packet level algorithms

Packet Level Algorithms

Michael Mitzenmacher

Goals of the Talk• Consider algorithms/data structures for

measurement/monitoring schemes at the router level.– Focus on packets, flows.

• Emphasis on my recent work, future plans.– “Applied theory”.

• Less on experiments, more on design/analysis of data structures for applications.

– Hash-based schemes • Bloom filters and variants.

Vision

• Three-pronged research data.• Low: Efficient hardware implementations of

relevant algorithms and data structures.• Medium: New, improved data structures and

algorithms for old and new applications.• High: Distributed infrastructure supporting

monitoring and measurement schemes.

Background / Building Blocks

• Multiple-choice hashing• Bloom filters

Multiple Choices: d-left Hashing

• Split hash table into d equal subtables.• To insert, choose a bucket uniformly for each subtable.• Place item in a cell in the least loaded bucket, breaking ties to the left.

Properties of d-left Hashing

• Analyzable using both combinatorial methods and differential equations.– Maximum load very small: O(log log n).– Differential equations give very, very accurate

performance estimates.• Maximum load is extremely close to

average load for small values of d.

Example of d-left hashing• Consider 3-left performance.

Average load 4Average load 6.4

Load 0 2.3e-05Load 1 6.0e-04Load 2 1.1e-02Load 3 1.5e-01Load 4 6.6e-01Load 5 1.8e-01Load 6 2.3e-05Load 7 5.6e-31

Load 0 1.7e-08Load 1 5.6e-07Load 2 1.2e-05Load 3 2.1e-04Load 4 3.5e-03Load 5 5.6e-02Load 6 4.8e-01Load 7 4.5e-01Load 8 6.2e-03Load 9 4.8e-15

Example of d-left hashing• Consider 4-left performance with average load of 6, using differential equations.

Insertions onlyAlternating insertions/deletions

Steady stateLoad > 1 1.0000Load > 2 1.0000Load > 3 1.0000Load > 4 0.9999Load > 5 0.9971Load > 6 0.8747Load > 7 0.1283Load > 8 1.273e-10Load > 9 2.460e-138

Load > 1 1.0000Load > 2 0.9999Load > 3 0.9990Load > 4 0.9920Load > 5 0.9505Load > 6 0.7669Load > 7 0.2894Load > 8 0.0023Load > 9 1.681e-27

Review: Bloom Filters• Given a set S = {x1,x2,x3,…xn} on a universe U,

want to answer queries of the form:

• Bloom filter provides an answer in– “Constant” time (time to hash).– Small amount of space.– But with some probability of being wrong.

• Alternative to hashing with interesting tradeoffs.

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Bloom FiltersStart with an m bit array, filled with 0s.

Hash each item xj in S k times. If Hi(xj) = a, set B[a] = 1.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0B

0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0B

To check if y is in S, check B at Hi(y). All k values must be 1.

0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0B

0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0BPossible to have a false positive; all k values are 1, but y is not in S.

n items m = cn bits k hash functions

False Positive Probability• Pr(specific bit of filter is 0) is

• If is fraction of 0 bits in the filter then false positive probability is

• Approximations valid as is concentrated around E[]. – Martingale argument suffices.

• Find optimal at k = (ln 2)m/n by calculus.– So optimal fpp is about (0.6185)m/n

kckkkk pp )e1()1()'1()1( /


pmp mknkn /e)/11('

Example

00.010.020.030.040.050.060.070.080.09

0.1

0 1 2 3 4 5 6 7 8 9 10

Hash functions

Fals

e po

sitiv

e ra

te m/n = 8

Opt k = 8 ln 2 = 5.45...


Handling Deletions

• Bloom filters can handle insertions, but not deletions.

• If deleting xi means resetting 1s to 0s, then deleting xi will “delete” xj.

0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0B

xi xj

Counting Bloom Filters

Start with an m bit array, filled with 0s.

Hash each item xj in S k times. If Hi(xj) = a, add 1 to B[a].

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0B

0 3 0 0 1 0 2 0 0 3 2 1 0 2 1 0B

To delete xj decrement the corresponding counters.

0 2 0 0 0 0 2 0 0 3 2 1 0 1 1 0BCan obtain a corresponding Bloom filter by reducing to 0/1.

0 1 0 0 0 0 1 0 0 1 1 1 0 1 1 0B

Counting Bloom Filters: Overflow

• Must choose counters large enough to avoid overflow.

• Poisson approximation suggests 4 bits/counter.– Average load using k = (ln 2)m/n counters is ln 2. – Probability a counter has load at least 16:

• Failsafes possible.• We assume 4 bits/counter for comparisons.

