INSERT(i): Start at leaf, set bi. Set corresponding bit in parent.
Repeat until root.
• FINDMIN(): Start at root. Find MSB (most-significant-bit) and
traverse sub-tree. Repeat until leaf.
• DELETE(i): Start at leaf, clear bi. If word = 0 (no more bits set),
clear corresponding bit in parent. Repeat until root.
Data Structure of PI with h = 3
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Time/Memory Complexity of PI
• TEST(i) takes constant time. All other operations take
Q(logwU) time, which is asymptotically not as “good”
as the Q(log2log2U) time complexity of a van Emde
Boas tree
• However, for W = 64, PI requires fewer or same
number of operations for U ≤ 64 billion (h ≤ 6), but
much simpler
• For PI of size U, memory size only 1.016U bits, Q(U)
space
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Motivation for Modified Structures
• PI is fast, but Q(logwU) time may still not be fast
enough for high-performance applications
– Want constant time operations (issue new
operation every cycle)
• But PI cannot be readily pipelined
– Some PI operations are top-down (e.g. FINDMIN),
but others are bottom-up (e.g. DELETE)
• Propose 2 modified structures
– Counting Priority Index (CPI)
– Pipelined Counting Priority Index (Pipelined CPI)
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Counting-Priority-Index
• In addition to having a bit set to indicate a child node has at
least one bit set, add counter to keep track of “how many”
bits in a child node are set. Enables all top-down operations.
Data Structure of CPI with h = 3
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Example CPI Operations
• FINDMIN(): Start at root. Find MSB (most-significant-bit) and
traverse sub-tree. Repeat until leaf. Same as before.
• DELETE(i): Start at root. Decrement counter. If count = 0, clear
bit. Go down corresponding sub-tree. Repeat until leaf.
Data Structure of CPI with h = 3
12
Time/Memory Complexity of CPI
• TEST(i) takes constant time. All other operations take
Q(logwU) time, same as basic PI structure.
• But all operations supported in top-down fashion
• For CPI of size U, memory size only 1.11U bits, still
Q(U) space
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Pipelined Counting-Priority-Index
• Reduced amortized time complexity to constant time
• At the expense of Q(logwU) pipeline stages
• Memory size also only 1.11U bits, Q(U) space
Data Structure of Pipelined CPI with h = 3
14
Operations Supported
• Operations supported by all 3 priority indexing structures
TEST(i)
INSERT(i)
DELETE(i)
FINDMIN
FINDMAX
EXTRACTMIN
EXTRACTMAX
SUCCESSOR(i)
PREDECESSOR(i)
EXTRACTSUCC(i)
EXTRACTPRED(i)
Test if index i is in set S
Insert a new index i to set S
Delete index i from set S
Find the smallest index in set S
Find the largest index in set S
Delete the smallest index in set S
Delete the largest index in set S
Find the successor of index i in set S
Find the predecessor of index i in set S
Delete the successor of index i in set S
Delete the predecessor of index i in set S
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Comparison
Time
Hardware
Memory
PI
Q(logwU)
constant
1.016 U
CPI
Q(logwU)
constant
1.11 U
Pipelined CPI
constant
Q(logwU)
1.11 U
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Hardware Complexity of Pipelined CPI
Number of Pipeline Stages in the Data Structures
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Summary
• Fast sorting data structures
– Fast and scalable succinct data structures for the
implementation of priority queues
– The Pipelined CPI supports constant time priority
management operations
– The hardware complexity is only Q(logwU) with
Q(U) memory space
18
Thank You
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