Teaching Graph Algorithms in the Field - Bipartite Matching in optical datacenters

© 2014 IBM Corporation
Bipartite Matching and Optical
Clouds - what?
Kostas Katrinis
Guest Lecture – CS2012 Programming Techniques
Trinity College Dublin
April 14, 2014

© 2014 IBM Corporation2
Outline
• Scope & Background
• Motivation & Challenges
• Hybrid Network Architecture
• MEMS Optical Switch Operation Brief
• Bipartite Graphs
• Matchings
• Make the connection!
• Hungarian Algorithm
Part I - Introduction
Part II – Modeling
Part III – Theory &
Algorithm

Scope Part I - Background
• Target Markets
– (Cloud) Datacenters - Θ(10K) Servers
– HPC Clusters (82% in Nov'12 Top-500)
• Target Systems:
– Data Network Fabric

DC Traffic Trends
• 76% of the traffic is
intra-datacenter *
• Total DC traffic CAGR
33% to 2015 *
• Traffic percentage exiting the rack
is high (up to 90%) **
– ...and we expect it to increase (scale-out workloads)
* Cisco Global Cloud Index: Forecast and Methodology, 2011–2016
** Benson et al., “Network Traffic Characteristics of Data Centers in the Wild”, IMC'10
Part I - Background

Design Trade-offs (Performance)
Performance
Cost
Highly
oversubcr.
High
Bisection
Trade-off
• We need high-capacity between any two points in the DC
• and at various scales (incremental deployment)
• ... we need $$$
Part I - Background

Motivating Example
Item Item List Price (USD)** Qty Total List Price (USD)
BNT G8264 (64-port switch) 30,000 5,120 153,600,000
BNT SFP+ SR Transceiver 665 262,164 174,325,760
MM Fiber Cable 28 131,072 3,670,016
Estimated List Prices
**Source: ibm.com
Fabric Price
331M USD
≈ Compute Price
(@5K/server)
• Full-bisection fat-tree @ 65k servers
• Building block: 64-port Ethernet switches (ala VL2*)
• Denser switches will not help you (e.g. 288-port Mellanox
Vantage)
Greenberg et al., “VL2: A Scalable and Flexible Data Center Network”, SIGCOMM 2009
Part I - Background

Motivating Example (cont.)
Total Price
331m USD
AND.....
#Cables to route
131,000
Can you count the birds in this nest?
Part I - Background

Paradigm Shift - Switch Light
Tiltable Mirrors implemented via MEMS (Micro-Electrical Mechanical Systems)
+ High-radix (320 ports you can buy, 1024 feasible)
+ No transceivers
+ Decreasing $/port
+ 50x less Watts/port vs. electronics
+ Can switch up to ~1Tbps
+ Protocol Agnostic
Electronic Switch (Ethernet) Optical MEMS switch
Price/Port (USD) 1100 (includes TxRx cost) 350
Bandwidth/Port 10Gbps “Rate-free”
Power/Port (W) 10 0.2
Requires TxRxs Yes No
x3
x∞
x50
Part I - Background

Hybrid Fabric Architecture Part I - Architecture

Dublin Prototype - Photos
Single-mode fiber
links, 25m length,
10GBASE-LR
96x96-Port 3D-MEMS Switch
Shamrock
Server Rack,
Chelsio T4 NIC
cards
G8264 TOR Switch
Redirect traffic from over-subscribed electrical
network to optical circuits.
 Packet forwarding removes bandwidth limitation of
over-subscribed electrical network.
Max.
Latency <13
ns
Part I - Architecture

High-level Functionality
• Bijective TE:
– Mice are routed via the 1G electronic fabric
– Elephants are routed via the 10G optical fabric
Optical Fabric is reconfigurable
– Centralized control optimizes topology against traffic pattern and
demand volume
Part I - Architecture

Optical MEMS Switch Part II – MEMS Switch
• Focus on the “Optical MEMS Switch”
• NxN ports (N inputs to connect to N outputs)
• Each input port is dynamically “crossconnected” to at most one output port
based on the rack pair we need to connect (decide every X seconds)
• E.g. Crossconnect port-13 to port-45 to connect rack-2 with rack-5 above

Bipartite Graphs
Definition (Bipartite Graphs)
G=(V,E) is bipartite if we can partition its vertex set V to two disjoing sets X
and Y, s.t. there are no edges between X and Y. We say that (X,Y) is a
bipartion of G.
Note: During this lecture, we will constrain our focus for the sake of
understanding to undirected graphs. Applying the concepts to directed graphs
as food for thought.
Part II – Bipartite

Matchings
Definition (Matching)
A matching M of a graph G=(V,E) is a set of edges of G such that no two
edges in M share a common vertex.
• If a vertex u is matched by M, then we say that u is saturated by M.
Otherwise, a vertex u is unsaturated by M
Example: In M1, v2 is saturated, whereas v1 is unsaturated
Part II – Matchings

Maximum Matchings
• Greediness strikes back: we want a matching to have as many edges as
possible
Definition (Maximum Matching)
A matching M of a graph G=(V,E) is a maximum matching if no matching in G
has more edges.
Example: M2 is a maximum matching (|G|=8 and |M2|=4). Since |M1|=3, M1 is
not maximum.
Part II – Matchings

Technology ↔ Theory: Connection Part II – Model

Maximum Matching – Optical Switch
• Remember that our Greediness keeps striking back, can't get rid of it!
• If we find a maximum matching (maximum number of edges), then we
have maximized the number of crossconnections in our optical switch
• .... thus we have maximized throughput (aggregate bandwidth that
crosses the optical switch)
Part II – Model