17E78.6!16/)2(ln 162ln e

Bloomier Filters

• Instead of set membership, keep an r-bit function value for each set element.– Correct value should be given for each set element.– Non-set elements should return NULL with high

probability.• Mutable version: function values can change.

– But underlying set can not.• First suggested in paper by Chazelle, Kilian,

Rubenfeld, Tal.

From Low to High

• Low– Hash Tables for Hardware– New Bloom Filter/Counting Bloom Filter Constructions

(Hardware Friendly)• Medium

– Approximate Concurrent State Machines– Distance-Sensitive Bloom Filters

• High– A Distributed Hashing Infrastructure

Low Level : Better Hash Tables for Hardware• Joint work with Adam Kirsch.

– Simple Summaries for Hashing with Choices.– The Power of One Move: Hashing Schemes for

Hardware.

Perfect Hashing Approach

Element 1 Element 2 Element 3 Element 4 Element 5

Fingerprint(4)Fingerprint(5)Fingerprint(2)Fingerprint(1)Fingerprint(3)

Near-Perfect Hash Functions

• Perfect hash functions are challenging.– Require all the data up front – no insertions or

deletions.– Hard to find efficiently in hardware.

• In [BM96], we note that d-left hashing can give near-perfect hash functions.– Useful even with insertions, deletions.– Some loss in space efficiency.

Near-Perfect Hash Functions via d-left Hashing

• Maximum load equals 1– Requires significant space to avoid all

collisions, or some small fraction of spillovers.• Maximum load greater than 1

– Multiple buckets must be checked, and multiple cells in a bucket must be checked.

– Not perfect in space usage.• In practice, 75% space usage is very easy.• In theory, can do even better.

Hash Table Design : Example

• Desired goals:– At most 1 item per bucket.– Minimize space.

• And minimize number of hash functions.– Small amount of spillover possible.

• We model as a constant fraction, e.g. 0.2%.• Can be placed in a content-addressable memory

(CAM) if small enough.

Basic d-left Scheme

• For hash table holding up to n elements, with max load 1 per bucket, use 4 choices and 2n cells.– Spillover of approximately 0.002n elements

into CAM.

Improvements from Skew• For hash table holding up to n elements, with max

load 1 per bucket, use 4 choices and 1.8n cells.– Subtable sizes 0.79n, 0.51n, 0.32n, 0.18n.– Spillover still approximately 0.002n elements into

CAM.– Subtable sizes optimized using differential equations,

black-box optimization.

xk

Summaries to Avoid Lookups• In hardware, d choices of location can be done by

parallelization.– Look at d memory banks in parallel.

• But there’s still a cost: pin count.• Can we keep track of which hash function to use for each

item, using a small summary?– Yes: use a Bloom-filter like structure to track.

• Skew impacts summary performance; more skew better.– Uses small amount of on-chip memory.– Avoids multiple look-ups.– Special case of a Bloomier filter.

Hash Tables with Moves

• Cuckoo Hashing (Pagh, Rodler)– Hashed items need not stay in their initial place.– With multiple choices, can move item to

another choice, without affecting lookups.• As long as hash values can be recomputed.

– When inserting, if all spots are filled, new item kicks out an old item, which looks for another spot, and might kick out another item, and so on.

Benefits and Problems of Moves

• Benefit: much better space utilization.– Multiple choices, multiple items per bucket, can

achieve 90+% with no spillover.• Drawback: complexity.

– Moves required can grow like log n. • Constant on average.

– Bounded maximum time per operation important in many settings.

– Moves expensive.• Table usually in slow memory.

Question : Power of One Move

• How much leverage do we get by just allowing one move?– One move likely to be possible in practice.– Simple for hardware.– Analysis possible via differential equations.

• Cuckoo hard to analyze.– Downside : some spillover into CAM.

Comparison, Insertions Only• 4 schemes

– No moves– Conservative : Place item if possible. If not, try to move earliest item that

has not already replaced another item to make room. Otherwise spill over.– Second chance : Read all possible locations, and for each location with an

item, check it it can be placed in the next subtable. Place new item as early as possible, moving up to 1 item left 1 level.

– Second chance, with 2 per bucket.

• Target of 0.2% spillover.• Balanced (all subtables the same) and skewed compared.• All done by differential equation analysis (and

simulations match).

