LogP Model

1
LogP Model

• Motivation
• BSP Model
• Limited to BW of Network (g) and Load of PE
• Requires large load per super steps.
• Need Better Models for Portable Algorithms
• Converging Hardware
• Independent from Network Topology
• Programming Models
• Assumption
• Number of PE much bigger than data elements

2
Parameters

• L Latency
• delay on the network
• o Overhead on PE
• g gap
• minimum interval between consecutive messages
  (due to bandwidth)
• P Number of PEs
• Note
• L,o,g independent from P or node distances
• Message length short message
• L,o,g are per word or per message of fixed
  length
• k word message k short messages (ko overhead)
• L independent from message length

3
Parameters (continue)

• Bandwidth 1/g unit message length
• Number of messages to send or receive for each
  PE L/g
• Send to Receive total time L2o
• if o gtgt g, ignore o
• Similar to BSP except no synchronization step
• No communication computation overlapping
• Speed-up factor at most two

4
Broadcast

• Optimal Broad cast tree

0
P1
P3
P4
10
14
18
22
20
24
24
P8, L6, g4, o2
5
Optimal Sum

• Given time T, how many items we can add?
• Approach recursive
• At root, if T lt L2o use a single PE (can add
  T1 items)
• If T gt L2o,
• Root should have data ready at T,
• and sender must have sum ready at T - L - 2o - 1
• Recursively construct the sum tree at the sender
• If T - g gt L2o, Root also can receive data, and
  compute the
• sum with T-g as the root.

6
Applications

• FFT on the Butterfly network
• Data Placement
• cyclic layout - First log n/P local comm, last
  log P global
• blocked layout - First log P global comm,
  remaining local
• hybrid After log (n/P) iteration, re-map to
  cyclic so that
• remaining can be also local
• Communication time g (n/P2) (P-1) L
• each PE has n/P data, each of 1/P goes to each
  other PE
• Total time is (1g/logn) optimal
• All to all Communication schedule
• Approach 1 each PE sends PE1, PE2, gt bottle
  neck at PE1, PE2 in this order
• Approach 2 (staggered re-map) -- no congestion
• PE1 sends PE2, PE3,..
• PE2 sends PE3, PE4, etc
  

7
Implementation on CM5

• CM
• 33MHz
• Fat Trees
• Global Control for scan/prefix/broadcast
• one CM-5 3.2 MFLOPs
• FFT on local 2.8 - 2.2 MFLOPs (cache effect)
• each cycle
• multiply and add 4.5 us
• o 2us
• L 6us
• g 4us
• load ans store overhaed per cycle 1us
• communication time n/P max (1us 2o, g) L
• bottleneck processing and overhead, not bw

8
LU decomposition

• Data arrangement critical

9
Matching machine with real machines

• Average Distance
• topology independent usually works for n1024
  nodes.
• The difference between average distance and
  max distance are not such different

10
Potential Concerns

• Algorithmic concern
• Theory?
• Too complex?
• Communication concerns
• how to use trivial comm such as local exchange
• topology dependencies?

11
Comparison with BSP

• Length of superstep
• message not usable till next step
• special hardware for sync
• virtual/physical large, context switching may be
  expensive