Maximum Matching in Bipartite Graphs
• So this is what we need (see title above) to maximize utilization of our
optical switch. We need an algorithm to find a maximum matching on a
bipartite graph. But how? Some theory first (and more definitions)
Definition (Alternating and Augmenting Paths)
Let G=(V,E) have a matching M. An alternating path P with respect to M is
any path whose edges are alternately in M and not in M (E-M). If the start and
end vertices of P are not saturated by M, then P is an augmenting path.
Example: P= (v1,v2,v3,v4,v5) is an alternating path
P'= (v1,v2,v3,v4,v5,v6) is an augmenting path
Part III – T&A

Matching expansion
Definition (Symmetric Difference)
Let E and E' be two edge sets. Then we define as the symmetric difference
M=E ◘ E' of the two sets the operation M = (E-E') U (E'-E) = (E U E') – (E ∩ E')
• Let G and M a matching (see below). P = (v1,v2,v3,v4,v5,v6) is an
augmentic path. E(P) are the edges of P.
• |M|=3
• Define M' = M ◘ E(P)
• Observe: |M'|=4 … wow! We obtained a larger matching M' starting from a
non-maximum one (M)
Part III – T&A

Expansion Lemma
Lemma –1 (Matching Expansion)
Let G have a matching M and let P be an augmenting path with respect to M.
Then M' = M ◘ E(P) is a matching with one more edge than M – or equivalently
– |M'|=|M|+1.
Proof
M' contains all the edges of M, plus all the edges in E(P) that are not in M.
Thus M' is per construction a matching of G.
Let u and v be the start and end vertices of P. Then u and v are unsaturated in
M, because P is an augmenting path. Because of the symmetric difference
operator, u and v are now saturated in M'. The rest of the edges of M that were
not in E(P) are still in M'. Thus, |M'|=|M|+1
Part III – T&A

Berge's Theorem
Theorem-1 (Berge's Theorem)
A matching M in G is maximum if and only if G has no augmenting path with
respect to M.
Proof
Left as an exercise (aka we don't have the time budget in class :-)). Reference
Textbook has a rigorous, easy-to-follow proof
• Theorems and lemmas are tools to an end. Let's use them!
• Back to our goal: We need an algorithm to find a maximum matching on
a bipartite graph
• Strategy:
1) Start with an arbitrary (even empty) matching M. If M is not maximum, there
will be an unsaturated vertex u.
2) Follow alternating paths from u until we identify another unsaturated vertex
v. Then the u-v path P is an augmentic path.
3) Set M'=M ◘ E(P). By the expansion Lemma, M' is a larger matching.
4) Repeat
Part III – T&A

Hungarian Algorithm - Preliminaries
Data Structures
• Integer Match[n]; // n is the number of vertices or n=|V| of G=(V,E)
– i.e. Match[u]=v if the matching matches vertex u to vertex v for all u in V
• Default: Match[u]=0 if u is unsaturated (not matched) by the matching
• For Breadth-First-Search (BFS) we use
Integer PrevPt[n]
– i.e. PrevPt[u]=v indicates that v is a parent of u in the BFS tree
• Alternating Paths: we use a combination of Match and PrevPt data
structures to store alternating paths. Example: P = (u, v2, v6, v3, v8)
• List S in X of vertices that are reachable
from a vertex u in X using alternate paths
• List NS in Y of vertices that are reachable
from a vertex u in X using alternate paths
Part III – T&A
PrevPt(v3)
PrevPt(v2)
Match(v6)
Match(v8)

Algorithm-1 (Augment subroutine)
Augment (y)
// Input: y is the end vertex of the augmenting path
// Output: augments the edges along the augmenting path “in place”
repeat
w = PrevPt[y]
Match[y] = w
u = Match[w]
Match[w]=y
y=u
until y=0 //reached root of BFS tree ie start vertex of augm.
//path
Part III – T&AHungarian Algorithm - Augmenting

Part III – T&AHungarian Algorithm – Augm. Example
y
w
v
u
augment(y)
rep.1
w = PrevPt[y]
Match[y] = w
v = Match[w]
Match[w]=y
y=v
rep.2
w = PrevPt[y]
Match[y] = w
v = Match[w]
Match[w]=y
y=v
Augmenting Path

Algorithm-2 (Maximum Matching of Graph G with bipartition (X,Y) and n vert.)
for i=1 to n do Match[i]=0 //init
for each u in X do {
  S.append(u);
  for i=1 to n do PrevPt[i]=0; //BFS rooted at u
   k=1;
   repeat
x = S[k];
for each y adjacent to x do {
         if y not in NS {
           NS.append(y);
           PrevPt[y]=x;
           if y is unsaturated {
             // found augmented path!!!
             augment(y);
             go to 1; // we saturated u
           } else S.append(Match[y]); // continue building the BFS tree
         }
     }
     k = k+1;
    until k > S.size(); // BFS rooted at u is now built out, no saturation :(
    Remove S and NS from the graph
}
Part III – T&AHungarian Algorithm

Part III – T&AHungarian Algorithm – Example (1)

Part III – T&AHungarian Algorithm – Example (2)

Food for Thought and Further Reading
Kocay and Kreher,
“Graphs, Algorithms,
and Optimization
(Discrete
Mathematics and Its
Applications)”,
Chapman and
Hall/CRC
How about faster algorithms?
Hints:
● Hopcroft-Karp Algorithm
● Edmond's Algorithm
How about weighted demand
(instead of edges with
implied unit demands)?
Intellectually appealing:
● Prove that Algorithm 2 finds indeed a maximum matching
●
Prove that the Hungarian Algorithm is O(n3
)

THANK YOU!
Q&A

Teaching Graph Algorithms in the Field - Bipartite Matching in optical datacenters

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