Results of Moves : Insertions Only

Space overhead, balanced

Space overhead, skewed

Fraction moved, skewed

No moves 2.00 1.79 0%Conservative

1.46 1.39 1.6%

Standard 1.41 1.29 12.0%Standard, 2 1.14 1.06 14.9%

Conclusions, Moves

• Even one move saves significant space.– More aggressive schemes, considering all

possible single moves, save even more. (Harder to analyze, more hardware resources.)

• Importance of allowing small amounts of spillover in practical settings.

From Low to High





Low- Medium: New Bloom Filters /

Counting Bloom Filters

• Joint work with Flavio Bonomi, Rina Panigrahy, Sumeet Singh, George Varghese.

A New Approach to Bloom Filters• Folklore Bloom filter construction.

– Recall: Given a set S = {x1,x2,x3,…xn} on a universe U, want to answer membership queries.

– Method: Find an n-cell perfect hash function for S.• Maps set of n elements to n cells in a 1-1 manner.

– Then keep bit fingerprint of item in each cell. Lookups have false positive < .

– Advantage: each bit/item reduces false positives by a factor of 1/2, vs ln 2 for a standard Bloom filter.

• Negatives:– Perfect hash functions non-trivial to find.– Cannot handle on-line insertions.

)/1(log2

Near-Perfect Hash Functions

• In [BM96], we note that d-left hashing can give near-perfect hash functions.– Useful even with deletions.

• Main differences– Multiple buckets must be checked, and multiple

cells in a bucket must be checked.– Not perfect in space usage.

• In practice, 75% space usage is very easy.• In theory, can do even better.

First Design : Just d-left Hashing

• For a Bloom filter with n elements, use a 3-left hash table with average load 4, 60 bits per bucket divided into 6 fixed-size fingerprints of 10 bits.– Overflow rare, can be ignored.

• False positive rate of – Vs. 0.000744 for a standard Bloom filter.

• Problem: Too much empty, wasted space.– Other parametrizations similarly impractical.– Need to avoid wasting space.

01171875.0212 10

Just Hashing : Picture

Bucket

1011011100000111010110101010000000111111

EmptyEmpty

Key: Dynamic Bit Reassignment

• Use 64-bit buckets: 4 bit counter, 60 bits divided equally among actual fingerprints.– Fingerprint size depends on bucket load.

• False positive rate of 0.0008937– Vs. 0.0004587 for a standard Bloom filter.

• DBR: Within a factor of 2.– And would be better for larger buckets.– But 64 bits is a nice bucket size for hardware.

• Can we remove the cost of the counter?

DBR : Picture

Bucket

Count : 4

000110110101111010100001101010101000101010110101010101101011

Semi-Sorting

• Fingerprints in bucket can be in any order.– Semi-sorting: keep sorted by first bit.

• Use counter to track #fingerprints and #fingerprints starting with 0.

• First bit can then be erased, implicitly given by counter info.

• Can extend to first two bits (or more) but added complexity.

DBR + Semi-sorting : Picture

Bucket

Count : 4,2

000110110101111010100001101010101000101010110101010101101011

DBR + Semi-Sorting Results• Using 64-bit buckets, 4 bit counter.

– Semi-sorting on loads 4 and 5.– Counter only handles up to load 6.– False positive rate of 0.0004477

• Vs. 0.0004587 for a standard Bloom filter.– This is the tradeoff point.

• Using 128-bit buckets, 8 bit counter, 3-left hash table with average load 6.4.– Semi-sorting all loads: fpr of 0.00004529– 2 bit semi-sorting for loads 6/7: fpr of 0.00002425

• Vs. 0.00006713 for a standard Bloom filter.

Additional Issues

• Futher possible improvements– Group buckets to form super-buckets that share

bits.– Conjecture: Most further improvements are not

worth it in terms of implementation cost.• Moving items for better balance?• Underloaded case.

– New structure maintains good performance.

Improvements to Counting Bloom Filter

• Similar ideas can be used to develop an improved Counting Bloom Filter structure.– Same idea: use fingerprints and a d-left hash table.

• Counting Bloom Filters waste lots of space.– Lots of bits to record counts of 0.

• Our structure beats standard CBFs easily, by factors of 2 or more in space.– Even without dynamic bit reassignment.

Deletion ProblemSuppose x and y have the same fingerprint z.

Insert x xx x

x

Insert y z

Delete x? z z

yy y

y

Deletion Problem

• When you delete, if you see the same fingerprint at two of the location choices, you don’t know which is the right one.– Take both out: false negatives.– Take neither out: false positives/eventual overflow.

Handling the Deletion Problem

• Want to make sure the fingerprint for an element cannot appear in two locations.

• Solution: make sure it can’t happen.– Trick: uses (pseudo)random permtuations

instead of hashing.

Two Stages• Suppose we have d subtables, each with 2b buckets, and

want f bit fingerprints.• Stage 1: Hash element x into b+f bits using a “strong” hash

function H(x).• Stage 2: Apply d permutations taking {0… 2b+f-1}

{0… 2b+f-1}

– Bucket Bi and fingerprint Fi for ith subtable given by ith permtuation.

– Also, Bi and Fi completely determine H(x).

),())(( iii FBxH

Handling the Deletion Problem• Lemma: if x and y yield the same fingerprint in the same

bucket, then H(x) = H(y). – Proof: because of permutation setup, fingerprint and bucket

determine H(x).• Each cell has a small counter.

– In case two elements have same hash, H(x) = H(y).– Note they would match for all buckets/fingerprints.– 2 bit counters generally suffice.

• Deletion problem avoided.– Can’t have two fingerprints for x in the table at the same time;

handled by the counter.

A Problem for Analysis

• Permutations implies no longer “pure” d-left hashing.– Dependence.– Analysis no longer applies.

• Some justification: – Balanced Allocation on Graphs (SODA 2006,

Kenthapadi and Panigrahy.)– Differential equations.

• Justified experimentally.

Other Practical Issues

• Simple, linear permtuations

– High order bits for bucket, low order for fingerprint.– Not analyzed, works fine in practice.

• Invertible permutations allow moving elements if hash table overflows.– Move element from overflow bucket to another choice.– Powerful paradigm…

• Cuckoo hashing and related schemes.– But more expensive in implemenation terms.

axaHxH fbi odd2mod)())((

Space Comparison : Theory

• Standard counting Bloom filter uses c counters/element = 4c bits/element.

• The d-left CBF using r bit remainders, 4 hash functions, 8 cells/bucket uses 4(r+2)/3 bits/element.

• Space equalized when c = (r+2)/3.

• Can change parameters to get other tradeoffs.

232ln 2242posfalseleftposfalseStandard

cc

d

Space Comparison : Practice

• Everything behaves essentially according to expectations.– Not surprising: everything is a “balls-and-bins”

process.• Using 4-left hashing:

– Save over a factor of 2 in space with 1% false postive rate.

– Save over a factor of 2.5 in space with 0.1% false positive rate.

From Low to High





Approximate Concurrent State Machines

• Joint work with Flavio Bonomi, Rina Panigrahy, Sumeet Singh, George Varghese.

• Extending the Bloomier filter idea to handle dynamic sets, dynamic function values, in practical setting.

Approximate ConcurrentState Machines

• Model for ACSMs– We have underlying state machine, states 1…X.– Lots of concurrent flows.– Want to track state per flow.– Dynamic: Need to insert new flows and delete

terminating flows.– Can allow some errors.– Space, hardware-level simplicity are key.

Motivation: Router State Problem• Suppose each flow has a state to be tracked.

Applications:– Intrusion detection– Quality of service– Distinguishing P2P traffic– Video congestion control– Potentially, lots of others!

• Want to track state for each flow.– But compactly; routers have small space.– Flow IDs can be ~100 bits. Can’t keep a big lookup table for

hundreds of thousands or millions of flows!

Problems to Be Dealt With

• Keeping state values with small space, small probability of errors.

• Handling deletions.• Graceful reaction to adversarial/erroneous

behavior.– Invalid transitions.– Non-terminating flows.

• Could fill structure if not eventually removed.– Useful to consider data structures in well-behaved

systems and ill-behaved systems.

ACSM Basics• Operations

– Insert new flow, state– Modify flow state– Delete a flow– Lookup flow state

• Errors– False positive: return state for non-extant flow– False negative: no state for an extant flow– False return: return wrong state for an extant flow– Don’t know: return don’t know

• Don’t know may be better than other types of errors for many applications, e.g., slow path vs. fast path.

ACSM via Counting Bloom Filters

• Dynamically track a set of current (FlowID,FlowState) pairs using a CBF.

• Consider first when system is well-behaved.– Insertion easy.– Lookups, deletions, modifications are easy

when current state is given.• If not, have to search over all possible states. Slow,

and can lead to don’t knows for lookups, other errors for deletions.

Direct Bloom Filter (DBF) Example

0 0 1 0 2 3 0 0 2 1 0 1 1 2 0 0

(123456,3) (123456,5)

0 0 0 0 1 3 0 0 3 1 1 1 1 2 0 0

Timing-Based Deletion

• Motivation: Try to turn non-terminating flow problem into an advantage.

• Add a 1-bit flag to each cell, and a timer.– If a cell is not “touched” in a phase, 0 it out.

• Non-terminating flows eventually zeroed.• Counters can be smaller or non-existent; since

deletions occur via timing.• Timing-based deletion required for all of our

schemes.

Timer Example

3 0 0 2 1 0 1 1

1 0 0 0 1 0 1 0

3 0 0 0 1 0 1 0

0 0 0 0 0 0 0 0

RESET

Timer bits

Stateful Bloom Filters• Each flow hashed to k cells, like a Bloom filter.• Each cell stores a state.• If two flows collide at a cell, cell takes on don’t know

value.• On lookup, as long as one cell has a state value, and there

are not contradicting state values, return state.• Deletions handled by timing mechanism (or counters in

well-behaved systems). • Similar in spirit to [KM], Bloom filter summaries for

multiple choice hash tables.

Stateful Bloom Filter (SBF) Example

1 4 3 4 3 3 0 0 2 1 0 1 4 ? 0 2

(123456,3) (123456,5)

1 4 5 4 5 3 0 0 2 1 0 1 4 ? 0 2

What We Need : A New Design

• These Bloom filter generalizations were not doing the job.– Poor performance experimentally.

• Maybe we need a new design for Bloom filters!

• In real life, things went the other way; we designed a new ACSM structure, and found that it led to the new Bloom filter/counting Bloom filter designs.

Fingerprint Compressed Filter

• Each flow hashed to d choices in the table, placed at the least loaded.– Fingerprint and state stored.

• Deletions handled by timing mechanism or explicitly.• False positives/negatives can still occur (especially in

ill-behaved systems).• Lots of parameters: number of hash functions, cells

per bucket, fingerprint size, etc.– Useful for flexible design.

Fingerprint Compressed Filter (FCF) Example

11111111110000000 400011110011101101 111110111001001011 211110101001000111 2

11100010010111110 101110010001011111 310101110010101011 2

x : 11110111001001011 : State 2 to State 4

01110010010101111 601110100100010111 110001110011111100 3

Fingerprint State

Experiment Summary

• FCF-based ACSM is the clear winner.– Better performance than less space for the

others in test situations.• ACSM performance seems reasonable:

– Sub 1% error rates with reasonable size.

Distance-Sensitive Bloom Filters

• Instead of answering questions of the form

we would like to answer questions of the form

• That is, is the query close to some element of the set, under some metric and some notion of close.

• Applications:– DNA matching– Virus/worm matching– Databases

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Distance-Sensitive Bloom Filters

• Goal: something in same spirit as Bloom filters.– Don’t exhaustively check set.

• Initial results for Hamming distance show it is possible. [KM]

• Closely related to locality-sensitive hashing.• Not currently practical.• New ideas?

From Low to High





A Distributed Router Infrastructure

• Recently funded FIND proposal.• Looking for ideas/collaborators.

The High-Level Pitch

• Lots of hash-based schemes being designed for approximate measurement/monitoring tasks.– But not built into the system to begin with.

• Want a flexible router architecture that allows:– New methods to be easily added. – Distributed cooperation using such schemes.

What We Need

On-ChipMemory

Hashing Computation

Unit

Off-ChipMemory CAM(s)

Programming Language

Memory

Unit for Other

ComputationComputation

Communication+ Control

ControlSystem

CommunicationArchitecture

Lots of Design Questions• How much space for various memory levels? How can we

dynamically divide memory among multiple competing applications?

• What hash functions should be included? How open should system be to new hash functions?

• What programming functionality should be included? What programming language to use?

• What communication is necessary to achieve distributed monitoring tasks given the architecture?

• Should security be a consideration? What security approaches are possible?

• And so on…

Related Theory Work• What hash functions should be included?

– Joint work with Salil Vadhan.– Using theory of randomness extraction, we show that

for d-left hashing, Bloom filters, and other hashing methods, choosing a hash function from a pairwise independent family is enough – if data has sufficient entropy.

• Behavior matches truly random hash function with high probability.

• Radnomness of hash function and data “combine”.• Pairwise independence enough for many applications.

Conclusions and Future Work• Low: Mapping current hashing techniques to hardware is

fruitful for practice.• Medium: Big boom in hashing-based algorithms/data

structures. Trend is likely to continue.– Approximate concurrent state machines: Natural progression from

set membership to functions (Bloomier filter) to state machines. What is next?

– Power of d-left hashing variants for near-perfect matchings.• High: Wide open. Need to systematize our knowledge for

next generation systems.– Measurement and monitoring infrastructure built into the system.

packet level algorithms

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